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\begin{document}
\label{firstpage}
\title{Longitudinal quantile regression in presence of informative drop-out through longitudinal-survival joint modeling}
\begin{abstract} We propose a joint model for a time-to-event outcome and a quantile of a continuous response repeatedly measured over time. The quantile and survival processes are associated via shared latent and manifest variables. Our joint model provides a flexible approach to handle informative drop-out in quantile regression. A general Monte Carlo Expectation Maximization strategy based on importance sampling is proposed, which is directly applicable under any distributional assumption for the longitudinal outcome and random effects, and parametric and non-parametric assumptions for the baseline hazard. Model properties are illustrated through a simulation study and an application to an original data set about dilated cardiomyopathies. \end{abstract}
{\bf Key Words:} Quantile Regression; Longitudinal Regression; Joint Models; Shared-parameter models.
\section{Introduction} \label{intro}
In longitudinal studies subjects may be lost to follow-up due to events, like death, which are associated with the outcome of interest. Failure to model drop-out may lead to biased estimates in such cases. From the reverse perspective, the time trend of a longitudinal measurement may predict the risk of an event (e.g., a steadily decreasing CD4 count is predictive of adverse events in HIV patients). A general account of related longitudinal and survival processes can be found in \cite{follmann:95}. Participation to a study can in general be described by a survival model of time to drop-out. The simpler way of taking into account informative drop-out is through pattern-mixture models (e.g., \cite{wu:bail:88,wu:bail:89,litt:wang:96}), where the outcome distribution is specified conditionally on the time to drop-out. Selection models on the other hand condition the drop-out mechanism to unobserved responses directly \citep{digg:kenw:94} or indirectly. A simple and effective indirect link between drop-out and unobserved outcomes is by assuming that these are independent conditionally on unobserved shared random effects, as in \cite{wu:carrol:88}. One can similarly assume that the risk of event at time $t$ is influenced by the \textit{expected value} of the longitudinal response, as in \cite{riz:10} and \cite{rizop:ghosh:11}. In the resulting Joint Model (JM), the hazard of drop-out is a function of the predicted longitudinal outcome, that is, of shared random and fixed effects, and related covariates. See also \cite{litt:95}, \cite{hend:et:al:00}, \cite{tsia:davi:04}, and references therein.
Our concern in this paper is that the expected value of the longitudinal outcome may not always be the summary of interest. Further, in some cases it might be difficult to find a suitable transformation to normality for the outcome, or some resistance to outliers may be desired. An effective solution to these issues is given by modeling conditional quantiles of the longitudinal outcome. Quantile regression models \citep{koen:05} are robust with respect to outliers, so that one can simply model the median rather than the mean. In many biomedical applications interest lies furthermore in at least one of the tails, and covariates may have different effects on different quantiles of a distribution. Examples include longitudinal fetal growth studies, which are usually focused on low and high quantiles of key anthropometric measurements. Finally, quantiles are invariant to transformations so it is never needed to transform the outcome.
Longitudinal quantile regression models are proposed among others in \cite{koen:04}, who maximizes a penalized version of the likelihood;
and \cite{gera:bott:07}, who introduce a random intercept and assume the outcome follows the asymmetric Laplace distribution (ALD). \cite{yuan:bott:09} extend the random intercept model to a general linear mixed quantile regression model. See also \cite{gera:bott:13}. The ALD assumption is also used in \cite{farc:12}, where random effects are time-varying and follow a discrete distribution. Informative missing data are ubiquitous in statistical applications, especially in longitudinal studies, but there are very few approaches to quantile regression with informative drop-out. In \cite{lips:et:al:97} and \cite{yi:he:09} estimating equations are weighted proportionally to the inverse of the probability of drop-out. \cite{yuan:yin:10}, in a Bayesian framework, model missingness as a binary time series sharing a random effect with the quantile regression process. A common limit of these approaches is that drop-out can occur only at one of the observation times of the longitudinal process. This does not hold in general (e.g., when measurements are scheduled according to a protocol, and death occurs between two visits). Moreover, all approaches proposed so far do not directly model the strength of association between the longitudinal and time-to-event processes. The latter is summarized by non-ignorability parameter(s) in JM as the one we propose. A longitudinal quantile regression model with ignorable missingness is outlined in \cite{gera:13}.
In this paper we propose a joint model for a right-censored time-to-event outcome and the quantile of a continuous response repeatedly measured over time. Drop-out is formally defined as a monotone missing pattern, that is, when the outcome is not measured for a subject, no further measurements take place for that subject until the end of the study.
In our approach the quantile and survival processes are associated not only via shared latent variables or the predicted longitudinal outcome. It is in fact assumed that the time-to-event outcome depends on a function, which for simplicity we assume linear, of both the latent variables and the predicted quantile of the longitudinal outcome. Two non-ignorability parameters are introduced, one for the fixed and the other for the random part of the linear predictor. Our joint model is therefore a flexible approach to handle informative drop-out in longitudinal quantile regression.
Maximum likelihood estimates are efficiently derived by setting up a Monte Carlo Expectation-Maximization (MCEM) algorithm based on importance sampling \citep{douc:et:al:01,levi:case:01}. There are two clear advantages of using importance sampling: first, the resulting MCEM is completely general and straightforward to use with any distributional assumption for the longitudinal observations or the random effects. Secondly, it is computationally efficient given that we evaluate the posterior distribution only once for each sample. The MCEM approach, unlike commonly used quadrature methods, is amenable also to moderate dimensional random effects.
The use of two non-ignorability parameters allows us also increase the flexibility of both the shared parameter model of \cite{wu:carrol:88}, by conditioning the drop-out process also on the residuals between the predicted longitudinal outcome and the random effects, and the JM of \cite{WULF:TSIAT:97} and \cite{riz:10}, by allowing a residual dependence on the random effects. In the first case, we can say that the two processes are not only linked by unmeasured heterogeneity, but that their dependence can also be in part explained through shared observed heterogeneity. In the second case, we can say that the two processes are linked not only via a quantile of the longitudinal outcome, but also by a residual unmeasured heterogeneity.
The rest of the paper is as follows: in the next section we describe the proposed model. In Section \ref{inference} we outline inference, we illustrate the approach through simulations in Section \ref{simu},
and in Section \ref{realdata} where we apply the method to an original data set about patients with cardiomyopathy. Finally, in Section \ref{concl} we conclude with a brief discussion.
\section{Joint longitudinal quantile and survival regressions}
Let $T_i=\min(T_i^*,C_i)$ denote the observed failure time for the $i$th individual, $i=1,\ldots,n$, taken as the minimum between the true event time $T_i^*$ and the censoring time $C_i$. Further, let $\Delta_i$ be the corresponding event indicator defined by $ \Delta_i = I(T_i^* \leq C_i)$, where $I(\cdot)$ is the indicator function. The continuous outcome $Y_{it}$ is repeatedly observed at times $t = 1,\ldots,n_i$ before $T_i$, and is missing for $t \geq T_i$. The longitudinal outcome at observation times is collected in $\b y_i = \{y_i(t) : t \leq T_i\}$. We assume that the longitudinal process is associated with $T_i^*$, i.e with the true event time, but, as customary in survival analysis, is independent of the censoring time $C_i$.
We let $\b X_{it}$ denote a vector of predictors used to model only the longitudinal outcome, $\b H_{it}$ a vector of shared predictors, and $\b W_i$ a vector of (time fixed) predictors used to model only the survival process. These are associated with fixed effects $\b\beta$, $\b\delta$ and $\b\gamma$, respectively. We then have a vector $\b Z_{it}$ of predictors associated with $q$ dimensional random effects $\b u_i$, and two non-ignorability parameters $\alpha_1$ (associated with fixed effects) and $\alpha_2$ (associated with random effects). Our model can be expressed by a set of two equations, one for the longitudinal and the other one for the survival outcome: \begin{eqnarray} \label{jqsm} \begin{cases} y_{it} = \b\beta' \b X_{it} + \b\delta' \b H_{it} + \b u_i'\b Z_{it} +\epsilon_{it} = \widetilde{\tau}_{it} + \epsilon_{it} \\
h(T_i | \mathcal{T}_{iT_i}, \b W_i ; \b \gamma, \alpha_1, \alpha_2) = h_0(T_i)\exp\{\b \gamma' \b W_i + \alpha_1 \b\delta'\b H_{iT_i} + \alpha_2\b u_i'\b Z_{iT_i} \} , \end{cases} \end{eqnarray} where the first equation gives the longitudinal model and the second the time-to-event model, and $h_0(s)$ is a baseline function. Specifically, the model for the longitudinal outcome $\widetilde{\tau}_{it}$ is formulated along the usual lines for mixed effects models \citep{verb:mole:00} and the model for the time-to-event outcome is based on the subject-specific hazard function $h(T_i)$ \citep{cox:72,and:82}. The risk of drop-out is conditional on $\mathcal{T}_{iT_i} =\{\widetilde{\tau}_{iu}: 0\leq u \leq T_i\}$, i.e. the error-free longitudinal process history up to time $T_i$. The model is completed by a distributional assumption for the shared latent distribution, that is, by specifying $\b u_i \sim f(\b u_i)$. Few options are discussed in Section \ref{ranef}.
The degree of dependence between the longitudinal and the survival processes is measured by the association parameters $\alpha_1$ and $\alpha_2$, which are introduced to assess potential non-ignorability of the missing data mechanism. In doing so, we admit two sources of non-ignorability: a part that can be explained through observed heterogeneity in $\b H_{it}$ (but not $\b X_{it}$) and a part that is due to unobserved heterogeneity. The log-hazard ratios associated with $\b H_{it}$ can be estimated as $\alpha_1\b\delta$, while those associated with $\b W_i$ are directly estimated as $\b\gamma$. All parameters are identifiable even if we multiply some of them in the survival model equation.
It is worth noting how our proposed model (\ref{jqsm}) generalizes previous work. If we fix $\alpha=\alpha_1=\alpha_2$, $\beta=0$, and assume a Gaussian distribution for the error, we obtain a usual formulation for the JM. Otherwise, with an ALD error distribution (see below), we obtain a joint quantile regression model. Here,
the link between the time-to-event and longitudinal processes is summarized by the $\alpha (\b\delta'\b H_{it} + \b u_i \b Z_{it}) = \alpha\widetilde{\tau_{it}}$ term in the model for the hazard function. If $\alpha_1 \neq \alpha_2$ and $\beta=0$, $$ \alpha_1 \b\delta' \b H_{it} + \alpha_2 \b u_i'\b Z_{it} = \alpha_1 \widetilde{\tau_{it}} + (\alpha_2-\alpha_1) \b u_i'\b Z_{it}, $$ that is, we generalize JM models by allowing for a residual dependence on unobserved heterogeneity as summarized by the shared random effects. When $\b\beta \neq 0$, we further assume that some predictors may be related only to the longitudinal outcome but may not contribute to explain non-ignorability. Similarly, a shared parameter model is obtained if we fix $\alpha_1=0$. Specifically, if $Z_{it}=(1\ t)$ and we have Gaussian error terms and random effects, we exactly obtain the classical model in \cite{WULF:TSIAT:97}. When $\alpha_1>0$ we condition the survival process also on the difference between the predicted longitudinal outcome and the random effects. It is straightforward to fully generalize (\ref{jqsm}) by including also random effects that are not shared in each part of the model, and by letting $W_i$ be time-dependent. In this way we would have, in each model, separate and shared covariates both for random and fixed effects. We do not pursue this explicitely to keep notation simple, but we note that our MCEM strategy can be directly adapted to this completely general case.
We start describing each part of the model separately, then we outline how they are linked by obtaining the observed likelihood.
\subsection{The longitudinal model}
The parametric assumption on the error distribution of the longitudinal outcome drives our target for inference. If we assume that $\varepsilon_{it}$ follows a zero-centered Gaussian, we work with the conditional expectation of the outcome. On the other hand, if we assume an ALD, $\widetilde{\tau}_{it}$ does not represent the conditional mean of $Y_{it}$ anymore, but its conditional $0 <\tau< 1$ quantile. Note that $\tau$ is pre-specified and fixed. The resulting density of $Y_{it}$, conditionally on covariates and random effects, is given by \begin{equation} \label{ald}
f(Y_{it}|\b X_{it},\b H_{it},\b Z_{it},\b \beta,\b \delta,\b u_i,\sigma) = \frac{\tau(1-\tau)}{\sigma} \exp\left\{-\rho\left(\frac{Y_{it}-\b\beta'\b X_{it} -\b\delta'\b
H_{it}-\b u_i'\b Z_{it}}{\sigma}\right)\right\}, \end{equation} where $\rho(u)=u\{\tau - I(u<0)\}$ is the quantile loss function and $\sigma>0$ is a scale parameter. When $Z_{it}=1$, we obtain a random intercept model \citep{gera:bott:07}. The ALD is justifiable since maximum likelihood is exactly equivalent to minimization of the quantile loss function, when no parametric assumptions are made on $\varepsilon$ \citep{yu:moye:01}. Further, it can be seen in simulation studies to lead to good estimates even when the residuals are not ALD distributed (see for instance \cite{yuan:bott:09}, \cite{farc:12}).
\subsection{The survival model}
The time-varying baseline risk function $h_0(T_i)$ in (\ref{jqsm}) can be seen as the risk obtained when all covariates and random effects are exactly zero. We may wish to specify a flexible parametric form for $h_0(t)$ (e.g., $h_0(t)=\eta t^{\eta-1}$, leading to a Weibull model), as in for instance \cite{rizop:ghosh:11}, \cite{riz:12}, \cite{vivi:alfo:rizo:13} or we may leave it unspecified as in \cite{follmann:95}. The inferential strategy for obtaining the MLE is slightly different in the two cases, as we discuss in the next section.
The time-to-event distribution can in both cases be written as \begin{eqnarray} \label{p.t} f(T_i, \Delta_i \mid \b u_i)&=& f(T_i \mid \mathcal{T}_{it},\b W_i)^{\Delta_i}S(T_i \mid \mathcal{T}_{it}, \b W_i)^{1-\Delta_i} \nonumber\\ &=&h(T_i \mid \mathcal{T}_{it},\b W_i)^{\Delta_i}S(T_i \mid \mathcal{T}_{it},\b W_i) , \end{eqnarray} where $S(\cdot)$ denotes the survival function, i.e. $$ S(T_i \mid \mathcal{T}_{it}, \b W_i) = \exp\left\{ - \int_0^{T_i} h_0(s)\exp\{\b\gamma' \b W_i + \alpha_1 \b\delta' \b H_{is} + \alpha_2\b u_i'\b Z_{is} \} ds \right \} $$ while $h(T_i \mid \mathcal{T}_{it},\b W_i)$ is given by the second equation in (\ref{jqsm}).
\subsection{The random effects model} \label{ranef}
A commonly used distribution for random effects is the multivariate normal. This is convenient to work with when a Gaussian assumption is formulated also for the longitudinal outcome. The multivariate normal may not be satisfactory anyway when the number of occasions is small (\cite{rizopoulos:08b},\cite{hsie:06}) and /or the dimensionality $q$ of $\b Z_{it}$ is large. Further, we may often expect a slower convergence of the posterior distribution of the random effects to a multivariate normal when modeling lower or upper quantiles. Two valid alternatives for the random effects distribution are given by a multivariate $T$ with $k$ degrees of freedom: $$
f(\b u_i|\b\Sigma) \propto |\b\Sigma|^{-1/2}\left(1+\frac{1}{k} \b u_i'\b\Sigma^{-1}\b u_i\right)^{-\frac{k+q}{2}}, $$ which can be used to capture fat tails of the random effects; and a multivariate ALD, which is often suggested in the quantile regression framework \citep{yuan:bott:09}: $$
f(u_i|\b\Sigma) \propto |\b\Sigma|^{-1/2}\left(\frac{\b
u_i'\b\Sigma^{-1}\b u_i}{2}\right)^{v/2} K_v\left(\sqrt{\b
u_i'\b\Sigma^{-1}\b u_i}\right) $$ where $v=(2-q)/2$ and $K_v(\cdot)$ is the modified Bessel function of the third kind. The most appropriate random effects distribution may be chosen for instance using the Bayesian Information Criterion (BIC) or the Akaike Information Criterion (AIC), see for instance \cite{verblesaf:96}.
\subsection{Observed likelihood} \label{sec:lik}
In what follows, $\b\theta$ is a short-hand notation for model parameters, that is, $\b\beta$, $\b\delta$, $\alpha_1$, $\alpha_2$, $\b\gamma$, $\sigma$, $\b\Sigma$ and any parameter associated with baseline hazard $h_0(s)$. The joint likelihood contribution of the longitudinal and survival processes for the $i$-th subject is obtained integrating the conditional distributions in (\ref{jqsm}) over the random effects space: \begin{eqnarray} \label{indlik}
f(T_i, \Delta_i,\b Y_i;\b\theta) = \int f(\b Y_i | \b u_i) f(T_i,
\Delta_i |\b u_i) f(\b u_i| \b\Sigma) d \b u_i, \end{eqnarray} where $f(T_i, \Delta_i \mid \b u_i)$ is given in equation (\ref{p.t}) and \begin{eqnarray*}
f(\b Y_i |\b u_i) = \prod_{t=1}^{n_i} f(Y_{it} |\b u_i), \end{eqnarray*} while $f(Y_{it} \mid \b u_i)$ is given in equation (\ref{ald}). The observed data log-likelihood for the joint quantile regression model is then \begin{eqnarray} \label{lik} \ell(\b\theta) = \sum_i \log f(T_i, \Delta_i, y_i;\b\theta). \end{eqnarray}
The integrals involved in (\ref{indlik}) are usually tackled in the JM context through quadrature methods (see for instance \cite{riz:12}, \cite{vivi:alfo:rizo:13}). While effective in one or two dimensions, quadrature methods tend to become too slow or less precise as the dimensionality of the random effect distribution grows. Furthermore, quadrature methods should have to be tailored to the random effects distribution (e.g., a Gauss-Hermite quadrature would be best for Gaussian random effects, while a Gauss-Laguerre may be better under other assumptions). In next section we propose a Monte Carlo strategy which is completely general, and allows us to set up an algorithm which is easily adapted to any assumption on the random effects and to any functional form for the two parts of the JM.
\section{Estimation of the proposed model} \label{inference}
We propose a MCEM algorithm for fitting the proposed model. This involves alternating two steps until convergence: (i) sampling from the posterior distribution for the random effects, given the data and current values of the parameters (Monte Carlo step), and obtaining the conditional expected value of the complete data log likelihood (E-step), (ii) maximizing the latter (M-step). The algorithm is guaranteed to converge to a local optimum. In order to increase the odds of obtaining the global optimum, we perform a multistart. The first run starts from estimates obtained from separate longitudinal and survival models, which are readily available. Other runs are initialized by randomly perturbing the deterministic initial solution. A second remark regards how to obtain standard errors and confidence intervals. When performing quantile regression on the longitudinal outcome, we use a non-parametric bootstrap strategy (\cite{buch:95}, \cite{andr:buch:00}). We preserve the dependency structure in the data by resampling subjects rather than separately resampling the outcomes (and related predictors). Under the usual regularity conditions, tests on the regression parameters may then be simply performed by using Wald statistics based on the standard errors.
The MCEM algorithm is based on the complete likelihood, that is, the likelihood we would have if we could observe the random effects. The individual contribution to the complete data log-likelihood can be obtained as \begin{equation} \label{IndcomplLik} \log f(T_i, \Delta_i, \b Y_i,\b u_i;\b\theta) = \log f(T_i, \Delta_i
|\b u_i;\b \theta) + \log f(\b Y_i |\b u_i;\b \theta)+ \log f(\b u_i|\b\Sigma). \end{equation} The MCEM algorithm is completely general and can be simply adapted to any distributional assumption for the longitudinal error and random effects distribution. Assuming an ALD for the error distribution of the longitudinal measurements, the complete data log-likelihood is as follows: \begin{eqnarray} \label{complLik} \ell_c(\b\theta)&=& \sum_i \log f(T_i, \Delta_i,\b Y_i,\b u_i;\b\theta)= \\ \nonumber
&=& \sum_i \log f(Y_i | \b u_i;\b \theta) + \sum_i \log f(T_i,
\Delta_i |\b u_i;\b \theta) + \sum_i \log f(\b u_i|\b\Sigma) \\ \nonumber &=& -\log \sigma\sum_i n_i - \sum_i\sum_{t=1}^{n_i} \rho\left(\frac{Y_{it} - \b\beta'\b X_{it} - \b\delta'\b H_{it}-\b
u_i'\b Z_{it}}{\sigma}\right) \\ \nonumber &+& \sum_i \Delta_i \log h_0(T_i) + \sum_i \Delta_i\b\gamma'\b W_i + \alpha_1 \sum_i \Delta_i \b\delta'\b H_{iT_i} + \alpha_2 \sum_i \Delta_i\b u_i'\b Z_{iT_i}\\ \nonumber &-& \sum_i \int_0^{T_i} h_0(s) \exp\{\b\gamma'\b W_i + \alpha_1 \b \delta'\b H_{is} + \alpha_2\b u_i'\b Z_{is}\} ds \\\nonumber
&+& \sum_i \log f(\b u_i|\b\Sigma). \end{eqnarray}
\subsection{Monte Carlo E-step} \label{e}
The conditional expected value of (\ref{IndcomplLik}) for the $i$-th subject at the $j$-th iteration of the algorithm is expressed as \begin{eqnarray} \label{Expect} \mathbb{E}[ \ell_c(\b\theta\mid T_i, \Delta_i, \b Y_i,\b u_i) ] &= \sum_i \int &[
\log f\left(\b Y_i | \b u_i;\b\theta\right) + \log f\left( T_i, \Delta_i |\b u_i; \b\theta\right) \nonumber \\ &+& \log f\left(\b u_i; \b\theta\right) ] f\left(\b u_i \mid T_i, \Delta_i,\b Y_i;\b \theta^{(j)}\right) d\b u_i , \end{eqnarray} where $\b \theta^{(j)}$ denotes the current value of the parameters. The posterior distribution of the random effects is
\begin{eqnarray} \label{post.b} f(\b u_i \mid T_i, \Delta_i,\b Y_i;\b \theta^{(j)}) &\propto & f(\b Y_i, T_i, \Delta_i,\b u_i;\b \theta^{(j)}) \nonumber\\
&=& f(T_i, \Delta_i |\b u_i;\b \theta^{(j)}) f(\b Y_i |\b u_i;\b \theta^{(j)})
f(\b u_i |\b \Sigma^{(j)}). \end{eqnarray} Straightforward algebra can be used to see that (\ref{post.b}) simplifies to \begin{eqnarray*} f(\b u_i \mid T_i, \Delta_i,\b Y_i;\b \theta^{(j)}) &\propto & \exp\left\{-\rho\left(\frac{Y_{it}-\b\beta^{(j)'}\b X_{it}
-\b\delta^{(j)'}\b H_{it}-
\b u_i'\b Z_{it}}{\sigma^{(j)}}\right)+ \Delta_i\alpha_2^{(j)}\b u_i'\b Z_{iT_i}\right\}\\ &\phantom{a}& \exp\left\{ -\int_{0}^{T_i} h_0^{(j)}(s)
\exp\{\alpha_2^{(j)}\b u_i'\b Z_{is}\}\right\} f(u_i |\b \Sigma^{(j)}) \end{eqnarray*}
In order to work with (\ref{Expect}), we need to marginalize the joint distribution with respect to the multivariate random effect posterior distribution. The resulting integral is conveniently approximated through Importance Sampling (IS). Importance sampling proceeds by obtaining a random sample $(v_{i1}, \ldots, v_{im_i^{(j)}})$ from a proposal distribution $g(\cdot)$. The IS identity $$ \int \ell_c(\b\theta) f(\b u_i \mid T_i, \Delta_i,\b Y_i;\b \theta^{(j)}) d\b u_i = \int \ell_c(\b\theta) \frac{f(\b u_i \mid T_i, \Delta_i,\b Y_i;\b \theta^{(j)})}{g(\b u_i)} g(\b u_i) d\b u_i $$ is used to approximate (\ref{Expect}). More formally, the expression in (\ref{Expect}) is approximated as \begin{eqnarray} \label{ExpApp} \mathbb{E}[ \ell_c(\b\theta) ] &\approx& \sum_i \sum_{b=1}^{m_i^{(j)}} [
\log f\left(\b Y_i |\b v_{ib};\b \theta^{(j)}\right) + \log f\left( T_i,
\Delta_i |\b v_{ib};\b \theta^{(j)}\right) \\ \nonumber &\phantom{+}& + \log f\left(\b v_{ib} \mid \b \theta^{(j)}\right)] w_{ib}, \end{eqnarray} where $$ \widetilde{w}_{ib} = \frac{f(\b v_{ib} \mid T_i,\Delta_i,\b Y_i;\b
\theta^{(j)})}{g(\b v_{ib})} $$ and $ w_{ib} = \frac{\widetilde{w}_{ib}}{\sum_b \widetilde{w}_{ib}}. $
In this work we proceed as in \cite{levi:case:01}, where we sample $(v_{i1}, \ldots, v_{im_i^{(j)}})$ from the posterior distribution at the initial parameter estimates using adaptive rejection Metropolis sampling \citep{gilks:95}, and then update the weights at each iteration. The MC sample size $m_i^{(j)}$ is increased to control the MC error \citep{levi:case:01,eichoff:04}. Finally, the observed likelihood (\ref{lik}) can be directly approximated as $\prod_i \sum_b \widetilde{w}_{ib}$. The latter is used to check convergence of the MCEM and after convergence for testing and computation of information criteria. In summary, the E-step is given by updating of $\widetilde{w}_{ib}$, $w_{ib}$, and evaluation of the likelihood.
\subsection{M-step} \label{m}
We outline here the M-step, which consists in maximizing the approximated conditional expected value of the complete likelihood with respect to $\b\theta$.
When $h_0(s)$ is left completely unspecified, we obtain a Nelson-Aalen \citep{nels:72,aale:78} type estimator as \begin{equation} \label{hath0} \widehat h_0(s) = \sum_{i=1}^n \frac{\Delta_i I(T_i=s)}{\sum\limits_{i:
T_i \geq s} \sum_{b=1}^{m_i^{(j)}} w_{ib}\exp\{\b\gamma'\b W_i +
\alpha_1 \b\delta' \b H_{is} + \alpha_2\b v_{ib} '\b Z_{is}\}}, \end{equation} by setting to zero the score equations for $h_0(s)$ \citep{WULF:TSIAT:97}. See also \cite{niel:gill:ande:sore:92} and \cite{gill:92}. The expression (\ref{hath0}) is not an explicit solution as it depends on other parameters. Nevertheless, it can be plugged in the conditional expected value of the complete likelihood, thus obtaining a profile complete likelihood (complete with respect to the random effects, and profiled with respect to the non-parametric baseline along the lines of \cite{cox:72}). If instead we specify a parametric form for $h_0(s)$, e.g, $h_0(s)=\eta s^{\eta-1}$, we can plug-in this expression and update any parameter involved in $h_0(s)$ within the rest of the M-step.
When a Gaussian distribution is assumed for the longitudinal measurements, regression coefficients and hazard ratios are updated through a one-step Newton-Raphson algorithm (which is easily adapted from \cite{WULF:TSIAT:97}), while the variance of the random term $\epsilon_{it}$ in (\ref{jqsm}) is estimated through the usual closed form expression.
When we assume an ALD for the longitudinal measurements, the M-step is complicated by the presence of the quantile check function. An estimator of $\sigma$, dependent on the other parameters, can be explicitely obtained as: \begin{equation} \label{hatsigma} \widehat \sigma = \frac{1}{\sum_i n_i} \sum_{i=1}^n\sum_{t=1}^{n_i} \sum_{b=1}^{m_i^{(j)}} w_{ib}\rho\left(y_{it}-\b\beta'\b X_{it} -\b\delta'
\b H_{it}-\b v_{ib}'\b Z_{it}\right). \end{equation} The vector of regression parameters and dispersion parameter for the ALD is block updated using one step of the \cite{neld:mead:65} numerical optimization algorithm, majorizing (\ref{ExpApp}) after plug-in of (\ref{hatsigma}) and (\ref{hath0}) or the parametric formula of $h_0(s)$. The resulting expected complete likelihood is \begin{eqnarray*} \sum_i \Delta_i \log({\widehat h}_0(T_i))+\sum_{i=1}^n \Delta_i\b \gamma'\b W_i + \alpha_1 \sum_{i=1}^n \Delta_i \b\delta'\b H_{iT_i} + \alpha_2 \sum_{i=1}^n \Delta_i\sum_{b=1}^{m_i^{(j)}} w_{ib}\b v_{ib}'\b Z_{iT_i}&\phantom{+}&\\ -\sum_{i=1}^n \sum_{b=1}^{m_i^{(j)}}w_{ib}\int_0^{T_i} \widehat h_0(s)\exp\{\b\gamma'\b W_i + \alpha_1 \b\delta' \b H_{is} + \alpha_2 \b v_{ib}'\b Z_{is}\}\ ds&\phantom{+}& \\ - \sum_{i=1}^n\sum_{t=1}^{n_i} \sum_{b=1}^{m_i^{(j)}}w_{ib} \rho\left(\frac{\b Y_{it} - \b\beta'\b X_{it} - \b\delta'
\b H_{it}- \b v_{ib}'\b Z_{it}}{\widehat\sigma}\right) -\log(\widehat\sigma)\sum_{i=1}^n n_i&\phantom{+}&. \end{eqnarray*} The integral involved in the expression above
(and similarly in any expression where $f(T_i,\Delta_i|\b v_{ib}; \b\theta)$ appears also at the E-step) reduce to sums when a non-parametric baseline is used, and can instead be approximated using one-dimensional Gauss-Kronrod quadrature (e.g., \cite{kaha:mole:nash:89}) when a parametric assumption is formulated for $h_0(s)$.
The M-step is concluded by maximizing the approximated conditional expected value of the complete likelihood with respect to parameters involved in the distribution of the random effects. This is readily accomplished under any of the assumptions we have proposed in Section \ref{ranef} using the method of moments. In all cases in fact $\b\Sigma$ can be updated as the weighted empirical covariance matrix of the random effects sampled at the E-step.
\section{Simulations} \label{simu}
In this section we illustrate our approach through a simulation study. We evaluate the bias and standard deviation of the estimates for our proposed model for data Missing Not At Random (MNAR), and compare with a model which ignores informative drop-out (Missing At Random - MAR model).
For $n=\{250,500\}$, $\alpha_1=\{0,1\}$, $\alpha_2=\{0,1\}$, $\tau=\{0.25,0.5,0.75\}$ we fix $\b\beta=\b\delta=\b\gamma=(1\ 1)$ and $\sigma=1$. We assume random effects arise from a centered bivariate normal distribution with standard deviations equal to 0.3 and correlation 0.16.
We let $\b Z_{it}=(1\ t)$, $\b H_{it}=(h_{i1}\ h_{i2}*t)$, $\b X_i=(1\ x_i)$; with $h_{i1}$, $h_{i2}$, $x_i$, $W_{i1}$ and $W_{i2}$ generated from independent standard normals. By also fixing $h_0(s)=1$ it is possible to exactly obtain the survival distribution as $$
S(t|\b u_i,\b H_i,\b W_i) = \exp\left\{ -\frac{e^{\alpha_1(\delta_1H_{i1}+\delta_2H_{i2t})+\alpha_2(u_{i1}+u_{i2}t)+
\b\gamma'\b W_i}-e^{\alpha_1\delta_1H_{i1}+\alpha_2u_{i1}+
\b\gamma'\b W_i}}{\alpha_2u_{i2}+\alpha_1\delta_2h_i}\right\} $$ when $\alpha_1 \neq 0$ or $\alpha_2 \neq 0$ and $$
S(t|U,H,W) = \exp\{-te^{\b\gamma'\b W_i}\} $$ when $\alpha_1=\alpha_2=0$. The expression above can be inverted to obtain $T_i$ after generating $n$ random variates uniformly distributed on the unit interval. We then let the censoring time $C_i/5$ be distributed according to a Beta with parameters 4 and 1, in order to obtain a censoring proportion around 25\%. We allow for a maximum of six observation times for each subject, at $t=0, 1/4, 1/2, 3/4, 1, 3$. Longitudinal observations before drop-out are independently generated from an ALD for the $\tau$-th quantile, centered on $$ \b\beta'\b X_i+\b\delta'\b H_{it}+\b u'\b Z_{it}, $$ and with dispersion parameter $\sigma$. We fit our joint model with parametric baseline distribution and two separate models (one for the longitudinal process and one for the time-to-event, therefore obtaining MAR estimates), based on each generated data set.
For each setting we report the bias and standard deviation of the estimates averaged over $B=1000$ replicates, and further averaged over groups of parameters for $\b\beta$, $\b\delta$ and $\b\gamma$. Results are shown in Table \ref{simures}, where it can be seen that the MNAR model has a very low bias and standard deviation of the estimates for all values of $\alpha_1$ and $\alpha_2$, with very few exceptions which are likely due to random fluctuation. Results are consistent across all quantiles, with a slightly larger MSE for quantiles distant from the median, as expected.
\begin{table}[ht] \centering \caption{Bias and standard deviation of the estimates of the proposed
model on simulated
data for different values of $n$, $\alpha_1$ and $\alpha_2$. Results are based on $B=1000$ replicates.} \label{simures} \begin{tabular}{rrrcccccccccc}
\hline \multicolumn{13}{c}{$\tau=0.25$} \\ &&& \multicolumn2c{$\b\beta$} & \multicolumn2c{$\b\delta$} &
\multicolumn2c{$\b\gamma$} & \multicolumn2c{$\alpha_1$} & \multicolumn2c{$\alpha_2$} \\ $n$ & $\alpha_1$ & $\alpha_2$ & bias & s.d. & bias & s.d & bias & s.d & bias & s.d. & bias & s.d \\
\hline 250 & 0 & 0 & -0.004 & 0.107 & 0.001 & 0.124 & 0.020 & 0.085 & -0.001 & 0.044 & -0.002 & 0.131\\ 250 & 0 & 1 & -0.013 & 0.104 & -0.002 & 0.119 & -0.023 & 0.100 & -0.001 & 0.066 & -0.076 & 0.177\\ 250 & 1 & 0 & -0.003 & 0.109 & 0.009 & 0.094 & 0.020 & 0.088 & 0.021 & 0.101 & -0.008 & 0.146\\ 250 & 1 & 1 & -0.012 & 0.105 & -0.003 & 0.108 & -0.021 & 0.106 & -0.025 & 0.108 & -0.068 & 0.189\\ 500 & 0 & 0 & -0.002 & 0.074 & -0.000 & 0.082 & 0.006 & 0.058 & -0.001 & 0.031 & -0.003 & 0.091\\ 500 & 0 & 1 & -0.011 & 0.072 & 0.002 & 0.082 & -0.034 & 0.072 & 0.001 & 0.046 & -0.076 & 0.120\\ 500 & 1 & 0 & -0.003 & 0.074 & 0.004 & 0.061 & 0.008 & 0.061 & 0.008 & 0.067 & -0.006 & 0.092 \\ 500 & 1 & 1 & -0.011 & 0.074 & -0.004 & 0.075 & -0.033 & 0.073 & -0.038 & 0.076 & -0.063 & 0.118\\ \hline \multicolumn{13}{c}{$\tau=0.5$} \\ &&& \multicolumn2c{$\b\beta$} & \multicolumn2c{$\b\delta$} &
\multicolumn2c{$\b\gamma$} & \multicolumn2c{$\alpha_1$} & \multicolumn2c{$\alpha_2$} \\ $n$ & $\alpha_1$ & $\alpha_2$ & bias & s.d. & bias & s.d & bias & s.d & bias & s.d. & bias & s.d \\ \hline 250 & 0 & 0 & -0.002 & 0.096 & -0.000 & 0.113 & 0.020 & 0.084 & -0.001 & 0.044 & -0.000 & 0.126 \\
250 & 0 & 1 & -0.014 & 0.092 & -0.003 & 0.107 & -0.020 & 0.100 & -0.001 & 0.066 & -0.064 & 0.164 \\
250 & 1 & 0 & -0.001 & 0.097 & 0.005 & 0.087 & 0.020 & 0.087 & 0.022 & 0.096 & -0.006 & 0.138 \\
250 & 1 & 1 & -0.015 & 0.094 & -0.005 & 0.101 & -0.019 & 0.105 & -0.030 & 0.107 & -0.074 & 0.175 \\
500 & 0 & 0 & 0.001 & 0.067 & -0.000 & 0.076 & 0.005 & 0.057 & -0.001 & 0.031 & -0.004 & 0.083 \\
500 & 0 & 1 & -0.008 & 0.063 & 0.000 & 0.074 & -0.028 & 0.071 & -0.001 & 0.045 & -0.060 & 0.106 \\
500 & 1 & 0 & -0.000 & 0.066 & 0.001 & 0.058 & 0.006 & 0.059 & 0.010 & 0.064 & -0.003 & 0.088 \\
500 & 1 & 1 & -0.008 & 0.066 & -0.006 & 0.069 & -0.031 & 0.072 & -0.035 & 0.072 & -0.056 & 0.119\\ \hline \multicolumn{13}{c}{$\tau=0.75$} \\ &&& \multicolumn2c{$\b\beta$} & \multicolumn2c{$\b\delta$} &
\multicolumn2c{$\b\gamma$} & \multicolumn2c{$\alpha_1$} & \multicolumn2c{$\alpha_2$} \\ $n$ & $\alpha_1$ & $\alpha_2$ & bias & s.d. & bias & s.d & bias & s.d & bias & s.d. & bias & s.d \\ \hline
250 & 0 & 0 & -0.001 & 0.108 & 0.002 & 0.122 & 0.019 & 0.085 & -0.003 & 0.044 & -0.007 & 0.134 \\
250 & 0 & 1 & -0.013 & 0.104 & -0.003 & 0.119 & -0.023 & 0.100 & -0.001 & 0.066 & -0.076 & 0.169 \\
250 & 1 & 0 & 0.003 & 0.111 & 0.004 & 0.091 & 0.021 & 0.088 & 0.024 & 0.099 & -0.002 & 0.145 \\
250 & 1 & 1 & -0.012 & 0.107 & -0.006 & 0.107 & -0.020 & 0.106 & -0.028 & 0.112 & -0.079 & 0.178 \\
500 & 0 & 0 & 0.004 & 0.075 & -0.003 & 0.083 & 0.006 & 0.058 & -0.001 & 0.031 & -0.004 & 0.088 \\
500 & 0 & 1 & -0.006 & 0.070 & -0.000 & 0.080 & -0.030 & 0.072 & -0.001 & 0.046 & -0.066 & 0.109 \\
500 & 1 & 0 & 0.004 & 0.075 & -0.002 & 0.062 & 0.007 & 0.060 & 0.014 & 0.068 & -0.004 & 0.092 \\
500 & 1 & 1 & -0.007 & 0.073 & -0.009 & 0.075 & -0.032 & 0.073 & -0.033 & 0.076 & -0.062 & 0.120\\
\hline \end{tabular} \end{table}
In Table \ref{simures2} we report the ratio of the bias and the variance of the estimates of the MAR over the MNAR model, based on the average bias for all parameters, separately for the longitudinal and survival parameters. These ratios are close to the unity when $\alpha_1=\alpha_2=0$, with our model generally performing slightly better given that the MNAR model assumes $\b\delta$ parameters are equal in the longitudinal and survival parts. When $\alpha_1$ or $\alpha_2$ are non-zero, the ratios of the variances of the estimates are still close to the unity, but the ratios of biases increase substantially. The bias of the MAR model may be up to 30 times the bias of our joint model. The effect of $\alpha_1$ is often stronger than the effect of $\alpha_2$, but this is likely only due to the fact that in all simulated settings there is a larger heterogeneity due to the shared covariates with respect to the unobserved heterogeneity due to random effects. As could be expected, the ratios are generally increasing with $n$, given that the MSE of the joint model is infinitesimal.
\begin{table}[ht] \centering \caption{Ratios of bias and variance of the estimates obtained with
the MAR model ($bias_{MAR}$, $var_{MAR}$) and with our model
($bias$, $var$) for the longitudinal $(Y)$ and survival $(T)$
part of the model. Results are shown for different values of $n$,
$\alpha_1$ and $\alpha_2$ and are based on $B=1000$ replicates.} \label{simures2} \begin{tabular}{rrrcccc}
\hline \multicolumn{7}{c}{$\tau=0.25$}\\ $n$ & $\alpha_1$ & $\alpha_2$ &
$\left|\frac{bias_{MAR}(Y)}{bias(Y)}\right|$ &
$\frac{var_{MAR}(Y)}{var(Y)}$ & $\left|\frac{bias_{MAR}(T)}{bias(T)}\right|$ &
$\frac{var_{MAR}(T)}{var(T)}$ \\
\hline 250 & 0 & 0 & 1.548 & 1.203 & 0.800 & 1.115 \\ 250 & 0 & 1 & 6.616 & 1.241 & 1.266 & 1.315 \\ 250 & 1 & 0 & 2.876 & 1.573 & 9.990 & 0.929 \\ 250 & 1 & 1 & 5.868 & 1.385 & 8.051 & 0.814 \\ 500 & 0 & 0 & 1.264 & 1.252 & 0.802 & 1.197 \\ 500 & 0 & 1 & 4.214 & 1.270 & 1.954 & 1.287 \\ 500 & 1 & 0 & 6.966 & 1.654 & 5.954 & 0.996 \\ 500 & 1 & 1 & 7.227 & 1.412 & 24.926 & 0.834 \\
\hline \multicolumn{7}{c}{$\tau=0.5$}\\ $n$ & $\alpha_1$ & $\alpha_2$ &
$\left|\frac{bias_{MAR}(Y)}{bias(Y)}\right|$ &
$\frac{var_{MAR}(Y)}{var(Y)}$ & $\left|\frac{bias_{MAR}(T)}{bias(T)}\right|$ &
$\frac{var_{MAR}(T)}{var(T)}$ \\
\hline 250 & 0 & 0 & 1.091 & 1.121 & 0.909 & 1.129 \\
250 & 0 & 1 & 1.679 & 1.204 & 1.464 & 1.333 \\
250 & 1 & 0 & 5.659 & 1.439 & 7.579 & 0.948 \\
250 & 1 & 1 & 4.646 & 1.289 & 10.572 & 0.804 \\
500 & 0 & 0 & 0.865 & 1.188 & 0.930 & 1.221 \\
500 & 0 & 1 & 6.936 & 1.221 & 1.208 & 1.339 \\
500 & 1 & 0 & 2.756 & 1.501 & 6.758 & 1.041 \\
500 & 1 & 1 & 5.797 & 1.290 & 29.194 & 0.851 \\
\hline \multicolumn{7}{c}{$\tau=0.75$}\\ $n$ & $\alpha_1$ & $\alpha_2$ &
$\left|\frac{bias_{MAR}(Y)}{bias(Y)}\right|$ &
$\frac{var_{MAR}(Y)}{var(Y)}$ & $\left|\frac{bias_{MAR}(T)}{bias(T)}\right|$ &
$\frac{var_{MAR}(T)}{var(T)}$ \\
\hline
250 & 0 & 0 & 1.308 & 1.223 & 1.010 & 1.151 \\
250 & 0 & 1 & 4.972 & 1.280 & 1.283 & 1.318 \\
250 & 1 & 0 & 3.895 & 1.586 & 7.153 & 0.934 \\
250 & 1 & 1 & 4.191 & 1.367 & 10.346 & 0.783 \\
500 & 0 & 0 & 1.866 & 1.255 & 1.304 & 1.201 \\
500 & 0 & 1 & 6.733 & 1.276 & 1.135 & 1.303 \\
500 & 1 & 0 & 4.370 & 1.672 & 5.414 & 1.014 \\
500 & 1 & 1 & 7.216 & 1.347 & 27.891 & 0.836 \\
\hline \end{tabular} \end{table}
\section{Application to dilated cardiomyopathy data} \label{realdata}
In this section we briefly illustrate the proposed approach on an original data set about patients with dilated cardiomyopathy. Data refers to $n=659$ consecutive patients who begun treatment for dilated cardiomyopathy in the cardiovascular department of ``Ospedali Riuniti'' in Trieste, Italy. Patients were enrolled at first treatment and scheduled for follow-up after 6 months, 1, 2, 3, 4, 6 and 10 years, with only 25\% of the patients having complete records. Maximum follow-up time before cardiovascular death or censoring due to loss at follow-up or transplant is 25 years, with a total of 212 events (32\%).
Dilation of the left ventricular is known to lead to hearth failure, and in many cases an ethiological basis cannot be identified \citep{merlo:11}. The goal of this study is to compare patients with mild dilation of the left ventricular (Mildly Dilated CardioMyopathy or MDCM) with respect to patients with a general dilation of unrecognized ethiology (Idiopatic Dilated CardioMyopathy or IDCM). MDCM patients are generally believed to be at a slightly lower risk (e.g., \cite{kere:90}), but the physiological reasons are still unrecognized.
Our longitudinal outcome of interest is the left ventricular ejection fraction (LVEF), that is, the volumetric fraction of blood pumped out of the ventricle with each heartbeat. Note that LVEF is a bounded outcome, but all measurements are far from the boundaries so that we unlike \cite{bott:cai:mcke:10} we can avoid any transformations. Dropout occurs due to cardiovascular death, and it can be easily expected that the two processes are related as patients with a lower ejection fraction are at higher risk of death. For example, a univariate Cox model for the baseline LVEF gives an hazard ratio (HR) of 0.95 for each percentage point, with $p<1e-16$. Furthermore, LVEF is skewed and its skewness seems to change over time. This leads to two issues: first, using a classical joint model after transformation of the LVEF would be awkward, as the optimal transformation is different at each time point. Secondly, modeling the mean of the transformed LVEF would not be as meaningful from a clinical perspective than directly modeling quantiles, which is also straightforward to interpret. We mostly are interested in low quantiles (like the 10th or the 15th), in the terziles or quartiles for the outcome. See for instance \cite{sand:03}, \cite{Clements01062005}, \cite{ndre:etal:07}, \cite{Cowie03102012}. Consequently, we explore the ejection fraction distribution by evaluating $\tau=0.1,0.15,0.25,0.33,0.5,0.66,0.75$.
We model the longitudinal outcome conditionally on an intercept and age at baseline ($\b X$ matrix), on the indicator of MDCM and its interaction with time ($\b H$ matrix). Besides covariates in the $\b H$ matrix, we let the hazard of death depend on gender (1 for males) and indicator of New York Hearth Association (NYHA) functional classes I or II at baseline ($\b W$ matrix). For more details on NYHA functional classes see for instance \cite{merlo:11} and references therein. We also include a shared subject-specific random intercept and a random slope, that is, $\b Z=(1\ t)$. Given that the number of follow-up times is slightly large, we use a normality assumption for $\b u_i$. We also have compared with a multivariate $T$ and multivariate Laplace, with analogous results which we do not report for reasons of space. We only mention that the multivariate normal distribution is chosen using AIC and BIC criteria. We estimate our proposed model both with a parametric Weibull baseline and with a non-parametric baseline. Likelihood, AIC, and BIC at each quantile are reported in Table \ref{aicandbic}.
\begin{table}[ht] \caption{Log-likelihood at convergence, AIC and BIC for our model fit with a parametric and non-parametric baseline, at
different quantiles of interest, on the dilated cardiomyopathy data.} \label{aicandbic} \fbox{ \begin{tabular}{ccccccc}
& \multicolumn3c{non-parametric baseline} & \multicolumn3c{parametric baseline} \\ $\tau$ & $\ell(\b\theta)$ & BIC & AIC & $\ell(\b\theta)$& BIC &AIC\\ .1& -10515.214& 21064.255& 21054.428& -10602.111& 21240.867& 21228.222\\ .15 &-10449.858& 20933.543& 20923.716& -10461.581& 20959.807& 20947.162\\ .25& -10289.721& 20613.269& 20603.442& -10322.538& 20681.721& 20669.076\\ .33 &-10197.313& 20428.453& 20418.626& -10235.107& 20506.859& 20494.214\\ .5 &-10154.227& 20342.281& 20332.454& -10175.175& 20386.995& 20374.350\\ .66 &-10223.996& 20481.819& 20471.992& -10229.257& 20495.159& 20482.514\\ .75& -10300.913& 20635.653& 20625.826& -10308.303& 20653.251& 20640.606\\ \end{tabular}} \end{table}
Based on those results we select the non-parametric baseline for all quantiles. We compare the estimates with those obtained under a MAR model in Table \ref{mdcm_res}.
\begin{table}[ht] \caption{Estimates for the MNAR and MAR models for the MDCM data. An
asterisk indicates that estimates in the column are significant
at the 5\% level for all quantiles $\tau$. The models are based on a non-parametric baseline
for the survival process and two dimensional shared random effects.} \label{mdcm_res}\fbox{ \begin{tabular}{ccccccccc} & \multicolumn8c{MNAR model} \\
& \multicolumn4c{Longitudinal outcome} &
\multicolumn4c{Survival outcome}\\ $\tau$ & Int* & Age* & MDCM* & MDCM:time* & Gender* & NYHA* & $\alpha_1$* &
$\alpha_2$* \\ .1& 22.535& -0.038& 7.754& -0.038& 0.538 &-0.912& -0.044 & -0.042\\ .15& 25.134 &-0.048& 7.768 &-0.028 &0.527 &-0.895& -0.037 & -0.057\\ .25& 29.255 &-0.068 & 7.752 &-0.042 &0.593 &-0.990& -0.017 & -0.068\\ .33& 32.813 &-0.095 & 7.752 &-0.018 &0.592 &-0.988& -0.018 & -0.087\\ .5& 39.086 &-0.124 & 7.749 &-0.093 &0.542 &-0.918& -0.031 & -0.063\\ .66& 44.437 &-0.148 & 7.734 &-0.136 &0.566 &-0.952& -0.011 & -0.097\\ .75& 47.355 &-0.156 & 7.744 &-0.044 &0.591 &-0.987& -0.021 & -0.094\\ \hline & \multicolumn8c{MAR model} \\
& \multicolumn4c{Longitudinal outcome} &
\multicolumn4c{Survival outcome}\\ & Int* & Age* & MDCM* & MDCM:time* & Gender* & NYHA* & $\alpha_1$ &
$\alpha_2$\\ .1& 29.700 &-0.141& 7.580& 0.041& 0.600& -1.096 & 0.000 & 0.000\\ .15& 36.266 &-0.251& 8.027& 0.076&0.600 &-1.096 &0.000& 0.000\\ .25& 35.175 &-0.130& 5.262& 0.011&0.600 & -1.096 &0.000& 0.000\\ .33& 35.246 &-0.109& 6.516& 0.068&0.600 & -1.096 &0.000& 0.000\\ .5& 40.048 &-0.137& 8.289& 0.028&0.600 &-1.096 &0.000& 0.000\\ .66& 44.541 &-0.123 &5.711& 0.022&0.600 &-1.096 & 0.000& 0.000\\ .75& 44.284 &-0.088& 5.558& 0.017&0.600 & -1.096 &0.000& 0.000\\ \end{tabular}} \end{table}
There is a stronger and stronger effect of age on LVEF as $\tau$ increases, while the effect of MDCM is slightly constant with $\tau$, with a negative interaction with time. Males are at slightly higher risk of death and patients in lower functional NYHA classifications are at a lower risk. After adjusting for these covariates, the significant and negative estimates for $\widehat\alpha_1$ lead us to conclude that MDCM is an {\it
independent} predictor of a slightly lower risk of death, even after considering its effect on LVEF. We could expect negative estimates for $\alpha_1$ and $\alpha_2$ as longitudinal measurements and survival time are positively dependent.
Ignoring drop-out may lead to an important bias. First of all, the intercepts estimated with the MAR models are slightly larger than those obtained with the MNAR models for all $\tau$, except $\tau=75\%$. This is in line with the expected consequences of drop-out in the low quantiles of the longitudinal outcome distribution. Secondly, under the MAR models a significant {\it
positive} interaction between MDCM and time is estimated. This may be due to the fact that subjects with higher LVEF tend to drop-out later and to be in the MDCM class more often, resulting in a positive bias when ignoring the informative drop-out. We conclude by noting that given the results in Table \ref{mdcm_res} we can conclude there is some sensitivity to drop-out for the data at hand within the proposed class of models. As clearly noted in \cite{mole:et:al:08} one can never test the MAR assumption.
\section{Conclusions} \label{concl}
Informative drop-out may bias parameter estimates both in mean and quantile regression if ignored. As our data example suggests, the problem may be stronger for quantiles corresponding to a higher rate of events. In our example we have checked that sensitivity to drop-out is milder in high quantiles than in low quantiles, for instance. Moreover, the brief simulation study reported confirms that, as long as the informative drop-out process is ignored, bias of the parameter estimates may be substantial and, more importantly, may not decrease with the sample size.
The proposed approach allows to simultaneously model the quantile of a longitudinal outcome and the hazard of drop-out, allowing them to share part of the observed and unobserved heterogeneity. Our model can be applied with right-censored event times occuring between two scheduled visits, when drop-out times coincide with observation times for the longitudinal process, and also when the observation times are not scheduled in advance. We have generalized shared-parameters and joint-models in different directions: first of all, we have proposed an alternative parametric assumption for the longitudinal error, the ALD, which allows to fit quantile regression models. Secondly, we have proposed two alternative random effects distributions. A general efficient MCEM strategy has been used to fit our model under any of those assumptions.
In our example we have specified different values for the quantile of interest $\tau$ for illustration. It can be argued that under conditional independence assumptions the total likelihood is the sum of the likelihood based on each quantile, hence this approach is equivalent to simultaneously fitting the model for different values of $\tau$, and $\tau$-specific parameters. When more than one quantile is of interest in applications, one could also allow dependence (e.g., over the random effects at each quantile) or $\tau$-homogeneity (e.g., for the variance of the random effects). Model estimation under these assumptions is at the moment grounds for further work.
\label{lastpage}
\end{document} | arXiv |
\begin{document}
\title[Computation of harmonic weak Maass forms]{Computation of harmonic weak Maass forms}
\date{\today} \author{Jan H. Bruinier and Fredrik Str\"omberg}
\address{Fachbereich Mathematik, Technische Universit\"at Darmstadt, Schlossgartenstrasse 7, D--64289 Darmstadt, Germany} \email{[email protected]}
\address{Fachbereich Mathematik, Technische Universit\"at Darmstadt, Schlossgartenstrasse 7, D--64289 Darmstadt, Germany} \email{[email protected]} \thanks{The authors are partially supported by DFG grant BR-2163/2-1. Hardware obtained through the NSF grant DMS-0821725 has been used for a large part of the computations.}
\subjclass[2000]{11Y35, 11Y40, 11F30, 11G05}
\begin{abstract} Harmonic weak Maass forms of half-integral weight are the subject of many recent works. They are closely related to Ramanujan's mock theta functions, their theta lifts give rise to Arakelov Green functions, and their coefficients are often related to central values and derivatives of Hecke $L$-functions. We present an algorithm to compute harmonic weak Maass forms numerically, based on the automorphy method due to Hejhal and Stark. As explicit examples we consider harmonic weak Maass forms of weight $1/2$ associated to the elliptic curves 11a1, 37a1, 37b1.
We made extensive numerical computations and the data we obtained is presented in the final section of the paper. We expect that experiments based on our data will lead to a better understanding of the arithmetic properties of the Fourier coefficients. \end{abstract}
\maketitle
\section{Introduction} \label{sect:intro}
Half-integral weight modular forms play important roles in arithmetic geometry and number theory. Their coefficients serve as generating functions for various interesting number theoretic functions, such as representation numbers of quadratic forms in an odd number of variables or class numbers of imaginary quadratic fields. Moreover, employing the Shimura correspondence \cite{Sh}, Waldspurger \cite{Wa}, and Kohnen and Zagier \cite{KZ,K} showed that the coefficients of half-integral weight cusp forms essentially are square-roots of central values of quadratic twists of modular $L$-functions.
In analogy with these works, Katok and Sarnak \cite{KS} used a Shimura correspondence to relate coefficients of weight 1/2 Maass forms to sums of values and sums of line integrals of Maass cusp forms.
In more recent work Zagier discovered that the generating function for the traces of singular moduli (the CM values of the classical $j$-function) is a weakly holomorphic modular form of weight $3/2$ \cite{Za1}. This result, which was generalized in various directions (see e.g. \cite{BO3}, \cite{BF2}, \cite{DJ}, \cite{Kim1}), demonstrates that also the coefficients of automorphic forms with singularities at the cusps carry interesting arithmetic information.
In a similar spirit, Ono and the first author proved that the coefficients of harmonic weak Maass forms of weight $1/2$ are related to both the values and central derivatives of quadratic twists of weight 2 modular $L$-functions \cite{BruO}. Harmonic weak Maass forms are also closely related to mock modular forms and to Ramanujan's mock theta functions, which have been the subject of various recent works (see e.g. \cite{BO1, BO2, O2, Za2, Z1, Z2}). In view of these connections, it is desirable to develop tools for the computation of such automorphic forms. In the present paper we propose an approach to this problem which yields an efficient algorithm. Moreover, we compute some harmonic weak Maass forms which are related to rational elliptic curves as in \cite{BruO}.
The non-holomorphic nature of harmonic weak Maass forms prevents the use of the well developed algorithms existing for (weakly) holomorphic modular forms, such as e.g. modular symbols. The use of Poincar{\'e} series does not work well either in small weights due to the poor convergence of the infinite series which appear in the explicit formulas for the coefficients. Instead we adapt the `automorphy method', originally developed by Hejhal for the computation of Maass cusp forms on Hecke triangle groups (see e.g. \cite{He}), to the setting of harmonic weak Maass forms.
We now describe the content of this paper in more detail. Let $k\in \frac{1}{2}\mathbb{Z}$, and let $N$ be a positive integer (with $4\mid N$ if $k\in \frac{1}{2}\mathbb{Z}\setminus \mathbb{Z}$). A {\em harmonic weak Maass form} of weight $k$ on $\Gamma_0(N)$ is a smooth function on $\mathbb{H}$, the upper half of the complex plane, which satisfies: \begin{enumerate} \item[(i)]
$f\mid_k\gamma = f$ for all $\gamma\in \Gamma_0(N)$; \item[(ii)] $\Delta_k f =0 $, where $\Delta_k$ is the weight $k$ hyperbolic Laplacian on $\mathbb{H}$ (see (\ref{deflap})); \item[(iii)] There is a polynomial $P_f=\sum_{n\leq 0} c^+(n)q^n \in \mathbb{C}[q^{-1}]$ such that $f(\tau)-P_f(\tau) = O(e^{-\varepsilon v})$ as $v\to\infty$ for some $\varepsilon>0$. Analogous conditions are required at all cusps. \end{enumerate} Throughout, for $\tau\in \mathbb{H}$, we let $\tau=u+iv$, where $u, v\in \mathbb{R}$, and we let $q:=e^{2 \pi i \tau}$. The polynomial $P_f$ is called the {\em principal part} of $f$ at $\infty$.
Such a harmonic weak Maass form $f$ has a Fourier expansion at infinity of the form \begin{equation}\label{fourier}
f(\tau)=\sum_{n\gg -\infty} c^+(n) q^n + \sum_{n<0} c^-(n)\Gamma\left(1-k, 4\pi |n| v\right) q^n, \end{equation} where $\Gamma(a,x)$ denotes the incomplete Gamma function. The series $\sum_{n\gg -\infty} c^+(n) q^n$ is called the {\it holomorphic part} of $f$, and its complement is called the {\it non-holomorphic part}. Naturally, $f$ has similar expansions at the other cusps. There is an antilinear differential operator, taking $f$ to the cusp form $\xi_{k}(f):=2iv^k\overline{\frac{\partial f}{\partial \bar \tau}}$ of weight $2-k$, see \eqref{defxi}. The kernel of $\xi_k$ consists of the space of {\it weakly holomorphic} modular forms, those meromorphic modular forms whose poles (if any) are supported at cusps.
Every weight $2-k$ cusp form is the image under $\xi_{k}$ of a weight $k$ harmonic weak Maass form. Ramanujan's mock theta functions correspond to those forms whose images under $\xi_{1/2}$ are weight 3/2 unary theta functions. Here we mainly consider those weight 1/2 harmonic weak Maass forms whose images under $\xi_{1/2}$ are orthogonal to the unary theta series. According to \cite{BruO}, their coefficients are related to both the values and central derivatives of quadratic twists of weight 2 modular $L$-functions.
We now briefly describe this result in the special case that the level is a prime $p$. Let $G\in S_2(\Gamma_0(p))$ be a normalized Hecke eigenform whose Hecke $L$-function $L(G,s)$ satisfies an odd functional equation. That is, the completed $L$-function $\Lambda(G,s)=p^{s/2}(2\pi)^{-s}\Gamma(s)L(G,s)$ satisfies $\Lambda(G,2-s)=\varepsilon_G \Lambda(G,s)$ with root number $\varepsilon_G=-1$. Therefore, the central critical value $L(G,1)$ vanishes. By Kohnen's theory of plus-spaces \cite{K}, there is a half-integral weight newform $ g\in S_{3/2}^{+}(\Gamma_0(4p))$, unique up to a multiplicative constant, which lifts to $G$ under the Shimura correspondence. We choose $g$ so that its coefficients are in $F_G$, the totally real number field generated by the Hecke eigenvalues of $G$.
There exists a weight 1/2 harmonic weak Maass form $f$ on $\Gamma_0(4p)$ in the plus space
whose principal part $P_{f}$ has coefficients in $F_G$, and such that \[
\xi_{1/2}(f)={\|g\|^{-2}}g, \]
where $\|g\|$ denotes the usual Petersson norm.
For a fundamental discriminant $\Delta$ let $\chi_{\Delta}$ be the Kronecker character for $\mathbb{Q}(\sqrt{\Delta})$, and let $L(G,\chi_{\Delta},s)$ be the quadratic twist of $L(G,s)$ by $\chi_{\Delta}$. One can show that the root number of $L(G,\chi_{\Delta},s)$ is equal to $\textrm{sign}(\Delta) \cdot \chi_{\Delta}(p)\, \varepsilon_G$.
\begin{theorem}[See \cite{BruO}] \label{Lvalues} Assume that $G$, $g$, and $f$ are as above, and let $c^{\pm}(n)$ denote the Fourier coefficients as in \eqref{fourier}. \begin{enumerate}
\item If $\Delta<0$ is a fundamental discriminant for which $\leg{\Delta}{p}=1$, then $$
L(G,\chi_{\Delta},1)=8\pi^2\|G\|^2 \|g\|^2 \sqrt{\frac{|\Delta|}{N}}\cdot c^{-}(\Delta)^2. $$ \item If $\Delta>0$ is a fundamental discriminant for which $\leg{\Delta}{p}=1$, then $L'(G,\chi_{\Delta},1)=0$ if and only if $c^{+}(\Delta)$ is algebraic. \end{enumerate} \end{theorem}
Note that the harmonic weak Maass form $f$ is uniquely determined up to the addition of a weight 1/2 weakly holomorphic modular form with coefficients in $F_G$. Furthermore, the absolute values of the nonvanishing coefficients $c^{+}(\Delta)$ are typically asymptotic to subexponential functions in $n$. For these reasons, the connection between $L'(G,\chi_{\Delta},1)$ and the coefficients $c^{+}(\Delta)$ in Theorem~\ref{Lvalues}(2) cannot be modified in a simple way to obtain a formula as in the first part of the Theorem. In fact, the proof of Theorem~\ref{Lvalues}(2) is rather indirect. It relies on the Gross-Zagier formula and on transcendence results of Waldschmidt and Scholl on periods of differentials on algebraic curves.
The above result is one of the main motivations for the present paper. Our goal is to carry out numerical computations for the involved harmonic weak Maass forms. In that way we hope to find more direct connections of the coefficients $c^+(\Delta)$ to periods or $L$-functions. When $L'(G,\chi_{\Delta},1)$ vanishes, meaning that $c^+(\Delta)$ is algebraic (actually contained in $F_G$), it would be interesting to see if $c^+(\Delta)$ carries any arithmetic information related to $G$. In a forthcoming paper \cite{Brperiods}, the coefficients $c^+(n)$ will be linked to periods of certain algebraic differentials of the third kind on modular curves. It leads to a conjecture on differentials of the third kind on elliptic curves, which is based on the numerical data presented in Section \ref{sect:results} of the present paper.
Our computations make use of an adaption of the so-called automorphy method. The key point of this method is to view an automorphic form on a non-co-compact (but co-finite) Fuchsian group $\Gamma$ as a function on the upper half-plane with certain transformation properties under the group $\Gamma$ as well as convergent Fourier series expansions at all cusps. This classical point of view, in terms of functions on the upper half-plane, stands in contrast to the more algebraic point of view, in terms of Hecke modules, usually taken when computing holomorphic modular forms.
By {\em computing} an automorphic form $\phi$ in this setting we mean that to any given (small) $\epsilon > 0$ we compute a sufficient number of Fourier coefficients, each to high enough precision, so that we are able to evaluate the function $\phi$ at any point in the upper half-plane with an error at most $\epsilon$.
To calculate these Fourier coefficients we truncate the Fourier series representing $\phi$ and view the resulting trigonometric sum as a finite Fourier series. Using the Fourier inversion theorem together with the automorphic properties of $\phi$ (which will additionally intertwine the Fourier series at various cusps) we are able to obtain a set of linear equations satisfied approximately by the coefficients. Cf.\,e.g.\,\cite{He,St,Av2}. The (surprising) effectiveness of this algorithm is closely related to the equidistribution properties of closed horocycles (cf.\, e.g.\,\cite{He1,S}). We describe the main algorithm in detail in Section \ref{sect:comp}. The implementation of the software package is briefly described in Section \ref{ssect:implementation}.
In Section \ref{sect:results} we describe our computational result in three cases of particular interest. We consider the elliptic curves 11a1, 37a1, and 37b1 and their corresponding weight $2$ newforms. For instance, the elliptic curve 37a1, is the curve of smallest conductor with rank 1. It corresponds to the unique weight two normalized newform $G$ on $\Gamma_0(37)$ whose $L$-function has an odd functional equation. We verified the statement of Theorem \ref{Lvalues} for all fundamental discriminants $\Delta$ which are squares modulo $148$ in the range $0<\Delta < 15000$. For eight of these fundamental discriminants the quantity $L'(G,\chi_{\Delta},1)$ vanishes. In all these cases we found a stronger statement then that of the Theorem \ref{Lvalues} to be true, namely, that the associated coefficient $c^+(\Delta)$ was an integer. For the corresponding data see Tables \ref{tab:37a1} and \ref{tab:37a1-zeros}. We conclude Section \ref{sect:results} by describing some analogous experiments for newforms $G$ of weight $4$, where $g$ is of weight $5/2$ and $f$ of weight $-1/2$.
The present paper is organized as follows. In Section \ref{sect:2} we recall some facts on (half integral weight) harmonic weak Maass forms. When working with arbitrary (not necessarily prime) level, it is convenient to use vector valued modular forms. In Section \ref{sect:2.3} we therefore recall from \cite{BruO} the vector valued version of Theorem \ref{Lvalues}. In Section \ref{sect:comp} we describe the automorphy method in the context of harmonic weak Maass forms. In Section \ref{sect:results} we collect our computational results. In particular, we present results for the elliptic curves 11a1, 37a1, and 37b1; cf., e.g.~Tables \ref{tab:11a1}, \ref{tab:37a1} and \ref{tab:37b1}. More extensive tables can be obtained from the authors on request.
\section{Preliminaries} \label{sect:2}
In order to be able to work with newforms of arbitrary level, it is convenient to work with vector valued modular forms of half integral weight for the metaplectic extension of $\operatorname{SL}_2(\mathbb{Z})$. We describe the necessary background in this section.
\subsection{A Weil representation}
Let $\mathbb{H}=\{\tau\in \mathbb{C};\;\Im(\tau)>0\}$ be the complex upper half plane. We write $\operatorname{Mp}_2(\mathbb{R})$ for the metaplectic two-fold cover of $\operatorname{SL}_2(\mathbb{R})$, realized as the group of pairs $(M,\phi(\tau))$, where $M=\kabcd\in\operatorname{SL}_2(\mathbb{R})$ and $\phi:\mathbb{H}\to \mathbb{C}$ is a holomorphic function with $\phi(\tau)^2=c\tau+d$. The multiplication is defined by \[ (M,\phi(\tau)) (M',\phi'(\tau))=(M M',\phi(M'\tau)\phi'(\tau)). \] We denote the inverse image of $\Gamma:=\operatorname{SL}_2(\mathbb{Z})$ under the covering map by $\tilde\Gamma:=\operatorname{Mp}_2(\mathbb{Z})$. It is well known that $\tilde\Gamma$ is generated by $T:= \left( \kzxz{1}{1}{0}{1}, 1\right)$, and $S:= \left( \kzxz{0}{-1}{1}{0}, \sqrt{\tau}\right)$.
Let $N$ be a positive integer. There is a certain representation $\rho$ of $\tilde \Gamma$ on $\mathbb{C}[\mathbb{Z}/2N\mathbb{Z}]$, the group ring of the finite cyclic group of order $2N$. For a coset $h\in \mathbb{Z}/2N\mathbb{Z}$ we denote by $\mathfrak e_h$ the corresponding standard basis vector of $\mathbb{C}[\mathbb{Z}/2N\mathbb{Z}]$. We write $\langle\cdot,\cdot \rangle$ for the standard scalar product (antilinear in the second entry) such that $\langle \mathfrak e_h,\mathfrak e_{h'}\rangle =\delta_{h,h'}$. In terms of the generators $T$ and $S$ of $\tilde \Gamma$, the representation $\rho$ is given by \begin{align} \label{eq:weilt} \rho(T)(\mathfrak e_h)&=e\left(\frac{h^2}{4N}\right)\mathfrak e_h,\\ \label{eq:weils} \rho(S)(\mathfrak e_h)&= \frac{1}{\sqrt{2iN}} \sum_{h' \; (2N)} e\left(-\frac{hh'}{2N}\right)
\mathfrak e_{h'}. \end{align} Here the sum runs through the elements of $\mathbb{Z}/2N\mathbb{Z}$ and we have put $e(a)=e^{2\pi i a}$. Note that $\rho$ is the Weil representation associated to the one-dimensional positive definite lattice $K=(\mathbb{Z},Nx^2)$ in the sense of \cite{Bor1}, \cite{Br}, \cite{BruO}. It is unitary with respect to the standard scalar product.
If $k\in \frac{1}{2}\mathbb{Z}$, we write $M^!_{k,\rho}$ for the space of $\mathbb{C}[\mathbb{Z}/2N\mathbb{Z}]$-valued weakly holomorphic modular forms of weight $k$ for $\tilde \Gamma$ with representation $\rho$. The subspaces of holomorphic modular forms and cusp forms are denoted by $M_{k,\rho}$ and $S_{k,\rho}$, respectively.
\subsection{Harmonic weak Maass forms}
\label{sect:2.2} In this subsection we assume that
$k\leq 1$. A twice continuously differentiable function $f:\mathbb{H}\to \mathbb{C}[\mathbb{Z}/2N\mathbb{Z}]$ is called a {\em harmonic weak Maass form} (of weight $k$ with respect to $\tilde \Gamma$ and $\rho$) if it satisfies: \begin{enumerate} \item[(i)] $f(M\tau) = \phi(\tau)^{2k}\rho(M,\phi) f(\tau)$
for all $(M,\phi)\in \tilde\Gamma$; \item[(ii)] $\Delta_k f=0$, \item[(iii)] there is a $\mathbb{C}[\mathbb{Z}/2N\mathbb{Z}]$-valued Fourier polynomial \[ P_f(\tau)=\sum_{h\;(2N)}\sum_{n\in \mathbb{Z}_{\leq 0}} c^+(n,h) q^{\frac{n}{4N}} \mathfrak e_h \] such that $f(\tau)-P_f(\tau)=O(e^{-\varepsilon v})$ as $v\to \infty$ for some $\varepsilon>0$. \end{enumerate}
Here we have that \begin{equation} \label{deflap} \Delta_k := -v^2\left( \frac{\partial^2}{\partial u^2}+ \frac{\partial^2}{\partial v^2}\right) + ikv\left( \frac{\partial}{\partial u}+i \frac{\partial}{\partial v}\right) \end{equation} is the usual weight $k$ hyperbolic Laplace operator (see \cite{BF}). The Fourier polynomial $P_f$ is called the {\em principal part} of $f$. We denote the vector space of these harmonic weak Maass forms by $H_{k,\rho}$ (it was called $H^+_{k,\rho}$ in \cite{BF}). Any weakly holomorphic modular form is a harmonic weak Maass form. The Fourier expansion of any $f\in H_{k,\rho}$ gives a unique decomposition $f=f^++f^-$, where \begin{subequations} \label{deff} \begin{align} \label{deff+} f^+(\tau)&= \sum_{h\;(2N)}\sum_{\substack{n\in \mathbb{Z}\\ n\gg-\infty}} c^+(n,h) q^{\frac {n}{4N}}\mathfrak e_h,\\ \label{deff-}
f^-(\tau)&= \sum_{h\; (2N)}\sum_{\substack{n\in \mathbb{Z}\\ n< 0}} c^-(n,h) \Gamma\left(1-k,4\pi\left|\frac{n}{4N}\right|v\right) q^{\frac{n}{4N}} \mathfrak e_h. \end{align} \end{subequations} We refer to $f^+$ as the {\em holomorphic part} and to $f^-$ as the {\em non-holomorphic part} of $f$. Note that $c^{\pm}(n,h)=0$ unless $n\equiv h^2\, (4N)$.
Recall that there is an antilinear differential operator $\xi= \xi_k:H_{k,\rho}\to S_{2-k,\bar\rho}$, defined by \begin{equation} \label{defxi} f(\tau)\mapsto \xi(f)(\tau):=2iv^k\overline{\frac{\partial f}{\partial \bar \tau}}. \end{equation} Here $\bar\rho$ denotes the complex conjugate of the representation $\rho$, which can be identified with the dual representation. The map $\xi$ is surjective and its kernel is the space $M^!_{k,\rho}$.
There is a bilinear pairing between $M_{2-k,\bar\rho}$ and $H_{k,\rho}$ defined by the Petersson scalar product \begin{equation}\label{defpair} \{g,f\}=\big( g,\, \xi(f)\big) :=\int_{\Gamma\backslash \mathbb{H}}\langle g,\, \xi(f)\rangle v^{2-k}\frac{du\,dv}{v^2}, \end{equation} for $g\in M_{2-k,\bar\rho}$ and $f\in H_{k,\rho}$. If $g$ has the Fourier expansion $g=\sum_{h,n} b(n,h) q^{n/4N}\mathfrak e_h$, and if we denote the Fourier expansion of $f$ as in \eqref{deff}, then by
\cite[Proposition 3.5]{BF} we have \begin{equation}\label{pairalt} \{g,f\}= \sum_{h\;(2N)} \sum_{n\leq 0} c^+(n,h) b(-n,h). \end{equation} Hence $\{g,f\}$ only depends on the principal part of $f$.
\subsection{The Shimura lift} \label{sect:2.3}
Let $k\in \frac{1}{2}\mathbb{Z}\setminus \mathbb{Z}$. According to \cite[Chapter 5]{EZ}, the space $M_{k,\bar\rho}$ is isomorphic to $J_{k+1/2,N}$, the space of holomorphic Jacobi forms of weight $k+1/2$ and index $N$. According to \cite{Sk1} and \cite{SZ}, $M_{k,\rho}$ is isomorphic to $J_{k+1/2,N}^{skew}$, the space of skew holomorphic Jacobi forms of weight $k+1/2$ and index $N$. There is an extensive Hecke theory for Jacobi forms (see \cite{EZ}, \cite{Sk1}, \cite{SZ}), which gives rise to a Hecke theory on $M_{k,\rho}$ and $M_{k,\bar\rho}$, and which is compatible with the Hecke theory on vector valued modular forms considered in \cite{BrSt}. In particular, there is an Atkin-Lehner theory for these spaces.
The subspace $S_{k,\rho}^{new}$ of newforms of $S_{k,\rho}$ is isomorphic as a module over the Hecke algebra to the space of newforms $S^{new,+}_{2k-1}(N)$ of weight $2k-1$ for $\Gamma_0(N)$ on which the Fricke involution acts by multiplication with $(-1)^{k-1/2}$. The isomorphism is given by the Shimura correspondence. Similarly, the subspace $S_{k,\bar\rho}^{new}$ of newforms of $S_{k,\bar\rho}$ is isomorphic as a module over the Hecke algebra to the space of newforms $S^{new,-}_{2k-1}(N)$ of weight $2k-1$ for $\Gamma_0(N)$ on which the Fricke involution acts by multiplication with $(-1)^{k+1/2}$ (see \cite{SZ}, \cite{GKZ}, \cite{Sk1}). Observe that the Hecke $L$-series of any $G\in S^{new,\pm}_{2k-1}(N)$ satisfies a functional equation under $s\mapsto 2k-1-s$ with root number $\varepsilon_G=\pm 1$.
\begin{comment} If $G\in S^{new,\pm}_{2k-1}(N)$ is a normalized newform (in particular a common eigenform of all Hecke operators), we denote by $F_G$ the number field generated by the Hecke eigenvalues of $G$. It is well known that we may normalize the preimage of $G$ under the Shimura correspondence such that all its Fourier coefficients are contained in $F_G$. \end{comment}
We now state the vector valued version of Theorem \ref{Lvalues}. Let $G\in S_{2}^{new}(N)$ be a normalized newform (in particular a common eigenform of all Hecke operators) of weight $2$ and write $F_G$ for the number field generated by the eigenvalues of $G$. If $\varepsilon_G=-1$ we put $\rho'=\rho$, and
if $\varepsilon_G=+1$ we put $\rho'=\bar \rho$.
There is a newform $g\in S_{3/2,\bar\rho'}^{new}$ mapping to $G$ under the Shimura correspondence. It is well known that we may normalize $g$ such that all its coefficients are contained in $F_G$. According to \cite[Lemma 7.3]{BruO}, there is a harmonic weak Maass form $f\in H_{1/2, \rho'}$ whose principal part has coefficients in $F_G$ with the property that \[
\xi_{1/2}(f)=\|g\|^{-2} g. \] This form is unique up to addition of a weakly holomorphic form in $M^!_{1/2, \rho'}$ whose principal part has coefficients in $F_G$.
In practice, the principal part of such an $f$ can be computed as follows: We may complete the weight $3/2$ form $g$ to an orthogonal basis $g, g_2,\dots , g_d$ of $S_{3/2,\bar\rho'}$ consisting of cusp forms with Fourier coefficients in $F_G$. Let $f\in H_{1/2, \rho'}$ such that \begin{align} \label{eq:conds} \{f,g\}=1, \quad \text{and $\{f,g_i\}=0$ for $i=2,\dots d$}. \end{align} Then $f$ has the required properties. In view of \eqref{pairalt} the conditions of \eqref{eq:conds} translate into an inhomogeneous system of linear equations for the principal part of $f$.
\begin{theorem} \label{Lval2} Let $G\in S_{2}^{new}(N)$ be a normalized newform.
Let $g\in S_{3/2,\bar\rho'}^{new}$, and $f\in H_{1/2, \rho'}$ be as above. Denote the Fourier coefficients of $f$ by $c^\pm(n,h)$ for $n\in \mathbb{Z}$ and $h\in \mathbb{Z}/2N\mathbb{Z}$. Then the following are true: \begin{enumerate} \item If $\Delta\neq 1$ is a fundamental discriminant and $r\in \mathbb{Z}$ such that $\Delta\equiv r^2\pmod{4N}$ and $\varepsilon_G \Delta >0$, then $$
L(G,\chi_{\Delta},1)=8\pi^2\|G\|^2 \|g\|^2 \sqrt{\frac{|\Delta|}{N}}\cdot c^{-}(\Delta)^2. $$ \item If $\Delta\neq 1$ is a fundamental discriminant and $r\in \mathbb{Z}$ such that $\Delta\equiv r^2\pmod{4N}$ and $\varepsilon_G \Delta <0$, then $$ L'(G,\chi_{\Delta},1)=0 \quad \Longleftrightarrow\quad c^{+}(-\varepsilon_G\Delta,r)\in \bar \mathbb{Q}\quad \Longleftrightarrow\quad c^{+}(-\varepsilon_G\Delta,r)\in F_G.$$ \end{enumerate} \end{theorem}
When $S_{1/2,\rho'}=\{0\}$ the above result also holds for $\Delta=1$, see also \cite[Remark 18]{BruO}. This is for instance the case when $N$ is a prime. If $N$ is a prime and $\varepsilon_G=-1$, then the space $H_{1/2, \rho'}$ can be identified with a space of scalar valued modular forms satisfying a Kohnen plus space condition. In that way one obtains Theorem \ref{Lvalues} stated in the introduction.
\section{Computational aspects} \label{sect:comp}
\subsection{The automorphy method for vector valued weak Maass forms}
To compute the Fourier coefficients of the harmonic weak Maass forms we use the so-called automorphy method, sometimes called ``Hejhal's method{}''. This is a general method which has been used to successfully compute various kinds of automorphic functions and forms on $\mbox{GL}_{2}\left(\mathbb{R}\right)$. It was originally developed by Hejhal in order to compute Maass cusp forms for the modular group and other Hecke triangle groups (cf.~e.g.~\cite{He}). The method was later generalized by the second author in \cite{St} to computations of Maass waveforms with non-trivial multiplier systems and arbitrary real weights, as well as to general subgroups of the modular group (see also \cite{St2}). Another generalization to automorphic forms with singularities (Eisenstein series, Poincar{\'e} series and Green's functions) was made by Avelin \cite{Av1,Av2}.
We will detail the adaptation of the algorithm to the case of vector-valued harmonic weak Maass forms for the Weil representation.
For simplicity consider the representation $\rho$ (the case of $\overline{\rho}$ is analogous) and $k\in\mathbb{Z}+\frac{1}{2}$. Furthermore, in order to avoid questions of uniqueness we assume that either $k<0$ or that $k=\frac{1}{2}$ and that $N$ is prime. In these cases, a harmonic weak Maass form is uniquely determined by its principal part. For computational purposes it is not feasible to use the definition of $\rho$ in terms of the action on the generators of the metaplectic group. We instead use formulas from {[}St1{]} to evaluate $\rho$ on the fixed (canonical) representative of $M=\left(\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\right)\in\operatorname{SL}_{2}(\mathbb{Z})$, i.e. $\rho\left(M\right):=\rho\left(M,j_{M}\left(\tau\right)\right)$ where $j_{M}\left(\tau\right)=\sqrt{c\tau+d}$ is defined by the principal branch of the argument.
\subsubsection{The algorithm -- phase 1}
Let $f\in H_{k,\rho}$ with a given (fixed) principal part $P_{f}\left(\tau\right)=\sum_{h}P_{f,h}\left(\tau\right)\mathfrak{e}_{h}$ where $P_{f,h}\left(\tau\right)=\sum_{n=-K}^{0}a\left(n,h\right)q^{\frac{n}{4N}}$ (for some finite $K\ge0$) and write $f=f^{+}+f^{-}$ (as in 2.3a and 2.3b) with $f^{+}=\sum_{h\left(2N\right)}f_{h}^{+}\mathfrak{e}_{h}$ and $f^{-}=\sum_{h\left(2N\right)}f_{h}^{-}\mathfrak{e}_{h}$ where \begin{align*} f_{h}^{+}\left(\tau\right) & = \sum_{n=-K}^{0}a\left(n,h\right)q^{\frac{n}{4N}}+\sum_{n>0}c^{+}\left(n,h\right)q^{\frac{n}{4N}}\quad\mbox{and}\\
f_{h}^{-}\left(\tau\right) & = \sum_{n<0}c^{-}\left(n,h\right)\Gamma\left(1-k,4\pi\left|\frac{n}{4N}\right|v\right)q^{\frac{n}{4N}} \end{align*} for $\tau=u+iv\in\mathbb{H}$. Our goal is to obtain numerical approximations to the coefficients $c^{\pm}(n,h)$.
To formulate our algorithm we prefer to separate the $u$- and the $v$-dependence in $f$ and therefore introduce the function $W$ defined by
$W(v)=e^{-2\pi v}$ if $v>0$ and $W(v)=e^{-2\pi v}\Gamma(1-k,4\pi|v|)$ if $v<0$. We also set $c\left(n,h\right)=c^{+}\left(n,h\right)$ for ~$n>0$ and $c^{-}\left(n,h\right)$ for $n<0$ and write $e_{4N}\left(u\right)=e^{\frac{2\pi iu}{4N}}$. With this notation
\[ f_{h}\left(\tau\right)=\sum_{n=-K}^{0}a\left(n,h\right) q^{\frac{n}{4N}}+ \sum_{n\ne0}c\left(n,h\right) W\left(\frac{nv}{4N}\right) e_{4N}(nu). \] By standard inequalities for the incomplete gamma function one can show that \[
|W(v)|<c_{k}\, e^{-2\pi|v|} \begin{cases} 1, & v>0,\\
\left(4\pi\left|v\right|\right)^{-k}, & v<0, \end{cases}\] where $c_{k}$ is an explicit constant only depending on $k$. To be able to determine a truncation point of the Fourier series above we also need bounds of the coefficients $c\left(n,h\right)$. Using {[}BruFu, Lemma 3.4{]} it follows that there exists an explicit constant $C>0$ such that \begin{align*} c\left(n,h\right) & = O\left(\exp\left(4\pi C\sqrt{n}\right)\right),\quad n\rightarrow+\infty,\\
c\left(n,h\right) & = O(|n|^{\frac{k}{2}}),\quad n\rightarrow-\infty. \end{align*} For $k<0$ we are able to make the implied constants explicit using non-holomorphic Poincar{\'e} series as in e.g.~\cite{Br} or \cite{He2}. For $k=\frac{1}{2}$ we rely on numerical a posteriori tests to assure ourselves that the truncation point was choosen correctly. See e.g.~Section \ref{heuristics}.
Let $\epsilon>0$ and fix $Y<Y_{0}=\frac{\sqrt{3}}{2}$. By the estimates above we can find an $M_{0}=M\left(Y,\epsilon \right)$ such that the function $\hat{f}=\sum_{h\,(2N)} \hat{f}_{h} \mathfrak{e}_h$ given by the truncated Fourier series \[
\hat{f}_{h}\left(\tau\right)=P_{f,h}\left(\tau\right)+\sum_{0<\left|n\right|\le M_{0}}c\left(n,h\right)W\left(\frac{nv}{4N}\right)e_{4N}\left(nu\right) \] satisfies \[
\left\Vert \hat{f}(\tau) - f(\tau) \right\Vert^{2} < \epsilon \]
for any $\tau \in \mathcal{H}_{Y}=\left\{\tau \in \mathcal{H}\,|\,\Im \tau \ge Y \right\}$. Here $\left\Vert z \right\Vert^{2}=\sum_{h=1}^{2N} \left| z_h \right|^2$ for $z \in \mathbb{C}^{2N}$. Let $A=\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right)\in \operatorname{SL}_{2}(\mathbb{Z})$ and set $z=x+iy = A \tau$. Then $
y = \Im A\tau = \frac{v}{\left|c\tau + d\right|^2} \le \frac{v}{c^2v^2}\le \frac{1}{v} $
and hence $\left|j_A(\tau)\right|^{4} = \left|c\tau + d\right|^{2} = \frac{v}{y} \le \frac{1}{y^2}$. Using the fact that $\rho$ is unitary it is now easy to see that if $\tau, A\tau \in \mathcal{H}_{Y}$ then \begin{align} \label{approx-autom} \left\Vert \hat{f}\left(A\tau\right) - j_{A}\left(\tau\right)^{2k}\rho\left(A\right)\hat{f}\left(\tau\right) \right\Vert^{2} < \epsilon \left( 1 + Y^{-2k} \right) < 2 \epsilon \cdot Y^{-2k}. \end{align} Consider now a horocycle at height $Y$ and a set of $2Q$ (with $Q>M_{0}$) equally spaced points \[ z_{m}=x_{m}+iY,\quad x_{m}=\frac{1-2m}{4Q},\quad1-Q\le m\le Q. \]
If we view the series $\hat{f}_{h}$ as a finite Fourier series we can invert it over this horocycle and it is easy to see that if $n$ is an integer with $0<\left|n\right|\le M_{0}$ and $n\equiv h^{2} \,(4N)$ then \begin{equation} \frac{1}{2Q}\sum_{m=1-Q}^{Q}\hat{f}_{h}\left(z_{m}\right)e_{4N}\left(-nx_{m}\right)=W\left(\frac{n}{4N} Y\right)c\left(n,h\right)+a\left(n,h\right)\,e^{-\frac{2\pi n}{4N}Y}.\label{eq:ffseries} \end{equation} One can also interpret the left-hand side as a Riemann-sum approximation to the integral \[\int_{-\frac{1}{2}}^{\frac{1}{2}}f_{h}\left(z\right)e_{4N}\left(-nx\right)dx. \] Let $z_{m}^{*}=x_{m}^{*}+iy_{m}^{*}=T_{m}^{-1}z_{m}$ ($T_{m}\in\operatorname{PSL}_{2}(\mathbb{Z})$) denote the pull-back of $z_{m}$ into the standard (closed) fundamental domain of $\operatorname{PSL}_{2}(\mathbb{Z})$,
$\mathcal{F}=\left\{ z=x+iy\,|\,\left|x\right|\le\frac{1}{2},\,\left|z\right|\ge1\right\} $.
Using (\ref{approx-autom}) we obtain
\[ \hat{f}_{h}\left(z_{m}\right) = j_{T_{m}}(z_{m}^{*}) \sum_{h'\,(2N)}\rho_{hh'}\left(T_{m}\right)\hat{f}_{h'}\left(z_{m}^{*}\right) + \llbracket 2\epsilon Y^{-2k} \rrbracket, \] where $\rho_{hh'}\left(T_{m}\right)$ is the $\left(h,h'\right)$-element of the matrix $\rho\left(T_{m}\right)$, and we use $\llbracket 2\epsilon Y^{-2k} \rrbracket$ to denote a quantity bounded in absolute value by $2\epsilon Y^{-2k}$.
Inserting this into (\ref{eq:ffseries}) we see that the left-hand side can be written as \begin{align}
\frac{1}{2Q} \sum_{m=1-Q}^{Q} j_{ T_{m}} \left(z_{m}^{*}\right) \sum_{h'\,(2N)} \rho_{hh'}\left(T_{m}\right)
& \left[ \sum_{l=-K}^{0} a (l,h') \exp\left(-\frac{2\pi l}{4N} y_{m}^{*}\right) e_{4N}\left(l x_{m}^{*}\right) \right. \nonumber \\
+ & \left.\sum_{0 < |l| \le M_{0} } c(l,h') W \left(\frac{l}{4N} y_{m}^{*}\right) e_{4N} \left(l x_{m}^{*}\right) \right] e_{4N}(-n x_{m})\nonumber \\
= & \sum_{h'\,(2N)}\sum_{0<\left|l\right|\le M_{0}} c\left(l,h'\right) \widetilde{V}_{nl}^{hh'}+\widetilde{W}_{n}^{h} + \llbracket 2\epsilon Y^{-2k} \rrbracket, \label{eq:fundamental_expression} \end{align} where \begin{align*} \widetilde{V}_{nl}^{hh'} & = \frac{1}{2Q}\sum_{m=1-Q}^{Q}j_{T_{m}}\left(z_{m}^{*}\right)\rho_{hh'}\left(T_{m}\right)W\left(\frac{l}{4N} y_{m}^{*}\right)e_{4N}(lx_{m}^{*}-nx_{m})\quad\mbox{and}\\ \widetilde{W}_{n}^{h} & = \frac{1}{2Q}\sum_{h'\,(2N)}\sum_{l=-K}^{0}a\left(l,h'\right)\sum_{m=1-Q}^{Q}j_{T_{m}}\left(z_{m}^{*}\right) \rho_{hh'}\left(T_{m}\right) \exp\left(-\frac{2\pi l}{4N} y_{m}^{*}\right) e_{4N}\left(lx_{m}^{*}-nx_{m}\right).\end{align*} We thus have an inhomogeneous system of linear equations which is (approximately)
satisfied by the coefficients $c\left(n,h\right)$. Let $\mathcal{D}=\left\{ \left(n,h\right)\,|\,0<\left|n\right|\le M_{0},\,0\le h < 2N \right\} $
(with a fixed ordering) and note that $\left|\mathcal{D}\right|=4M_{0}N$. If we set $\vec{D}=\left(d\left(n,h\right)\right)_{\left(n,h\right)\in\mathcal{D}}$, \begin{align*} V &= V\left(Y\right)=\left(V_{nl}^{hh'}\right)_{\left(h,n\right),\left(h',l\right)\in\mathcal{D}},& V_{nl}^{hh'}&=\widetilde{V}_{nl}^{hh'}-\delta_{nl}\delta_{hh'}W\left(\frac{n}{4N}Y\right) \quad\text{and} \\ \vec{W} &= \vec{W}\left(Y\right)=\left(W_{n}^{h}\right)_{\left(h,n\right)\in\mathcal{D}}, & W_{n}^{h}&=\widetilde{W}_{n}^{h}-a\left(n,h\right)e^{-\frac{2\pi n}{4N}Y}, \end{align*}
we can write this linear system as $\left|\mathcal{D}\right|$ linear equations in $\left|\mathcal{D}\right|$ variables: \begin{equation} V\vec{D}+\vec{W}=\vec{0}.\label{eq:linsyst}\end{equation} In practice it turns out that the the matrix $V$ is non-singular as soon as the subspace of $H_{k,\rho}$ consisting of functions with a given singular part is one-dimensional. In these cases we can immediately obtain the solution as
\[ \vec{D}=-V^{-1}\vec{W}, \] and since we know that the vector of the ``true'' coefficients, $\vec{C}=\left(c\left(n,h\right)\right)_{\left(n,h\right)\in\mathcal{D}}$, satisfies \[
\left\Vert V\vec{C}+\vec{W} \right\Vert_{\infty} \le 2\epsilon Y^{-2k}, \] we see that \[
\left\Vert \vec{C}-\vec{D} \right\Vert_{\infty} = \left\Vert \vec{C}+V^{-1} \vec{W} \right\Vert_{\infty} \le
\left\Vert V^{-1} \right \Vert_{\infty} \cdot \left\Vert V\vec{C}+\vec{W} \right\Vert_{\infty} \le 2\epsilon Y^{-2k} \left\Vert V^{-1} \right \Vert_{\infty}. \] To obtain a theoretical error estimate we would thus need to estimate $\left\Vert V^{-1} \right \Vert_{\infty}$ from below. Unfortunately this does not seem to be possible from the formulas above and we have to use numerical methods to estimate this norm. Hence, to obtain the Fourier coefficients up to a (proven) desired precision we might have to go back and decrease the original $\epsilon$ or increase either of $M_0$ or $Q$.
At this point one should also remark that the error bound $\left\Vert V^{-1} \right \Vert_{\infty}$ is in general much worse than the actual apparent error, as verified by studying coefficients known to be integers. The reason for this is that the sums $\widetilde{V}_{nl}^{hh'}$ exhibit massive cancellation and are therefore overpowered by the terms $W\left( \frac{n}{4N} Y\right)$ on the diagonal.
\subsubsection{The algorithm -- phase 2} Returning to (\ref{eq:fundamental_expression}) and solving for $c\left(n,h\right)$ we see that \begin{equation}
c\left(n,h\right)=W\left(\frac{n}{4N} Y\right)^{-1}\left[\sum_{h'\,(2N)}\sum_{\left|l\right|\le M_{0}}c\left(l,h'\right)\widetilde{V}_{nl}^{hh'}+W_{n}^{h} + \llbracket 2\epsilon Y^{-2k} \rrbracket \right]\label{eq:phase2} \end{equation}
for \emph{any} $n$, i.e. also when $\left|n\right|>M_{0}$, provided that $Q>M\left(Y\right)$. If we first choose $Y$ such that $W\left(\frac{n}{4N} Y\right)$ is not too small then we can in fact use this equation to compute $c\left(n,h\right)$ with an error of size $\epsilon\,W\left(\frac{n}{4N} Y\right)^{-1}$. In this manner, we may produce long stretches of coefficients (before we need to decrease $Y$ again) at arbitrary intervals $N_{A}\le n\le N_{B}$ without the need of computing intermediate coefficients above the initial set up to $n=M_{0}$.
\begin{remark} The exact same algorithm, with the non-holomorphic parts set to zero, also lets one compute holomorphic vector-valued modular forms for the Weil representation. This has been exploited by the second author, in verifying computations of holomorphic Poincar\'e series in \cite{RSS}. \end{remark}
\subsection{Heuristic error estimates} \label{heuristics}
For $k<0$ all implied constants and therefore all error estimates can be made explicit. In the remaining case which interests us, $k=\frac{1}{2}$, the known bounds for the twisted Kloosterman sums are not enough to prove the necessary explicit bounds for the Fourier coefficients of the associated Poincar\'e series. We are therefore not able to give effective theoretical error estimates in this case. However, this is not a serious problem since there are a number of tests we may perform on the resulting coefficients to assure ourselves of their accuracy. We list a few tests which we have used. \begin{itemize}
\item First of all, one can simply use two different values of $Y$ and verify that the resulting vectors $\vec{D}=\vec{D}(Y)$ are independent of $Y$. \end{itemize} This test is completely general and can be used for all instances where the algorithm can be applied. Suppose now that we have a harmonic weak Maass form $f\in H_{k,\rho}$ of half-integral weight $k$ such that
$\xi_{k}\left(f\right)=\|g\|^{-2} g$, with $g\in S_{2-k,\bar\rho}$. We then know the following. \begin{itemize}
\item The coefficients $\sqrt{\left|\Delta\right|} c^{-}(-\varepsilon_G\cdot\Delta)$ are proportional to the coefficients $b(\varepsilon_G\cdot\Delta)$ of $g$ (cf.~e.g.~\cite[p.~3]{BruO}). \end{itemize} If additionally the Shimura lift of $g$ is a newform $G\in S^{new}_{3-2k}\left(\Gamma_0(N)\right)$ then we can predict that certain coefficients $c^{+}(\Delta)$ are algebraic (cf.~e.g.~\cite[Sect.~7]{BruO}) and if we are able to identify these coefficients as algebraic numbers to a certain precision this can be used as another measure of the accuracy.
\subsection{Implementation} \label{ssect:implementation} The first implementation of the above described algorithm was made in Fortran 90, using the package ARPREC \cite{AR} for arbitrary (fixed) precision arithmetic.
The second and more recent implementation was done in Sage \cite{SA}, using the included package mpmath for arbitrary (fixed) precision arithmetic. The algorithms are currently under development but can be obtained on request from the authors. The final format we intend for these algorithms are standard classes for computing with vector and scalar-valued harmonic weak Maass forms in Sage or Purple Sage.
\section{Results} \label{sect:results}
\subsection{Harmonic Maass forms corresponding to elliptic curves} In this section we present the numerical results we have obtained for harmonic weak Maass forms corresponding to weight two holomorphic forms associated to elliptic curves. We have concentrated on three particular examples. In Cremona's notation, these correspond to the curve $11a1$ of level 11 and the two curves $37a1$ and $37b1$ of level 37.
Recall that if the holomorphic weight $2$ newform $G$ of level $N$ has Atkin-Lehner eigenvalue $\pm1$ then the $L$-function $L(G,s)$ has root number $\varepsilon_G=\mp1$. Furthermore, since the root number of the twisted $L$-function $L(G,\chi_{\Delta},s)$ is $\textrm{sign}(\Delta)\chi_{\Delta}(N)\varepsilon_G$ and we always consider fundamental discriminants for which $\chi_\Delta(N)=1$ we see that the central value $L(G,\chi_{\Delta},1)$ vanishes if $\textrm{sign}(\Delta)\varepsilon_G=-1$, i.e., if $L(G,s)$ has an even functional equation we consider $\Delta<0$ and otherwise $\Delta>0$.
For each of these examples we computed a large set of central derivatives of the twisted $L$-functions with the appropriate $\Delta$ using Sage and the standard algorithms there which were developed by Dokchitser. We then fixed a harmonic weak Maass form with non-zero principal part $P_f$ such that $\xi_{\frac{3}{2}}\left(f \right)$ maps to $G$ under the Shimura lift. In all cases we took a Poincar\'e series $P_{-\Delta}$ having principal part $q^{-\frac{\Delta}{4N}}$ and computed an initial set of Fourier coefficients for this function using the methods described in the previous section. We then used the second phase of the algorithm and computed more Fourier coefficients.
Note that for the results in this section, all initial ``phase 1'' computations were all performed using the new Sage package and all further, ``phase 2'', computations were done in Fortran 90.
We would like to give a flavour of the cpu-times involved. The initial computations, using our Sage code, took in all cases approximately 2 hours on a 2.66GHz Xeon processor. On the same processor, the cpu time for a single stretch of phase 2 calculations range between less than an hour for the smallest discriminant up to several days for the largest discriminant.
As a measure of the accuracy of our computations one can consider the difference between the coefficients in Tables \ref{tab:11a1-zeros}, \ref{tab:37a1-zeros} and \ref{tab:37b1-zeros}
and the nearest integer (the third column). To further support the correctness we also list, in Tables \ref{tab:11a1-cminus}, \ref{tab:37a1-cminus} and \ref{tab:37b1-cminus}, normalized coefficients of the non-holomorphic parts, i.e. $\sqrt{\left|\Delta\right|}c^{-}(\Delta)/\sqrt{\left|\Delta_0\right|}c^{-}(\Delta_0)$ by some fixed non-zero coefficient of index $\Delta_0$.
\subsubsection{11a1} Here the unique newform of weight two and level 11 is given by \[G=\eta(\tau)^2 \eta(11\tau)^2=q-2q^2- q^3 + 2q^4 + q^5 +\cdots \in S^{new}_{2}\left(\Gamma_0(11)\right) \] and the corresponding $L$-function $L(G,s)$ has an even functional equation. Using Sage we computed all values of $L'(G,\chi_{\Delta},1)$ for fundamental discriminants $\Delta<0$ such that
$\left(\frac{\Delta}{11}\right)=1$ and $|\Delta|\le19703$. This set consists of $2749$ fundamental discriminants and amongst these we found $14$ discriminants for which $L'(G,\chi_{\Delta},1)$ vanished up to the numerical precision (see Table \ref{tab:11a1-zeros}).
As a representative for the harmonic weak Maass form in the space $H_{1/2,\bar\rho}$ corresponding to $G$, we choose the Poincar\'e series $P_{-5}$ with the principal part $q^{-\frac{5}{44}}(\mathfrak e_7-\mathfrak e_{-7})$. To compute the Fourier coefficients of $P_{-5}$ we used the method described in the previous section with an initial $\varepsilon=10^{-40}$ and $Y=0.5$, which gave us a truncation point of $M_0=42$, corresponding to $\Delta$ between $-1847$ and $1885$. For a short selection of computed values of $c^+(\Delta)$ see Table \ref{tab:11a1} and for a table of coefficients corresponding to all vanishing $L'(G,\chi_{\Delta},1)$ see Table \ref{tab:11a1-zeros}. The first few normalized ``negative'' coefficients are displayed in Table \ref{tab:11a1-cminus}. These values should be compared to the list in \cite[p.\,505]{Sk2}.
\subsubsection{37a1} Consider the newform of weight two and level 37 which has an odd functional equation. The $q$-expansion is given by \[ G=q - 2q^2 - 3q^3 + 2q^4 - 2q^5 + 6q^6 - q^7 + 6q^9 + 4q^{10} - 5q^{11} + \cdots \in S^{new}_{2}\left(\Gamma_0(37)\right). \]
Using Sage we computed all values of $L'(G,\chi_{\Delta},1)$ for fundamental discriminants $\Delta>0$ such that $\left(\frac{\Delta}{37}\right)=1$ and $|\Delta|\le 15000$. This set consists of $2217$ fundamental discriminants and amongst these we found $8$ discriminants for which $L'(G,\chi_{\Delta},1)$ vanished up to the numerical precision (see Table \ref{tab:37a1-zeros}). For the corresponding harmonic weak Maass form in $H_{1/2,\rho}$ we took $P_{-3}$, which has a principal part $q^{-\frac{3}{148}}(\mathfrak e_{21}+\mathfrak e_{21})$. The initial computation was done in Sage, using $\varepsilon=1\cdot10^{-35}$, which gave a value of $M_0=30$, corresponding to discriminants in the range $-4440 \le \Delta \le 4585$. For examples of the coefficients $c^{+}(\Delta)$ see Tables \ref{tab:37a1} and \ref{tab:37a1-zeros}. The first few normalized ``negative'' coefficients are displayed in Table \ref{tab:37a1-cminus}.
\subsubsection{37b1} In this case we consider the newform of weight two and level 37 which has an even functional equation. The $q$-expansion is given by \[ G=q + q^3 - 2q^4 - q^7 - 2q^9 + 3q^{11} + \cdots \in S^{new}_{2}\left(\Gamma_0(37)\right). \] Using Sage we computed all values of $L'(G,\chi_{\Delta},1)$ for fundamental discriminants $\Delta<0$ such that
$\left(\frac{\Delta}{37}\right)=1$ and $|\Delta|\le 12000$. This set consists of $1631$ fundamental discriminants and amongst these we found $15$ discriminants for which $L'(G,\chi_{\Delta},1)$ vanished up to the numerical precision (see Table \ref{tab:37b1-zeros}). For the corresponding harmonic weak Maass form in $H_{1/2,\bar\rho}$ we took $P_{-12}$, which has a principal part $q^{-\frac{12}{148}}(\mathfrak e_{30}-\mathfrak e_{30})$.
The initial computation was done in Sage, using $\varepsilon=1\cdot10^{-30}$, which gave a value of $M_0=33$, corresponding to discriminants in the range $-4883 \le \Delta \le 5029$. For examples of the coefficients $c^{+}(\Delta)$ see Tables \ref{tab:37b1} and \ref{tab:37b1-zeros}. The first few normalized ``negative'' coefficients are displayed in Table \ref{tab:37b1-cminus}.
\subsection{Conclusions of the numerical experiments for weight two}
In each of the examples of weight two newforms that we studied we saw agreement with the theorem, i.e. the coefficients $c^{+}(\Delta)$ (for fundamental discriminants with the appropriate property) were only algebraic when the corresponding central derivative $L'(G,\chi_{\Delta},1)$ vanished. Furthermore, we observed that in the cases we considered, the algebraic coefficients $c^{+}(\Delta)$ were in fact even rational {\em integers}.
\subsection{Further computations}
To investigate whether a result analogous to Theorem \ref{Lvalues} also holds for newforms of weight $4$, we computed $L'(2,G,\chi_{\Delta})$ for all newforms $G$ of weight $4$ on $\Gamma_0(N)$ whith $5 \le N\le 150$ and fundamental discriminants
$\Delta$ with $|\Delta|\le 300$ and the property that the twisted
$L$-function $L(s,G,\chi_{\Delta})$ has an odd functional equation. For $5\le N \le 10$ we additionally computed these values for fundamental discriminants $\Delta$ with $|\Delta|\le5000$. Amongst all these values we did not find a single example of a vanishing derivative. Even though we did not get any positive case where we could test the theorem we still wanted to make sure that there was no easily accesible counter example.
We therefore computed the Fourier coefficients, up to 40 digits precision, of the associated weight $-\frac{1}{2}$ harmonic Maass form corresponding to all weight $4$ newforms defined over $\mathbb{Q}$ for $N$ up to $100$. To test the accuracy (and making sure that the implementation was correct) we did not only rely on the provable error bounds, but also checked algebraicity of certain coefficients corresponding to non-fundamental discriminants. These coefficients were indeed all found to be integers or rational with fairly small denominators. In contrast to this, the Fourier coefficients corresponding to fundamental discriminants were found not to be similarly ``simple'' rational numbers.
The $L$-value computations were performed in Sage \cite{SA}, using the included version of Rubinstein's lcalc library \cite{L}.
\subsection{Tables}
\begin{sidewaystable}
\caption[]{$E=11a1$, $P_{-5}\in H_{\frac{1}{2},\bar\rho}$} \label{tab:11a1} \begin{tabular}{rll} \multicolumn{1}{c}{$\Delta$} & \multicolumn{1}{c}{$c^{+}(\Delta)$} & \multicolumn{1}{c}{$L'(G,\chi_{\Delta},1)$}\\ \hline\noalign{
} $-7 $&$ \hphantom{-}2.8463370190285980186651576519711751393948073861988\cdot 10^{00} $&$ 1.22556687406888\cdot 10^{00}$\\ $-8 $&$ \hphantom{-}2.5138482002575729165892711124435774460030012341762\cdot 10^{00} $&$ 1.88791354720393\cdot 10^{00}$\\ $-19$&$ \hphantom{-}4.8192428148963255861870924437043119000673519519042\cdot 10^{00} $&$ 7.51391655667451\cdot 10^{00}$\\ $-24$&$ \hphantom{-}5.1018494088339703187507177747597327598232414476402\cdot 10^{00} $&$ 4.02744559024300\cdot 10^{00}$\\ $-35$&$ \hphantom{-}4.7515892636101723769649079639675162004017245362399\cdot 10^{00} $&$ 7.64786334637073\cdot 10^{00}$\\ $-39$&$ -1.6466690697010481166272091028219327677356442914804\cdot 10^{01} $&$ 2.97721567216550\cdot 10^{00}$\\ $-40$&$ \hphantom{-}1.1470941388138074683747768314723689292860539962900\cdot 10^{01} $&$ 5.58789208952436\cdot 10^{00}$\\ $-43$&$-1.7622439638503327722737780360046423237568367805048\cdot 10^{01} $&$ 1.18814465355690\cdot 10^{01}$\\ $-51$&$ \hphantom{-}2.0736222999878741718629718432682995552582880786065\cdot 10^{01} $&$ 1.30416363302768\cdot 10^{01}$\\ $-52$&$ \hphantom{-}1.5723528683914990387103216700146317562411438497615\cdot 10^{01} $&$ 5.14853759817659\cdot 10^{00}$\\ $-68$&$ \hphantom{-}9.6889673322938493992006043404127469979247370067926\cdot 10^{00} $&$ 3.80344864881298\cdot 10^{00}$\\ $-79$&$ \hphantom{-}1.7557351755436160739388564340027760291317089229254\cdot 10^{01} $&$ 4.75620653690677\cdot 10^{00}$\\ $-83$&$ -7.1767664383427675609861242907417950544611683162859\cdot 10^{01} $&$ 6.43843846621214\cdot 10^{00}$\\ $-84$&$ \hphantom{-}6.1666200626587315159968126799603650525586539365601\cdot 10^{01} $&$ 6.53746327159376\cdot 10^{00}$\\ $-87$&$ -7.7230036424433334541697484050338439023979647483280\cdot 10^{01} $&$ 2.35584785481347\cdot 10^{00}$\\ $-95$&$ \hphantom{-}7.8467572084064151556661839046504144426199227897994\cdot 10^{01} $&$3.03660486030085\cdot 10^{00}$\\ $-811$&$\hphantom{-}3.0046247983067285336553431175489765847382042907105\cdot 10^{06} $&$1.25949136911120\cdot 10^{01}$\\ $-820$&$-6.0493754250387304262091147332158578046510749019315\cdot 10^{06} $&$1.19119437485937\cdot 10^{01}$\\ $-824$&$-5.7985199999999999999999999999999999999999999999999\cdot 10^{06} $&$-6.6\cdot 10^{-24}$\\ $-827$&$ \hphantom{-}1.8535489407871222859528059423736067736521222528554\cdot 10^{06} $&$1.60961273159300\cdot 10^{01}$\\ $-831$&$-6.7911392225835416131083026699411310608420151994986\cdot 10^{06} $&$3.36744068632019\cdot 10^{00}$\\ $-996$&$-3.5516294505685820400211045047063422129168082892941\cdot 10^{07}$ & $1.15828152096335\cdot 10^{01}$\\ $-1003$&$ \hphantom{-}1.0811934742079303073802766406181476437608668928409\cdot 10^{07} $&$2.53076681967579\cdot 10^{01}$\\ $-1007$&$-3.9469248000000000000000000000000000000000000000000\cdot 10^{07} $& $-1.1\cdot 10^{-22}$\\ $-1011$&$-3.7685140824429636488934775010060106547341275101054\cdot 10^{07} $&$ 1.84592490209627\cdot 10^{01}$\\ $-1019$&$ \hphantom{-}3.3790315957549749442218769650593817997888628818338\cdot 10^{07}$ &$ 1.68145450009782\cdot 10^{01}$ \\
\end{tabular} \end{sidewaystable}
\begin{table} \caption[]{$E=11a1$, $P_{-5} \in H_{\frac{1}{2},\bar\rho}$} \label{tab:11a1-zeros} \begin{tabular}{llc} \multicolumn{1}{c}{$\Delta$} & \multicolumn{1}{c}{$c^{+}(\Delta)$} &
\multicolumn{1}{c}{$\left|c^{+}(\Delta)-[c^{+}(\Delta)]\right|$} \\ \hline\noalign{
} $-824 $&$ -5798520 $&$ 3.0\cdot 10^{-76}$ \\
$-1799 $&$-2708450784 $&$ 2.7\cdot 10^{-46}$\\ $-4399 $&$-68135748249936640 $&$ 2.3\cdot 10^{-21}$ \\ $-8483 $&$ \hphantom{-}214445760716391388216704 $&$ 9.1\cdot 10^{-28}$ \\ $-11567 $&$ -12412267149099919205092899456 $&$ 1.6 \cdot 10^{-25} $\\ $-14791 $&$ \hphantom{-} 66850179291021019012709832099520 $&$ 3.1\cdot 10^{-30}$\\ $-15487 $&$ -478732239405182448762415030881280 $&$ 5.6\cdot 10^{-30}$ \\ $-15659 $&$ -804489814454597618648064770159415 $&$ 6.8\cdot 10^{-30}$ \\ $-15839 $&$ -1162122495004344641799524116135680 $&$ 7.0\cdot 10^{-30}$ \\ $-16463 $&$ \hphantom{-}4542575922533728228643934862230144 $&$ 1.5\cdot 10^{-30} $\\ $-17023 $&$ -23302350713109514450879400185948800 $&$ 2.0\cdot 10^{-29}$\\ $-17927 $&$ \hphantom{-}110133238181959291703634158808374784 $&$ 1.2\cdot 10^{-29}$\\ $-18543 $&$ \hphantom{-}464726791864282489334104058164482624 $&$ 1.9\cdot 10^{-29}$\\ \end{tabular} \end{table}
\begin{table}
\caption[]{$E=11a1$, $P_{-5}\in H_{\frac{1}{2},\bar\rho}$. Coefficients are scaled by $ c^{-}(1)$.} \label{tab:11a1-cminus} \begin{tabular}{llc} \multicolumn{1}{c}{$\Delta$} & \multicolumn{1}{c}{$\sqrt{\Delta}\,c^{-}(\Delta)$} &
\multicolumn{1}{c}{$\left|c^{-}(\Delta)-[c^{-}(\Delta)]\right|$} \\ \hline\noalign{
}
$ 4 $&$-3$ &$2.0 \cdot 10^{-100}$\\ $ 5 $&$\hphantom{-}5$ &$2.1 \cdot 10^{-99\hphantom{0}}$\\ $ 9 $&$-2$ & $1.7 \cdot 10^{-100}$\\ $ 12$&$\hphantom{-}5$ &$8.0 \cdot 10^{-100}$\\ $ 16$&$\hphantom{-}4$ &$1.5 \cdot 10^{-99\hphantom{0}}$\\ $ 20$&$\hphantom{-}5$ &$1.1 \cdot 10^{-100}$\\ $ 25$&$\hphantom{-}0$ &$1.0 \cdot 10^{-100}$\\ $ 36$&$\hphantom{-}6$ &$1.0 \cdot 10^{-99\hphantom{0}}$\\ $ 37$&$\hphantom{-}5$ &$4.2 \cdot 10^{-99\hphantom{0}}$\\ $ 45$&$\hphantom{-}0$ &$6.4 \cdot 10^{-99\hphantom{0}}$\\ \end{tabular} \end{table}
\begin{sidewaystable}
\caption[]{$E=37a1$, $P_{-3}\in H_{\frac{1}{2},\rho}$} \label{tab:37a1} \begin{tabular}{rll} \multicolumn{1}{c}{$\Delta$} & \multicolumn{1}{c}{$c^{+}(\Delta)$} & \multicolumn{1}{c}{$L'(G,\chi_{\Delta},1)$}\\ \hline\noalign{
} $ 1 $&$-2.8176178498959956879756075537515493438922975370716\cdot 10^{-01}$ & $3.05999773834052\cdot 10^{-01}$\\ $ 12 $&$-4.8852723826201225228227029607337071669095284814788\cdot 10^{-01}$ & $4.29861479867736\cdot 10^{00}$\\ $ 21 $&$-1.7273925723265275652082007397068992218426924398791\cdot 10^{-01}$ & $9.00238680032537\cdot 10^{00}$\\ $ 28 $&$\hphantom{-}6.7819399530394779828450578400669420938246859928076\cdot 10^{-01}$ & $4.32726024966011\cdot 10^{00}$\\ $ 33 $&$\hphantom{-}5.6630232015906998168220545669245622604190884430064\cdot 10^{-01}$ & $3.62195679113882\cdot 10^{00}$\\ $ 37 $&$-9.1326561374611652958506448407204050631184401026129\cdot 10^{-01}$ & $3.47328771649229 \cdot 10^{00}$\\ $ 40 $&$\hphantom{-}4.0098509269543637915254766073122850557290259963615\cdot 10^{-01}$ & $3.70588717878444\cdot 10^{00}$\\ $ 41 $&$\hphantom{-}6.5637495744757231959699415722023547525400239084778\cdot 10^{-01}$ & $5.93680171871573\cdot 10^{00}$\\ $ 44 $&$\hphantom{-}9.6886404434506397321859573425794267139455920322171\cdot 10^{-01}$ & $1.01334656625280\cdot 10^{01}$\\ $ 53 $&$-5.6688852568232517859984506645723944339238503996096\cdot 10^{-01}$ & $2.61746665637296\cdot 10^{01}$\\ $ 65 $&$-6.0328072889521477971996798071175156059595671497733\cdot 10^{-01}$ & $7.67818286326206\cdot 10^{00}$\\ $ 73 $&$\hphantom{-}3.4874711835362408853804154923777452565842552803223\cdot 10^{-01}$ &$2.92507284795068\cdot 10^{00}$\\ $ 77 $&$\hphantom{-}2.2699132373705254600799448087564660809534768699467\cdot 10^{-01}$ &$3.42067600398534\cdot 10^{10}$\\ $ 85 $&$-7.6894617048676272061865441758881289699552927122551\cdot 10^{-01}$ & $9.90133670369251\cdot 10^{00}$\\ $ 1481$&$-3.2715595098273932057423414526408419506801164996884\cdot 10^{00}$ &$5.26994449124823\cdot 10^{00}$ \\ $ 1484$&$-1.3432792297590353562651264178555980321660674399890\cdot 10^{01}$ &$3.86746474997364\cdot 10^{01}$\\ $ 1489$& $\hphantom{-}8.9999999999999999999999999999999999999999999999999\cdot 10^{00}$ &$-3.7\cdot 10^{-23}$\\ $ 1496$&$\hphantom{-}1.1199440423162819213593329208218112792285448658029\cdot 10^{01}$ &$2.27616829409607\cdot 10^{01}$\\ $ 1501$&$-5.8188238119388864901078937905792951783427273426771\cdot 10^{02}$ &$ 6.06007663972706\cdot 10^{00}$\\ $ 4376$&$-3.6731327299348159991042234350611468700145535059868\cdot 10^{02}$ &$2.03155740209437\cdot 10^{01}$\\ $ 4377$&$-5.0062522276143084997015960658866819832113068397294\cdot 10^{02}$ &$2.27150950159608\cdot 10^{00}$\\ $ 4393$&$\hphantom{-}6.6000000000000000000000000000000000000000000001468\cdot 10^{01}$ &$5.8\cdot 10^{-23}$\\ $ 4396$&$-2.3023069110811173762943326075771836710063196221488\cdot 10^{02}$ &$2.00437958330233\cdot 10^{00}$\\ $ 4412$&$-3.1500483730098996665306117169085504925545562420809\cdot 10^{02}$ &$3.73011222569745\cdot 10^{01}$\\ \end{tabular} \end{sidewaystable}
\begin{table} \caption[]{$E=37a1$, $P_{-3}\in H_{\frac{1}{2},\rho}$} \label{tab:37a1-zeros} \begin{tabular}{llc} \multicolumn{1}{c}{$\Delta$} & \multicolumn{1}{c}{$c^{+}(\Delta)$} &
\multicolumn{1}{c}{$\left|c^{+}(\Delta)-[c^{+}(\Delta)]\right|$} \\ \hline\noalign{
} $1489$&$\hphantom{-}9 $ &$ 1.6\cdot 10^{-72}$\\ $4393$&$\hphantom{-}66 $ &$ 1.5\cdot 10^{-45}$\\ $5116$&$-746$ &$ 8.5\cdot 10^{-23}$\\ $5281$&$\hphantom{-}153$ &$ 8.2\cdot 10^{-23}$\\ $5560$&$-1124$ &$ 1.2\cdot 10^{-22}$\\ $5761$&$-974$ &$ 1.1\cdot 10^{-22}$\\ $6040$&$-1404$ &$ 4.2\cdot 10^{-23}$\\ $6169$&$\hphantom{-}336$ &$ 1.1\cdot 10^{-22}$\\ \end{tabular} \end{table}
\begin{table} \caption[]{$E=37a1$, $P_{-3}\in H_{\frac{1}{2},\rho}$. Coefficients are scaled by $\sqrt{3}\,c^{-}(-3)$.}
\label{tab:37a1-cminus} \begin{tabular}{ccc} \multicolumn{1}{c}{$\Delta$} &
\multicolumn{1}{c}{$\sqrt{|\Delta|}\,c^{-}(\Delta)$} &
\multicolumn{1}{c}{$\left|c^{-}(\Delta)-[c^{-}(\Delta)]\right|$} \\ \hline\noalign{
}
$ -4$&$\hphantom{-}1$ &$4.0 \cdot 10^{-84}$\\ $ -7$&$-1$ &$5.0 \cdot 10^{-84}$\\ $-11$&$\hphantom{-}1$ &$4.5 \cdot 10^{-84}$\\ $-12$&$-1$ &$2.0 \cdot 10^{-84}$\\ $-16$&$-2$ &$1.1 \cdot 10^{-83}$\\ $-27$&$-3$ &$1.3 \cdot 10^{-83}$\\ $-28$&$\hphantom{-}3$ &$1.4 \cdot 10^{-83}$\\ $-36$&$-2$ &$1.0 \cdot 10^{-83}$\\ $-40$&$\hphantom{-}2$ &$3.6 \cdot 10^{-85}$\\ $-44$&$-1$ &$1.1 \cdot 10^{-83}$\\ \end{tabular} \end{table}
\begin{sidewaystable}
\caption[]{$E=37b1$, $P_{-12}\in H_{\frac{1}{2},\bar\rho}$} \label{tab:37b1} \begin{tabular}{rll} \multicolumn{1}{c}{$\Delta$} & \multicolumn{1}{c}{$c^{+}(\Delta)$} & \multicolumn{1}{c}{$L'(G,\chi_{\Delta},1)$}\\ \hline\noalign{
} $ -3 $&$\hphantom{-}1.0267149116920354474451980654263083626994977508118 \cdot 10^{00}$&$1.47929949208 \cdot 10^{00}$\\ $ -4 $&$\hphantom{-}1.2205364009670316625279102409757685190938519711297 \cdot 10^{00}$&$1.81299789722 \cdot 10^{00}$\\ $ -7 $&$\hphantom{-}1.6900297463200076214148752932012403965170838158011 \cdot 10^{00}$&$2.11071898018 \cdot 10^{00}$\\ $-11 $&$\hphantom{-}5.8849982354849175483779961900586424239744874288522 \cdot 10^{-01}$&$3.65679089534 \cdot 10^{00}$\\ $-40 $&$\hphantom{-}1.2669706585839831188366862729215921230412462308976 \cdot 10^{00}$&$4.16362898338 \cdot 10^{00}$\\ $-47 $&$\hphantom{-}3.0756790552662277517712909874001702657447701023621 \cdot 10^{00}$&$5.26739088546 \cdot 10^{00}$\\ $-67 $&$\hphantom{-}2.1608356105538234382282266707128748893591830597455 \cdot 10^{00}$&$4.98143961845 \cdot 10^{00}$\\ $-71 $&$-1.5945418432752378367351454423028372659103804842831\cdot 10^{00}$&$5.33295381308 \cdot 10^{00}$\\ $-83 $&$\hphantom{-}2.9631171578917930530100644900469583789690213329975 \cdot 10^{00}$&$7.30522465208 \cdot 10^{00}$\\ $-84 $&$-3.8773494709413749500399075799371212202544017987791\cdot 10^{00}$&$1.00026475317 \cdot 10^{01}$\\ $-95 $&$-2.6554862688645143792016861758519887392731139540185 \cdot 10^{00}$&$5.83606039003 \cdot 10^{00}$\\ $-132$&$\hphantom{-}4.1944733541115532186541330550136737538249181082859 \cdot 10^{00}$&$9.99216716471 \cdot 10^{00}$\\ $-136$&$-4.8392675993443437829864850814885823635034770966657 \cdot 10^{00}$&$5.73824076491 \cdot 10^{00}$\\ $-139$&$-5.9999999999999999999999999999999999999999999999991 \cdot 10^{00}$&$-8.5 \cdot 10^{-23}$\\ $-151$&$-8.3135688179267692046624844818371994826339638923811 \cdot 10^{-01}$&$6.69750855159 \cdot 10^{00}$\\ $-152$&$\hphantom{-}4.3274351625459058613812696410805017025617476195953 \cdot 10^{00}$&$7.95190347996 \cdot 10^{00}$\\ $-811$&$-1.4731293182498551151700589944493338505308298027148 \cdot 10^{02}$&$5.32436617837 \cdot 10^{00}$\\ $-815$&$\hphantom{-}1.2194410312093092058885476868302234805268383148338 \cdot 10^{02}$&$4.74925836935 \cdot 10^{00}$\\ $-823$&$\hphantom{-}3.1200000000000000000000000000000000000000000000000 \cdot 10^{02}$&$-1.5 \cdot 10^{-23}$\\ $-824$&$-3.2299860660409750567356931348586086493552010570382 \cdot 10^{02}$&$1.75028741141 \cdot 10^{01}$\\ $-835$&$-2.4035736526655124690110045874885626910322384359422 \cdot 10^{02}$&$8.64359690730 \cdot 10^{00}$\\ \end{tabular} \end{sidewaystable}
\begin{table} \caption[]{$E=37b1$, $P_{-12}\in H_{\frac{1}{2},\bar\rho}$} \label{tab:37b1-zeros} \begin{tabular}{llc} \multicolumn{1}{c}{$\Delta$} & \multicolumn{1}{c}{$c^{+}(\Delta)$} &
\multicolumn{1}{c}{$\left|c^{+}(\Delta)-[c^{+}(\Delta)]\right|$} \\ \hline\noalign{
} $ -139 $ & $ -6 $ & $ 1.5 \cdot 10^{-85} $ \\
$ -823 $ & $\hphantom{-}312 $ & $ 9.1 \cdot 10^{-80} $ \\
$ -2051 $ & $ -26724 $ & $ 1.0 \cdot 10^{-67} $ \\
$ -2599 $ & $\hphantom{-}122048 $ & $ 3.4 \cdot 10^{-63} $ \\
$ -3223 $ & $ -472416 $ & $ 3.2 \cdot 10^{-57} $ \\
$ -3371 $ & $ -674712 $ & $ 7.4 \cdot 10^{-56} $ \\
$ -5227 $ & $\hphantom{-}5816 $ & $ 5.5 \cdot 10^{-31} $ \\
$ -5307 $ & $ -5192 $ & $ 4.6 \cdot 10^{-31} $ \\
$ -6583 $ & $ -13320 $ & $ 4.6 \cdot 10^{-31} $ \\
$ -7892 $ & $ -79552 $ & $ 1.2 \cdot 10^{-30} $ \\
$ -7951 $ & $\hphantom{-}28152 $ & $ 4.0 \cdot 10^{-31} $ \\
$ -9112 $ & $ -224548 $ & $ 1.6 \cdot 10^{-30} $ \\
$ -9715 $ & $\hphantom{-}236934 $ & $ 2.8 \cdot 10^{-32} $ \\
$ -11444 $ & $ -1437956 $ & $ 2.0 \cdot 10^{-33} $ \\
$ -11651 $ & $\hphantom{-}563716 $ & $ 7.0 \cdot 10^{-34} $ \\ \end{tabular} \end{table}
\begin{table} \label{tab:37b1-cminus} \caption[]{$E=37b1$, $P_{-12}\in H_{\frac{1}{2},\bar\rho}$. Coefficients are scaled by $c^{-}(1)$.}
\begin{tabular}{ccc} \multicolumn{1}{c}{$\Delta$} & \multicolumn{1}{c}{$\sqrt{\Delta}\,c^{-}(\Delta)$} &
\multicolumn{1}{c}{$\left|c^{-}(\Delta)-[c^{-}(\Delta)]\right|$} \\ \hline\noalign{
}
$ 4 $&$-1$ &$1.6 \cdot 10^{-85}$\\ $ 9 $&$\hphantom{-}0$ &$3.2 \cdot 10^{-85}$\\ $ 12$&$\hphantom{-}3$ &$9.6 \cdot 10^{-85}$\\ $ 16$&$-2$ &$1.6 \cdot 10^{-85}$\\ $ 21$&$\hphantom{-}3$ &$2.7 \cdot 10^{-85}$\\ $ 25$&$-1$ &$1.7 \cdot 10^{-85}$\\ $ 28$&$\hphantom{-}3$ &$3.9 \cdot 10^{-85}$\\ $ 33$&$\hphantom{-}3$ &$3.6 \cdot 10^{-85}$\\ $ 36$&$\hphantom{-}0$ &$3.9 \cdot 10^{-85}$\\ $ 40$&$\hphantom{-}0$ &$5.6 \cdot 10^{-85}$\\ \end{tabular} \end{table}
\end{document} | arXiv |
\begin{document}
\title{A remark on the Laplacian flow and the modified Laplacian co-flow in ${
m G} \begin{abstract} We observe that the DeTurck Laplacian flow of ${\rm G}_2$-structures introduced by Bryant and Xu as a gauge fixing of the Laplacian flow can be regarded as a flow of ${\rm G}_2$-structures (not necessarily closed) which fits in the general framework introduced by Hamilton in \cite{positive}. \end{abstract}
\section{Introduction} In \cite{Bryant} Bryant introduced a geometric flow in ${\rm G}_2$-geometry which evolves an initial closed ${\rm G}_2$-structure $\varphi_0$ in the direction of its Laplacian.
Given a compact $7$-dimensional manifold with a closed ${\rm G}_2$-structure $(M,\varphi_0)$, a {\em Laplacian flow} is a solution to the evolution equation \begin{equation}\label{LF}
\tfrac{\partial}{\partial t}\varphi_t=\Delta_{\varphi_t}\varphi_t\,,\quad d\varphi_t=0\,,\quad \varphi_{|t=0}=\varphi_0\,. \end{equation} The well-posedness of equation \eqref{LF} is proved in \cite{BryantXu} by applying the Nash-Moser theorem to the gauge fixing \begin{equation}\label{DTLF}
\tfrac{\partial}{\partial t}\varphi_t=\Delta_{\varphi_t}\varphi_t+\mathcal{L}_{V(\varphi_t)}\varphi_t\,,\quad d\varphi_t=0\,,\quad \varphi_{|t=0}=\varphi_0\,, \end{equation} where $\mathcal L$ is the Lie derivative and $V\colon C^{\infty}(M,\Lambda^3_+)\to C^{\infty}(M,TM)$ is a first order differential operator which depends on the choice of a connection on $M$. Here $\Lambda^3_+$ denotes the open subbundle of $\Lambda^3$ of ${\rm G}_2$-structures on $M$. A solution to \eqref{DTLF} is usually called a {\em DeTurck Laplacian flow}.
A DeTurck Laplacian flow $\varphi_t$ is also a solution to \begin{equation}\label{flow}
\tfrac{\partial}{\partial t}\varphi_t=dd^*_{\varphi_t}\varphi_t+d\iota_{V(\varphi_t)}\varphi_t\,,\quad \varphi_{|t=0}=\varphi_0\,. \end{equation}
In the present note we observe that equation \eqref{flow} fits in the general framework introduced by Hamilton in \cite{positive}. As a direct consequence we have the following theorem which in particular implies the well-posedness of \eqref{DTLF} \begin{theorem}\label{main1} Let $(M,\varphi_0)$ be a compact $7$-dimensional manifold with a ${\rm G}_2$-structure. Then equation \eqref{flow} has a unique short-time solution. \end{theorem}
In \cite{K} Karigiannis, McKay and Tsui introduced the {\em Laplacian co-flow} as the solution to the evolution equation \begin{equation}\label{LCF} \tfrac{\partial}{\partial t}(*_{\varphi_t}\varphi_t)=-\Delta_{\varphi_t}*_{\varphi_t}\varphi_t\,,\quad d*_{\varphi_t}\varphi_t
=0\,,\quad \varphi_{|t=0}=\varphi_0\,. \end{equation} where in this case $\varphi_0$ is supposed to be co-closed with respect to the metric induced by itself. The well-posedness of this last equation is still an open problem and Grigorian introduced in \cite{Gri} the following modification \begin{equation}\label{MLCF} \tfrac{\partial}{\partial t}(*_{\varphi_t}\varphi_t)= \Delta_{\varphi_t}*_{\varphi_t}\varphi_t+2d((A-{\rm Tr}(T(\varphi_t))\varphi_t )\,,\quad d*_{\varphi_t}\varphi_t
=0\,,\quad \varphi_{|t=0}=\varphi_0\,, \end{equation} where $A$ is a constant and $T(\varphi_t)$ is the torsion of $\varphi_t$. In \cite{Gri} it is proved the well-posedness of \eqref{MLCF} following the same approach of Bryant in \cite{Bryant} by applying the Nash-Moser theorem to the gauge fixing \begin{equation}\label{DTMLCF}
\tfrac{\partial}{\partial t}(*_{\varphi_t}\varphi_t)=\Delta_{\varphi_t}*_{\varphi_t}\varphi_t+2d((A-{\rm Tr}(T(\varphi_t))\varphi_t )+\mathcal{L}_{V(\varphi_t)}\varphi_t\,,\quad d*_{\varphi_t}\varphi_t=0\,,\quad \varphi_{|t=0}=\varphi_0\,, \end{equation} Any solution to this last equation \eqref{DTMLCF} satsfies \begin{equation}\label{DTMLCF}
\tfrac{\partial}{\partial t}(*_{\varphi_t}\varphi_t)=dd^{*}_{\varphi_t}*_{\varphi_t}\varphi_t+2d((A-{\rm Tr}(T(\varphi_t))\varphi_t )+d\iota_{V(\varphi_t)}\varphi_t\,,\quad \varphi_{|t=0}=\varphi_0\,, \end{equation} Analogously to theorem \ref{main1} we have \begin{theorem}\label{main2} Let $(M,\varphi_0)$ be a compact $7$-dimensional manifold with a ${\rm G}_2$-structure. Then equation \eqref{DTMLCF} has a unique short-time solution. \end{theorem}
\section{Proof of the results} Both theorems \ref{main1} and \ref{main2} can be proved by using the following set-up introduced by Hamilton in \cite{positive}.
Let $M$ be an oriented compact manifold, $F$ a vector bundle over $M$, $U$ an open subbundle of $F$ and $$ E\colon C^{\infty}(M,U)\to C^{\infty}(M,F) $$ a second order differential operator. For $f\in C^{\infty}(M,U)$, we denote by $D E(f)\colon C^{\infty}(M,F)\to C^{\infty}(M,F)$ the linearization of $E$ at $f$ and by $\sigma D E(f)$ the principal symbol of $D E(f)$.
\begin{definition} {\em An integrability condition for $E$ is a first order linear differential operator $$ L\colon C^{\infty}(M,F)\to C^{\infty}(M,G)\,, $$ where $G$ is another vector bundle over $M$, such that $L(E(f))=0$ for all $f\in C^{\infty}(M,U)$, and all the eigenvalues of $\sigma DE(f)$ restricted to $\ker \sigma L$
have strictly positive real part. } \end{definition}
\begin{theorem}[Hamilton {\cite[Theorem 5.1]{positive}}]\label{Ham_int} Assume that $E$ admits
an integrability condition. Then for every $f_0\in C^{\infty}(M,U)$ the geometric flow \begin{equation}\label{flow_Ham} \frac{\partial f}{\partial t}=E(f)\,,\quad f(0)=f_0\,, \end{equation} has a unique short-time solution. \end{theorem}
Now we can focus on the set-up of theorem \ref{main1}. Here we consider $$ F=\Lambda^3\,,\quad U=\Lambda^3_+\,,\quad G=\Lambda^4\,,\quad E(\varphi)= dd^*_{\varphi}\varphi+d\iota_{V(\varphi)}\varphi\,,\quad L=d\colon C^{\infty}(M,\Lambda^3)\to C^{\infty}(M,\Lambda^4)\,. $$ From \cite{BryantXu} it follows that for every $\varphi\in C^{\infty}(M,U)$ and every {\em closed} $\psi \in C^{\infty}(M,\Lambda^{3})$ we have $$ DE(\varphi)(\psi)=-\Delta_{\varphi} \psi+{\rm l.o.t.} $$ Hence all the assumptions of Hamilton's theorem \ref{Ham_int} are satisfied and theorem \ref{main1} follows.
Notice that if the starting form $\varphi_0$ is closed, then the solution to \eqref{flow} is closed for every $t$ since $$ d\tfrac{\partial}{\partial t}\varphi=0\,. $$
Therefore if $\varphi_0$ is closed,
the unique solution $\varphi_t$ to \eqref{flow} solves also the DeTurck-Laplacian flow \eqref{DTLF} and the short-time existence of the DeTurck-Laplacian flow \eqref{DTLF} can be deduced from Theorem \ref{main1}.
About the proof of theorem \ref{main2} we set
$$ \begin{aligned} & F=\Lambda^4\,,\quad U=\Lambda^4_+\,,\quad G=\Lambda^5\,,\quad E(*_\varphi\varphi)= dd^*_{\varphi}\varphi+d\iota_{V(\varphi)}+2d((A-{\rm Tr}(T(\varphi))\varphi)\,,\\ &L=d\colon C^{\infty}(M,\Lambda^4)\to C^{\infty}(M,\Lambda^5)\,. \end{aligned} $$ From \cite{Gri} it follows $$ DE(*_\varphi\varphi)(\psi)=-\Delta_{\varphi} \psi+{\rm l.o.t.} $$ for every closed $\psi\in C^\infty(M,\Lambda^4)$ and the proof of theorem \ref{main2} follows.
\noindent {\bf Acknowledgments.} We would like to thank Jason Lotay for useful conversations.
\end{document} | arXiv |
Snub disphenoid
In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron is a convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some vertices have four faces and others have five. It is a dodecahedron, one of the eight deltahedra (convex polyhedra with equilateral triangle faces), and is the 84th Johnson solid (non-uniform convex polyhedra with regular faces). It can be thought of as a square antiprism where both squares are replaced with two equilateral triangles.
Snub disphenoid
TypeJohnson
J83 – J84 – J85
Faces4+8 triangles
Edges18
Vertices8
Vertex configuration4(34)
4(35)
Symmetry groupD2d
Dual polyhedronElongated gyrobifastigium
Propertiesconvex, deltahedron
Net
The snub disphenoid is also the vertex figure of the isogonal 13-5 step prism, a polychoron constructed from a 13-13 duoprism by selecting a vertex on a tridecagon, then selecting the 5th vertex on the next tridecagon, doing so until reaching the original tridecagon. It cannot be made uniform, however, because the snub disphenoid has no circumscribed sphere.
History and naming
This shape was called a Siamese dodecahedron in the paper by Hans Freudenthal and B. L. van der Waerden (1947) which first described the set of eight convex deltahedra.[1] The dodecadeltahedron name was given to the same shape by Bernal (1964), referring to the fact that it is a 12-sided deltahedron. There are other simplicial dodecahedra, such as the hexagonal bipyramid, but this is the only one that can be realized with equilateral faces. Bernal was interested in the shapes of holes left in irregular close-packed arrangements of spheres, so he used a restrictive definition of deltahedra, in which a deltahedron is a convex polyhedron with triangular faces that can be formed by the centers of a collection of congruent spheres, whose tangencies represent polyhedron edges, and such that there is no room to pack another sphere inside the cage created by this system of spheres. This restrictive definition disallows the triangular bipyramid (as forming two tetrahedral holes rather than a single hole), pentagonal bipyramid (because the spheres for its apexes interpenetrate, so it cannot occur in sphere packings), and icosahedron (because it has interior room for another sphere). Bernal writes that the snub disphenoid is "a very common coordination for the calcium ion in crystallography".[2] In coordination geometry, it is usually known as the trigonal dodecahedron or simply as the dodecahedron.
The snub disphenoid name comes from Norman Johnson's 1966 classification of the Johnson solids, convex polyhedra all of whose faces are regular.[3] It exists first in a series of polyhedra with axial symmetry, so also can be given the name digonal gyrobianticupola.
Properties
The snub disphenoid is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. The other three polyhedra with this property are the regular octahedron, the pentagonal bipyramid, and an irregular polyhedron with 12 vertices and 20 triangular faces.[4]
The snub disphenoid has the same symmetries as a tetragonal disphenoid: it has an axis of 180° rotational symmetry through the midpoints of its two opposite edges, two perpendicular planes of reflection symmetry through this axis, and four additional symmetry operations given by a reflection perpendicular to the axis followed by a quarter-turn and possibly another reflection parallel to the axis.[5] That is, it has D2d antiprismatic symmetry, a symmetry group of order 8.
Spheres centered at the vertices of the snub disphenoid form a cluster that, according to numerical experiments, has the minimum possible Lennard-Jones potential among all eight-sphere clusters.[6]
Up to symmetries and parallel translation, the snub disphenoid has five types of simple (non-self-crossing) closed geodesics. These are paths on the surface of the polyhedron that avoid the vertices and locally look like a shortest path: they follow straight line segments across each face of the polyhedron that they intersect, and when they cross an edge of the polyhedron they make complementary angles on the two incident faces to the edge. Intuitively, one could stretch a rubber band around the polyhedron along this path and it would stay in place: there is no way to locally change the path and make it shorter. For example, one type of geodesic crosses the two opposite edges of the snub disphenoid at their midpoints (where the symmetry axis exits the polytope) at an angle of π/3. A second type of geodesic passes near the intersection of the snub disphenoid with the plane that perpendicularly bisects the symmetry axis (the equator of the polyhedron), crossing the edges of eight triangles at angles that alternate between π/2 and π/6. Shifting a geodesic on the surface of the polyhedron by a small amount (small enough that the shift does not cause it to cross any vertices) preserves the property of being a geodesic and preserves its length, so both of these examples have shifted versions of the same type that are less symmetrically placed. The lengths of the five simple closed geodesics on a snub disphenoid with unit-length edges are
$2{\sqrt {3}}\approx 3.464$ (for the equatorial geodesic), ${\sqrt {13}}\approx 3.606$, $4$ (for the geodesic through the midpoints of opposite edges), $2{\sqrt {7}}\approx 5.292$, and ${\sqrt {19}}\approx 4.359$.
Except for the tetrahedron, which has infinitely many types of simple closed geodesics, the snub disphenoid has the most types of geodesics of any deltahedron.[7]
Construction
The snub disphenoid is constructed, as its name suggests, as the snub polyhedron formed from a tetragonal disphenoid, a lower symmetry form of a regular tetrahedron.
Disphenoid Snub disphenoid
The snub operation produces a single cyclic band of triangles separating two opposite edges (red in the figure) and their adjacent triangles. The snub antiprisms are analogous in having a single cyclic band of triangles, but in the snub antiprisms these bands separate two opposite faces and their adjacent triangles rather than two opposite edges.
The snub disphenoid can also constructed from the square antiprism by replacing the two square faces by pairs of equilateral triangles. However, it is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids.
A physical model of the snub disphenoid can be formed by folding a net formed by 12 equilateral triangles (a 12-iamond), shown. An alternative net suggested by John Montroll has fewer concave vertices on its boundary, making it more convenient for origami construction.[8]
Cartesian coordinates
Let $q\approx 0.16902$ be the positive real root of the cubic polynomial
$2x^{3}+11x^{2}+4x-1.$
Furthermore, let
$r={\sqrt {q}}\approx 0.41112,$
$s={\sqrt {\frac {1-q}{2q}}}\approx 1.56786,$
and
$t=2rs={\sqrt {2-2q}}\approx 1.28917.$
The eight vertices of the snub disphenoid may then be given Cartesian coordinates
$(\pm t,r,0),\,(0,-r,\pm t),$
$(\pm 1,-s,0),\,(0,s,\pm 1).$[6]
Because this construction involves the solution to a cubic equation, the snub disphenoid cannot be constructed with a compass and straightedge, unlike the other seven deltahedra.[9]
With these coordinates, it's possible to calculate the volume of a snub disphenoid with edge length a as $\xi a^{3}$, where $\xi \approx 0.85949$, is the positive root of the polynomial
$5832x^{6}-1377x^{4}-2160x^{2}-4.$[10]
The exact form of $\xi $ can be expressed as,
$\xi ={\frac {1}{6{\sqrt {6}}}}{\sqrt {17+{\sqrt[{3}]{155249-28848i{\sqrt {237}}}}+{\sqrt[{3}]{155249+28848i{\sqrt {237}}}}}},$
$\xi ={\frac {1}{6{\sqrt {6}}}}{\sqrt {17+2{\sqrt {6049}}\cos \left({\frac {1}{3}}\tan ^{-1}\left({\frac {28848{\sqrt {237}}}{155249}}\right)\right)}},$
where $i$ is the imaginary unit.
Related polyhedra
Another construction of the snub disphenoid is as a digonal gyrobianticupola. It has the same topology and symmetry, but without equilateral triangles. It has 4 vertices in a square on a center plane as two anticupolae attached with rotational symmetry. Its dual has right-angled pentagons and can self-tessellate space.
Digonal anticupola
Digonal gyrobianticupola
(Dual) elongated gyrobifastigium
Partial tessellation
References
1. Freudenthal, H.; van d. Waerden, B. L. (1947), "On an assertion of Euclid", Simon Stevin, 25: 115–121, MR 0021687.
2. Bernal, J. D. (1964), "The Bakerian Lecture, 1962. The Structure of Liquids", Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, 280 (1382): 299–322, Bibcode:1964RSPSA.280..299B, doi:10.1098/rspa.1964.0147, JSTOR 2415872, S2CID 178710030.
3. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, S2CID 122006114, Zbl 0132.14603.
4. Finbow, Arthur S.; Hartnell, Bert L.; Nowakowski, Richard J.; Plummer, Michael D. (2010), "On well-covered triangulations. III", Discrete Applied Mathematics, 158 (8): 894–912, doi:10.1016/j.dam.2009.08.002, MR 2602814.
5. Cundy, H. Martyn (1952), "Deltahedra", The Mathematical Gazette, 36 (318): 263–266, doi:10.2307/3608204, JSTOR 3608204, MR 0051525, S2CID 250435684.
6. Sloane, N. J. A.; Hardin, R. H.; Duff, T. D. S.; Conway, J. H. (1995), "Minimal-energy clusters of hard spheres", Discrete and Computational Geometry, 14 (3): 237–259, doi:10.1007/BF02570704, MR 1344734.
7. Lawson, Kyle A.; Parish, James L.; Traub, Cynthia M.; Weyhaupt, Adam G. (2013), "Coloring graphs to classify simple closed geodesics on convex deltahedra." (PDF), International Journal of Pure and Applied Mathematics, 89 (2): 123–139, doi:10.12732/ijpam.v89i2.1, Zbl 1286.05048.
8. Montroll, John (2004), "Dodecadeltahedron", A Constellation of Origami Polyhedra, Dover Origami Papercraft Series, Dover Publications, Inc., pp. 38–40, ISBN 9780486439587.
9. Hartshorne, Robin (2000), Geometry: Euclid and Beyond, Undergraduate Texts in Mathematics, Springer-Verlag, p. 457, ISBN 9780387986500.
10. Wolfram Research, Inc. (2020). "Wolfram|Alpha Knowledgebase". Champaign, IL. MinimalPolynomial[PolyhedronData[{"Johnson", 84}, "Volume"], x] {{cite journal}}: Cite journal requires |journal= (help)
External links
• Weisstein, Eric W. "Snub disphenoid". MathWorld.
Johnson solids
Pyramids, cupolae and rotundae
• square pyramid
• pentagonal pyramid
• triangular cupola
• square cupola
• pentagonal cupola
• pentagonal rotunda
Modified pyramids
• elongated triangular pyramid
• elongated square pyramid
• elongated pentagonal pyramid
• gyroelongated square pyramid
• gyroelongated pentagonal pyramid
• triangular bipyramid
• pentagonal bipyramid
• elongated triangular bipyramid
• elongated square bipyramid
• elongated pentagonal bipyramid
• gyroelongated square bipyramid
Modified cupolae and rotundae
• elongated triangular cupola
• elongated square cupola
• elongated pentagonal cupola
• elongated pentagonal rotunda
• gyroelongated triangular cupola
• gyroelongated square cupola
• gyroelongated pentagonal cupola
• gyroelongated pentagonal rotunda
• gyrobifastigium
• triangular orthobicupola
• square orthobicupola
• square gyrobicupola
• pentagonal orthobicupola
• pentagonal gyrobicupola
• pentagonal orthocupolarotunda
• pentagonal gyrocupolarotunda
• pentagonal orthobirotunda
• elongated triangular orthobicupola
• elongated triangular gyrobicupola
• elongated square gyrobicupola
• elongated pentagonal orthobicupola
• elongated pentagonal gyrobicupola
• elongated pentagonal orthocupolarotunda
• elongated pentagonal gyrocupolarotunda
• elongated pentagonal orthobirotunda
• elongated pentagonal gyrobirotunda
• gyroelongated triangular bicupola
• gyroelongated square bicupola
• gyroelongated pentagonal bicupola
• gyroelongated pentagonal cupolarotunda
• gyroelongated pentagonal birotunda
Augmented prisms
• augmented triangular prism
• biaugmented triangular prism
• triaugmented triangular prism
• augmented pentagonal prism
• biaugmented pentagonal prism
• augmented hexagonal prism
• parabiaugmented hexagonal prism
• metabiaugmented hexagonal prism
• triaugmented hexagonal prism
Modified Platonic solids
• augmented dodecahedron
• parabiaugmented dodecahedron
• metabiaugmented dodecahedron
• triaugmented dodecahedron
• metabidiminished icosahedron
• tridiminished icosahedron
• augmented tridiminished icosahedron
Modified Archimedean solids
• augmented truncated tetrahedron
• augmented truncated cube
• biaugmented truncated cube
• augmented truncated dodecahedron
• parabiaugmented truncated dodecahedron
• metabiaugmented truncated dodecahedron
• triaugmented truncated dodecahedron
• gyrate rhombicosidodecahedron
• parabigyrate rhombicosidodecahedron
• metabigyrate rhombicosidodecahedron
• trigyrate rhombicosidodecahedron
• diminished rhombicosidodecahedron
• paragyrate diminished rhombicosidodecahedron
• metagyrate diminished rhombicosidodecahedron
• bigyrate diminished rhombicosidodecahedron
• parabidiminished rhombicosidodecahedron
• metabidiminished rhombicosidodecahedron
• gyrate bidiminished rhombicosidodecahedron
• tridiminished rhombicosidodecahedron
Elementary solids
• snub disphenoid
• snub square antiprism
• sphenocorona
• augmented sphenocorona
• sphenomegacorona
• hebesphenomegacorona
• disphenocingulum
• bilunabirotunda
• triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)
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Home > Journals > J. Symbolic Logic > Volume 78 > Issue 2 > Article
June 2013 On colimits and elementary embeddings
Joan Bagaria, Andrew Brooke-Taylor
J. Symbolic Logic 78(2): 562-578 (June 2013). DOI: 10.2178/jsl.7802120
We give a sharper version of a theorem of Rosický, Trnková and Adámek [13], and a new proof of a theorem of Rosický [12], both about colimits in categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as $\alpha$-strongly compact and $C^{(n)}$-extendible cardinals.
Joan Bagaria. Andrew Brooke-Taylor. "On colimits and elementary embeddings." J. Symbolic Logic 78 (2) 562 - 578, June 2013. https://doi.org/10.2178/jsl.7802120
Digital Object Identifier: 10.2178/jsl.7802120
Primary: 03E55, 03E75, 18A15, 18C35
Secondary: 18A30, 18A35
Keywords: accessible categories, Colimits, elementary embeddings, strongly compact cardinals, Vopěnka's Principle
Rights: Copyright © 2013 Association for Symbolic Logic
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Joan Bagaria, Andrew Brooke-Taylor "On colimits and elementary embeddings," Journal of Symbolic Logic, J. Symbolic Logic 78(2), 562-578, (June 2013) | CommonCrawl |
Henry Charles McWeeney
Henry Charles McWeeney (1867–1935) was an Irish mathematician, who was Professor of Mathematics at University College Dublin (UCD) from 1891 until his death in 1935. From 1909 on he served as vice president of UCD.[1]
Education and career
McWeeney was born in Dublin and was educated at University College Dublin (BSc 1887, MSc 1890) and Trinity College Dublin (BA 1889). He won a Royal University of Ireland Travelling Studentship award in 1891, and spent his entire career at UCD. He also taught at St Patrick's College, Drumcondra, from 1892 to 1910.[2]
Speaking of his days as a student at UCD in the mid 1880s, McWeeney remembers listening to the Professor of Mathematics there, John Casey, who had a mixed class of students. "Although the book-work which he communicated would not have been adequate for the training of a higher mathematician, yet his explanations, so far as they went, were marked by extreme lucidity."[3]
It is an interesting fact that on McWeeney’s going first to the Catholic University he had been sent by M'Grath to attend some lectures by Robert Graham, a well-known and successful Tutor of Trinity College, who had joined Fr~Delany's staff. He was author of well-known handbooks of Algebra and of Factors, and was known to be a marvelous expert in preparing for examinations, being familiar with the sort of tricky questions that were often set. But McWeeney found him to be also a first-rate teacher, and attributes chiefly to him the foundation of his own mathematical career.[3]
In the article Mathematics in U.C.D. 1854 to 1974 by J. R. Timoney recalls McWeeney and Egan giving the traditional rite of passage lectures to brand new first year students in the autumn of 1927:[1]
He started to do questions off the entrance scholarship paper, indeed the more outlandish bits of the toughest questions. Fr Egan gave the second lecture. He described it later as his most unintelligible lecture on irrational numbers. This was the honours class clearance act, of course, and things moderated for the hard necks who stuck it out.
Timoney went on to say, "He was a magnificent teacher and a geometer of great elegance. A favourite expression of he was 'if you attack it judiciously it will come out in a line'."[1]
References
1. Mathematics in U.C.D. 1854 to 1974 by J. R. Timoney, page 22
2. A New History of Ireland: Ireland Under the Union, 1870–1921, Hither education, 1793–1908, page 568
3. A Page of Irish History: Story of University College, Dublin 1883–1909 Fathers of the Society of Jesus (Talbot Press, 1930)
External links
• "McWeeny, Henry C." . Thom's Irish Who's Who . Dublin: Alexander Thom and Son Ltd. 1923. p. 157 – via Wikisource.
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Research, Policy and Practice
Pastoral livelihood pathways transitions in northern Kenya: The process and impact of drought
Leonard K. Kirui ORCID: orcid.org/0000-0003-0785-24951,
Nathaniel D. Jensen2,
Gideon A. Obare1,
Isaac M. Kariuki1,
Philemon K. Chelanga2 &
Munenobu Ikegami3
Pastoralism volume 12, Article number: 23 (2022) Cite this article
Recurrent droughts and long-term changes to climate, social structures, and the economy in the world's arid and semi-arid lands have impacted pastoralists' livelihood activities over time, resulting in different livelihood pathways. Some pastoralists continue to follow more traditional strategies of herd accumulation, while others frequently engage in livestock markets and value addition activities, and others still are dropping out of pastoralism. Using data collected over 6 years from 924 households in northern Kenya and applying a generalized structural dynamic multinomial logit model, this study quantitatively determined the dynamic transitions between livelihood categories conditional on drought incidences. From the results, there were considerable and frequent transitions between livelihood pathways within the panel period (2009–2015). Notably, many households that started in the low-cash income, larger herd size category denoted as hanging in, were transitioning to lower cash income with small herds (dropping out). At the same time, there was a great deal of back and forth between the category with low-cash income and small herd size (dropping out) and the category with higher cash income and small herd size (moving out), indicating that moving out was the only way out of poverty. Also, an increase in vegetation index from a drought season where the index was at a 10% level to a good season where the level was 90% decreased the likelihood of households dropping out from a predicted probability of 37.9 to 28.7% and increased the likelihood of moving up and moving out from 22.2 to 25.0% and 22.6 to 34.3%, respectively, unconditionally. The study findings imply that any livelihood interventions aimed at reducing the impact of drought and alleviating poverty among pastoral households should support the transition to market-oriented, relatively successful pathways and also protect households from falling back into the ranks of poverty by dropping out.
The sustainability of pastoralists' ability to generate income and earn a livelihood through extensive livestock grazing in drylands has been a subject of a global debate among development agencies, policymakers, and governments due to the challenges it faces (McPeak et al. 2012). Over time, long-term climatic, social, land-use, and economic changes have posed dynamic implications on the livelihoods of pastoralists. While dependence on livestock prevails, these disruptions have led to transitions into livelihood options with varying degrees of reliance on livestock. For instance, poor households pursue low-return marginal activities such as charcoal burning and firewood sales, among others (Ellis 2000; Little et al. 2001). This happens when herds are no longer viable to support their livelihoods (Lind et al. 2016). On the other hand, herd-sufficient households diversify to lucrative livestock business activities (Little et al. 2001). These tangents from traditional pastoralism pursued by the poor and wealthy households represent pastoralist livelihood pathways. In fragile production systems like pastoralism, expanded knowledge of these dynamics provides the foundation necessary to develop impactful context-specific interventions.
Pastoralists build their herds over the years through breeding from their stock, purchases from markets, and other socially instituted intake mechanisms (Hauck and Rubenstein 2017). It is a wealth-building process preferred by most pastoralists as the benefits through births and milk are perceived to outweigh returns from live animals sold (Lind et al. 2020). However, covariate and idiosyncratic shocks inhibit growth by causing adverse impacts, whose outcomes systematically vary across households and regions. These outcomes have long-run implications that are difficult to capture using cross-sectional surveys or even short-period longitudinal surveys accurately. Also, these disruptions and subsequent recovery dynamics imply that pastoralists would systematically oscillate across the income and herd size thresholds over time resulting in a change in livelihood pathways pursued. However, there is little understanding of the transition process between the various pastoral livelihood pathways. This study, therefore, sought to fill in the existing knowledge gaps with respect to how different groups of pastoralists transition between livelihood pathways over time and the impact of drought.
Earlier studies on pastoralist livelihoods have used different approaches while providing contextual dynamics in response to notable economic and environmental shocks. This includes pastoral risk management and social change (McPeak et al. 2012), pastoral resilience in a time characterized by drought and recovery phases (McPeak and Little 2017), policy-driven sedentarization initiatives like irrigation agriculture and growth of industries (Dai et al. 2020), options outside the livestock economy (Watete et al. 2016), and diversification to crop income-based livelihoods (Rufino et al. 2013). These studies used frameworks developed using household socio-economic indicators to generate insights on livelihood changes as occasioned by these shocks. For instance, and the most relevant to this study, McPeak et al. (2012) and McPeak and Little (2017) clustered 190 households in northern Kenya and 153 households in southern Ethiopia into four livelihood groups based on whether households were below or above the median thresholds of livestock assets from the baseline survey in March 2000 and cash income values in a repeat June 2000 survey. The four clusters were named 'staying with' referring to those maintaining traditional pastoralism, 'combining' and 'moving from', implying households deviating on opposite sides of the median herd size and 'left out' group referring to cases pushed out of pastoralism into other low-income livelihood options. These livelihood groups were used to demonstrate how households in the four identified livelihood clusters strive to achieve economic security and differ in exposure, sensitivity, and ability to cope with shocks. Similarly, using household-level data, the groups can be created and used to map pastoralist livelihood trajectories over time (Lind et al. 2020).
This study further advances the above literature by using a 6-year annual household-level panel data to map pastoralist livelihood pathways trends, transitions, and the impact of drought over time and space. The relatively more extended period allowed the analyses of transitions between categories in a way that other works have not. Accordingly, households were stratified based on relative thresholds of per capita cash income per day from all sources and livestock holdings per capita, calculated once over the panel period (2009–2015), but allowed to vary from one insurable unit to another. Insurable/index units are areas created for selling insurance to herders. The thresholds were allowed to vary by index units to focus on the changes in livelihood groups over time within units rather than using a single shared threshold which would have focused the study on more stationary differences between index units. By looking at livelihood dynamics among pastoralists and the impact of drought, this study provides a basis for the implementation of policy interventions for transitioning households into sustainable livelihoods. This is important considering the number of investments focusing on drought mitigation and the sustenance of the pastoralist way of life.
The study was conducted in Marsabit County in northern Kenya (Fig. 1). The county covers 70,944 km2 representing 12% of the national territory. It borders Samburu County to the south, Isiolo and Wajir to the east, and Turkana County to the west. Ethiopia also borders the county to the north. It is divided into four sub-counties (North Horr, Saku, Moyale, and Laisamis), sub-divided into 20 electoral wards (KNBS 2019a).
Map of the study area
The county receives an annual rainfall of between 200 and 1000 mm with an average precipitation of 254 mm, making it one of Kenya's driest counties (NDMA 2019). The area is also characterized by poor infrastructure, frequent droughts, low market access, and remote settlements. To cope with these harsh conditions, the communities in this area primarily practice semi-nomadic pastoralism, where livestock are moved during the dry season searching for pastures and water (McPeak et al. 2012).
Marsabit has 447,150 people and 77,495 households with an average household size of about 5.8 (Kenya National Bureau of Statistics 2019a). The county is home to several pastoral communities, including Borana, Somali, Samburu, Rendile, and Gabra. These communities rely on livestock as their primary livelihood source. The residents depend on milk from livestock, most commonly from cattle and camels, for consumption, and they also trade in animals to buy other foods and meet their different daily needs (Mahmoud 2013). Other livelihood activities include casual labour, wage employment, petty trading, charcoal burning, and small agro-pastoralism.
Sampling and data collection
This study used six rounds of household panel data collected by the International Livestock Research Institute (ILRI) between 2009 and 2015 to track the impacts of the Index-Based Livestock Insurance (IBLI) intervention. Briefly, IBLI was formulated as an innovation that seeks to bring the benefits of formal insurance to help pastoralists of Northern Kenya manage drought-induced livestock losses (Mude et al. 2010). The IBLI project initially started in 2010 as a strategy to help herders quickly recover from losses caused by frequent catastrophic droughts (Mude et al. 2010). As of writing, IBLI is commercially sold across the drylands of Kenya and in southern Ethiopia. In 2015, Kenya's government started using it as the first government livestock insurance scheme in Africa under the Kenya Livestock Insurance Program (KLIP).
During the survey period, information was collected from 16 sub-locations from a proportionately stratified sample based on the 2009 census with a minimum sample size of 30 households and a maximum of 100 per sub-location. Households were placed in three wealth categories based on their livestock holdings in tropical livestock units (TLUs): 1 TLU = 250 kg of animal's live weight, equal to 0.7 of a camel, 1 head of cattle, and 10 shoats (Schwartz et al. 1991). The wealth groups were low (< 10 TLU), medium (≥ 10 ≤ 20), and high (> 20 TLU). In each of the 16 sub-locations, one-third of the location-specific sample was selected randomly from each of the three wealth groups, and the samples were then used to generate a list of 924 households.
To address attrition across the survey rounds, missing households were replaced from the same TLU class in the same sub-location. In the first four survey rounds, the replacement strategy yielded a consistent sample of 924. In round 5, the sample reduced to 923, and in the sixth and final round in 2015, the sample reduced to 919 households. A total of 770 respondents were consistently interviewed during the survey period in the six survey rounds conducted from 2009 to 2015.
The IBLI data is one of the most recent datasets in Kenya, with relevant variables needed to perform analyses on pastoralist livelihoods and change over time. According to Lind et al. (2016), empirical studies on pastoralists' livelihoods and changes over time should capture key variables on actual access and use of resources and markets. As of 2021, the only publicly available/accessible datasets in Kenya that had such variables were IBLI 2009–2015; Hunger Safety Net Program (HSNP) data; Pastoral Risk Management (PARIMA) 2000–2002 data; the Maasai South Rift's Homewood, Kristjanson, and Trench, Staying Maasai 1998–2004; and Grandin Maasai Systems Study 1981–1985. Therefore, this study used the relatively most recent primary data to contribute to the existing knowledge on pastoralist livelihoods and changes over time. Most importantly, it incorporated livelihood transition conditional on previous household and local circumstances to pastoral livelihood analyses.
In the analyses, Normalized Difference Vegetation Index (NDVI) data were used to measure drought. NDVI is a satellite-generated indicator of the amount and vigour of vegetation cover based on the observed level of photosynthetic activity (Tucker et al. 2005). Lower values of NDVI indicate forage scarcity, while higher ones reflect better vegetative greenness. NDVI image data are computed at high spatial resolution and consistent quality from advanced very high-resolution radiometer (AVHRR) onboard the US National Oceanic and Atmospheric Administration (NOAA) satellite (Chantarat et al. 2013).
The NDVI data are commonly used to compare the current vegetation state against long-term average conditions to detect any anomalies and anticipate drought (Bayarjargal et al. 2006). Due to their reliability in ASALs, NDVI data are widely used in studies that use remote sensing data for drought management. IBLI also uses the data. To account for differences in regions, agro-ecological conditions, climatic patterns, and herd composition, NDVI values specific for different index areas were extracted, namely Central and Gadamoji, Maikona, Laisamis, and Loiyangalani in each survey year. The Cumulative z-values of Normalized Difference Vegetation Index (CZNDVI) for these regions are provided in Table 1.
Table 1 Standardized normalized difference vegetation indices during survey rounds
Livelihood classification
Studies on livelihoods often use different approaches based on various theoretical foundations to derive livelihood categories for insightful analyses. One method gaining popularity, especially in studies on changes in pastoralist livelihoods over time, is an approach used in McPeak et al. (2012), which was further improved in McPeak and Little (2017). In this approach, households are categorized into four livelihood groups distinguished by their access to markets and resources. The variable or proxy for market access is household cash income, and the proxy for access to key resources (mainly pastures and water) is livestock ownership. The four livelihood categories (hanging in, dropping out, moving up, and moving out), also termed the long-term pastoralism pathways, represent a simple schema for thinking broadly about change in pastoral areas over time (Lind et al. 2020). This study used this schema as illustrated in Fig. 2 to study the transition processes over time and the effects of drought on pastoral livelihood pathways.
Pastoralist livelihood pathways. Source: Lind et al. (2020)
In this study, livelihood pathways' relative terms were defined using area-specific thresholds. Accordingly, the sampled households from Marsabit County in northern Kenya were classified into the four livelihood groups using herd size in TLUs per capita (all adults) and cash income per capita (all adults) per day. More specifically, the means of household TLUs per capita and income per capita per day specific to five insurable units were calculated once over the 6 years and used as thresholds for categorizing households into the four livelihood pathways. Insurable units are geographical areas created to sell index insurance to herders. The use of these units as opposed to administrative units such as sub-locations is that these units reflect socio-ecological patterns including rainfall patterns, agro-ecological factors, and herding/migration patterns among pastoralists in the area (Chelanga et al. 2017).
The rationale behind this study's classification approach is that pastoral areas are characterized by a high degree of spatial heterogeneity of land use. Therefore, insurable units better reflect socio-ecological patterns in the drylands compared to the standard administrative units. Indeed, household herd size and cash income thresholds that were derived showed a striking difference from one insurable unit to another, key factors in establishing a projection of likely individual household's livelihood pathways in the face of drought and drought-related stresses.
Modelling strategy
Given the response variable (livelihood pathways) is a discrete variable that assumes four unordered outcomes, this study applied a generalized structural dynamic multinomial logit model to determine the effect of drought on pastoralist livelihood pathways.
The generalized structural dynamic multinomial logit model follows the traditional multinomial logit (MNL) model but incorporates change over time (dynamism) by using covariates in period t−1 to predict livelihood outcomes in period t. This approach reduces reverse causality and allows these covariates to infer and predict future pastoralist livelihood outcomes. In this model, the probability that a household i is following the jth livelihood pathway in the current period (t) as a function of drought among a set of other factors as measured in the previous period (t−1) is given as follows:
$$ \Pr \left({y}_{ij}=\left.j\right|{x}_{it-1},{y}_{it-1}\right)=\frac{\exp \left({x}_{it-1}^{\prime }{\beta}_j+{y}_{it-1}\right)}{1+{\sum}_{r=1}^3\exp \left({x}_{it-1}^{\prime }{\beta}_r+{y}_{it-1}\right)}\forall j=1,2,3,4 $$
where j = 1 is hanging in; j = 2 is dropping out; j = 3 is moving up; j = 4 is moving out, xit − 1and yit − 1are explanatory variables and livelihood pathways in the previous period (t−1), respectively; and βj are parameters to be estimated. The major interest in the analysis was the effect of drought on livelihood pathways transitions. So, margins were calculated and reported which were interpreted as the effect of a unit change in vegetation index (drought indicator) in the previous period (t−1), on the probability of a household remaining or transitioning to another livelihood category in the current period (t) relative to the likelihood of hanging-in (the reference group).
Thresholds for livelihood classification
Households were classified into the four livelihood categories based on relative thresholds of cash income per capita per day controlled for consumer price indices (CPI) and livestock holdings per capita in tropical livestock units. The thresholds (pooled means) varied between insurable units but were fixed over the 6 years. CPI for the periods 2009, 2010, 2011, 2012, 2013, and 2015 were 96, 100, 114, 125, 132, and 150, respectively (Kenya National Bureau of Statistics 2019b). The base period chosen for the adjustment of cash income for inflation for this study was 2010 (CPI of 100). Accordingly, incomes were adjusted based on CPI ratios derived. Table 2 provides the thresholds used for the classification of households in the five insurable units.
Table 2 Thresholds (pooled means 2009–2015) for livelihood classification
As shown in Table 2, the highest TLU per capita threshold was 3.16, while the lowest was 0.71. On the other hand, the highest income per capita per day threshold was KES 47.44 (0.47 USD), while the lowest was KES 25.62 (0.25 USD). The disparity between these thresholds demonstrates the rationale for using relative thresholds rather than general ones. For instance, a household in Gadamoji with 1 TLU per capita and KES 30 (0.3 USD) cash income per capita per day would have been categorized as dropping out in Central when, in fact, they are moving up relative to other households in Gadamoji.
Proportions and trends of pastoralist livelihood pathways over time
The proportions of the four pathways had patterns of increase and decrease over the panel period. The proportion of households in the hanging in pathway generally decreased over time from 28% in 2009 to 12% in 2015. The proportion of households dropping out increased over time from 28% in 2009 to 32% in 2015. Moving up was a little bit stable, but it slightly decreased in proportion from 29% in 2009 to 24% in 2015. Finally, the trend displayed by households in the moving out group was a general growth from 16% in 2009 to 32% in 2015. Figure 3 provides the respective distribution and proportions of pastoralist livelihood pathways from 2009 to 2015.
Trends in pastoralist livelihood pathways between 2009 and 2015
Socio-economic characteristics by livelihood pathways
Various socio-economic characteristics were compared across the four livelihood pathways. Heads of households pursuing moving up strategies were relatively younger compared to their counterparts in the hanging in, dropping out, and moving out pathways. They also had more years of education and more savings (Table 3). Household settlement status, access to credit, livestock insurance, and land ownership showed significant variation across the four pathways (Table 4). The highest percentage of settled households were those in the moving out pathway, while the least were those in the hanging in group. More households in the moving up group had access to credit, while those in the dropping out pathway had the least access. Most households that practised moving out activities privately owned land compared to their counterparts in the other groups.
Table 3 Descriptive summary statistics of continuous variables by livelihood pathways
Table 4 Summary statistics (percentages) for discrete and dummy variables by livelihood pathways
Proportionate transitions between pathways
The transition of households from one pathway to another was very dynamic over the panel period. There were no clear pathways or sets of transitions that seemed much more common than others. The proportionate transitions also indicated that dropping out, moving up, and moving out pathways were very resilient. That is, each year, more than 50% of households categorized into these three livelihood pathways in the previous year remained in the same category in the current year (Table 5).
Table 5 Proportionate transitions between livelihood pathways
Effects of drought on pastoralist livelihood pathways
The effect of drought as indicated by CZNDVI was significant across all the four pastoralist livelihood pathways, unconditionally, controlling for the previous pathway and controlling for other covariates. Generally, an increase in CZNDVI from a drought season where the vegetation index was at 10% level to a good season where the index was at 90% level reduced the likelihood of households dropping out, increased the likelihood of moving up, and also increased the likelihood of households moving out (Table 6).
Table 6 The impact of CZNDVI on livelihood pathways
Households in the hanging in livelihood pathway earn a living from traditional pastoralism and small agro-pastoralism. The highest percentage of their earnings comes from livestock, making other sources negligible. A striking trend of this group was their continued decrease in numbers over time. In 2009, 27.81% of households were categorized as hanging in. By 2015, that figure had fallen to 11.53%. According to Lind et al. (2020), this group of pastoralists is common in areas where there is good access to rangeland and water sources, but market access is limited. People in such areas are hanging in customary pastoralism, and their incomes drive demand for other non-pastoral products or services such as crop produce, natural products, constructions services, and others (Headey et al. 2012).
The dropping out pathway represents households with the lowest herd size and cash incomes. It was the largest livelihood group that cumulatively represented 28.96% of the panel respondents. In the base year, 28.35% of households were categorized into this group. The proportion of households grew over the years, implying that relatively more households became dropouts over time. This group's striking feature was its drastic rise from 22.40% in 2010 to 39.07% in 2011, a trend attributable to the devastating 2010/2011 drought that began in late 2010 and peaked in 2011. Drought not only deteriorates livestock health or results in high mortality, but it also pushes pastoralists out of their production system, forcing them to seek alternative sources of livelihood. Therefore, the increasing number of households dropping out depicts the increasing frequency and the severity of the drought. For instance, the 2010/2011 drought was thought to be the worst in the last 60 years, but the subsequent ones were even more severe, with 2017 having been the worst of all. Such a trend threatens pastoralism as a livelihood. As evidenced by the results of this study, the increasing number of households dropping out and decreasing number of households hanging in on pastoralism is an indication of a general exit from pastoralism over time.
Households in the moving up group represent the wealthiest in terms of both herd size and cash income. Also, from the income profiles of these households, much of their income comes from the sale of livestock and livestock products. Over time, the number of households categorized as being in this pathway decreased to 24.37% of households from the initial proportion of 27.81%. However, the rate of decrease over time is slower with patterns of increase, followed by a decrease, for instance, from 36.69% in 2010 to 23.59% in 2011 and then up to 28.35% in 2012. In northern Kenya, moving up activities are evident in Moyale where pastoralists are engaging in cross-border trade and other high-value livestock marketing activities with their counterparts from Ethiopia's eastern lowlands (Mahmoud 2013).
Moving out is the second largest group among the four livelihood pathways. Furthermore, it is also the second group (after dropping out) that depicts a trend of growth in numbers over time. At the starting period (2009), 16.02% of the sampled households were categorized into this group. Although there were instances of a slight decrease in some periods, the group generally registered an upward trend reaching 31.66% in the final year. Also, households in this group had high incomes with most of it coming from sources not directly linked to pastoralism such as salaried employment, sale of crops, business, and petty trading. Improved connections with large centres, small-town expansion, and acceptance of non-traditional livelihoods by the younger generation are some of the factors that promote household engagement in moving-out activities (Lind et al. 2020). The general trends observed from the proportions of these four livelihood pathways are illustrated in Fig. 3.
Socio-economic characteristics of pastoralist households by livelihood pathways
A number of socio-economic characteristics were associated with the four pastoralist livelihood pathways. Table 3 provides the descriptive summary statistics for continuous household characteristics by livelihood pathways. The F-statistic values included in the table are the analysis of variance (ANOVA) tests for significant differences in the means of selected variables between the four livelihood pathways. As a precaution against extreme values including many zero-valued observations in some variables such as household head's education years, household education stock, amount of savings, remittances, and size of land irrigated, the study used inverse hyperbolic sine transformation (IHS) as used by Bellemare and Wichman (2020). This procedure, to some extent, is the same as taking the log of the variable, but it allows the retention of observations with zero values.
As provided in the summary statistics in Table 3, the mean age of household heads varied significantly across the four livelihood pathways. The dropping out group had the highest mean age while moving up had the lowest, reflecting participation in market-oriented keeping and moving of large herds among young pastoralists. Dropping out had the highest mean of household size, while hanging in had the smallest. These results indicate that a relatively older head leads an average drop-out household with many household members who increase pressure on an already small income and fewer animals due to severe droughts and other factors.
Schooling years of household head and household education stock (sum of schooling years over all household members) showed a significant difference between the four livelihood pathways. Moving up had the highest mean of schooling years of the household head while dropping out had the lowest average years of schooling. On the other hand, moving out had the highest average household education stock, while hanging in had the lowest average. This highlights a vulnerability among drop-outs concerning access to or their search for education. According to Teshome and Bayissa (2014), dropping out of pastoralism is mainly due to unexpected events such as drought, conflict, and chronic poverty, and households that drop out often end up in old peri-urban impoverished areas where access to education and other services is a challenge. Such households' activities are generally inferior survival strategies that yield low income that might not be sufficient for them to seek a better social status and education (Little et al. 2010). On the other hand, higher education years in the moving up and out pathways imply more cash income opportunities and greater access to education among those groups. Generally, moving up and out livelihood activities are common in urban areas where better living conditions, job opportunities, and access to education are among the pull factors that attract households to such areas (Lind et al. 2016).
The amount of savings significantly differed across the four livelihood pathways. The moving up group had the highest mean of household savings, while the dropping out group had the lowest mean. This finding highlights the vulnerability faced by households in the dropping out group. Should they face an income shortfall, these households have fewer resources to fall back on. Few savings are likely a result, in part, of their low income and could also be a signal of less access to savings institutions.
Moving out had the highest mean of irrigated land while hanging in had the lowest mean. Moving out involves participation in value-added activities such as micro-dairying, small-holder crop farming, and other activities; hence, irrigation projects play a very vital role for households in this category, especially when it comes to the growing of fodder, watering crops, and other high income-generating activities that may not do well in some of the world's harshest lands that hardly receive adequate rainfall throughout the year.
To further show the difference in socio-economic characteristics by livelihood pathways, Table 4 provides the summary statistics for discrete and dummy variables.
The hanging in group had the highest percentage of households headed by females while moving out had the lowest. The highest percentage of female-headed households among traditional mobile pastoralists reflects a common practice of sending male heads to satellite camps while females take care of the basecamp (Jensen et al. 2017).
Settlement status varied significantly across the four livelihood pathways. Households practising traditional pastoralism (hanging in) had the highest mobility, while those in the moving out group had the least. These results affirm that the traditional adaptability and risk management strategy of mobility still plays a significant role among households that depend on livestock as their primary livelihood. This finding is consistent with the observation of McPeak et al. (2012) regarding livestock mobility among pastoralists that it must be supported and not hindered by policy or other changes.
Access to credit provides financial capital to venture into both livestock-based livelihood activities and those that are non-livestock-based. The majority of households that accessed credit were in the moving up group, with those dropping out having the lowest access rate to credit. Arguably, credit provides financial capital that increases the likelihood of household engagement in highly remunerative strategies such as moving up livelihood activities. Indeed, as Umeta and Temesgen (2013) note, helping pastoralists get credit enables them to participate in lucrative livestock and even non-livestock livelihood ventures.
Index-based livestock insurance is an intervention that aims to help pastoralists manage drought-related shocks. Moving out and up livelihood pathways had the highest insurance uptake, while the hanging in and dropping out groups had the lowest. This illustrates the idea that buying insurance enables pastoralists to build herds for trade, and any losses likely to be incurred in the event of drought will be catered for by indemnity received once the index reaches strike points set for payments to be released to all insured clients. Matsuda et al. (2019) also observed that insurance could increase herd size.
Land ownership varied significantly across the livelihood pathways. The findings of this study indicate that the moving out group had the highest percentage of land ownership while the hanging in group had the lowest. Ideally, traditional mobile pastoralism (hanging in) requires extensive and free access to commonly managed pastures. Individual land ownership is thus not expected in such groups. On the other hand, having higher productive physical assets holdings is necessary for households transitioning into or engaging in more remunerative market-oriented livelihood activities such as moving out.
Pastoralist livelihood pathway transitions
In the context of change over time, it is possible to estimate the probability that a household categorized under a given livelihood pathway in the previous period (t−1) is likely to remain in the same group or transition to another category in the current period (t). Table 5 presents the computed proportionate transitions between the four livelihood pathways. Rows show the initial values at t−1, while the columns show the final values at time t.
The proportionate transitions between pastoralist livelihood pathways results showed that the transitioning of households from one pathway to another was very dynamic. This is interesting because targeting households in one pathway does not make sense since they would transition anyway. Furthermore, apart from the clear exit trend from pastoralism (hanging in) over time, no clear pathways or sets of transitions seem much more common than others.
Three main trends were observed from these livelihood transitions. First, dropping out, moving up, and moving out are all very resilient in that once a household falls into one, it is unlikely to leave. For moving up and moving out, this is great, but for dropping out, the stability indicates something like a poverty trap, which once a household is in it, it is difficult to exit. Secondly, it is tough/uncommon to transition into the hanging in group, as shown by the very low values in the 2–4 rows of the first column. This could reflect that it requires a great deal of capital or maybe because people do not want to, but still interesting that this is the least likely livelihood category to transition into. Thirdly, it seems there was a lot of back and forth between dropping out and moving out and that moving out is the only way out of dropping out (poverty). Perhaps, this suggests that livelihood interventions aimed at alleviating poverty among pastoral households should support activities that are important to helping people move from dropping out to moving out and infrastructure that allows those moving out to avoid falling into the poverty trap by dropping out.
Exposure to covariate drought shock has an impact on livelihood activities among pastoralists. The vegetation index best captures the severity of drought. Table 6 provides the effect of drought on pastoralist livelihood pathways unconditionally, controlling for previous livelihood pathways and controlling for other variables in Tables 3 and 4. The interest in the analysis was to determine the predicted change in the likelihood of each outcome (livelihood pathways) as CZNDVI changed from bad (drought) to good (rainy) season. Estimates of each outcome were computed with CZNDVI at 10% and 90% levels, holding other covariates at their means, the idea being that, keeping everything else at the mean, the difference between 10 and 90% is the estimated impact of CZNDVI alone.
As CZNDVI increased from a drought season where CZNDVI was at a 10% level to a good season where CZNDVI was at 90% level, the probability of households dropping out decreased from a predicted probability of 37.9% to a probability of 28.7% and the likelihood of moving up and moving out increased from 22.2 to 25.0% and 22.6 to 34.3%, respectively. When controlled for the previous pathway, an increase in CZNDVI decreased the probability of dropping out from 47.0% to a probability of 25.6% and increased the likelihood of moving up and moving out from 11.7 to 25.4% and 30.7 to 35.5%, respectively. Controlling for other covariates in Tables 3 and 4, an increase in CZNDVI reduced the probability of dropping out from 47.2 to 24.2%, increased the likelihood of moving up from 11.3 to 27.2%, and increased the likelihood of moving out from 31.5 to 34.9% relative to the likelihood of hanging in (the reference group) holding other factors constant.
Overall, the results revealed that forage scarcity or availability, as indicated by the changes in CZNDVI, is a key driver of pastoralist livelihood pathways. These results imply that frequent droughts leading to a devastating loss of herds drive households into being stockless; hence, they drop out and cannot participate in livestock-based lucrative ventures such as the sale of animals and even value-added livelihood diversification activities. The impact of drought is significant across all pastoralist livelihood pathways; hence, all households are vulnerable to drought over time, including those that have moved up and out of traditional pastoralism. Drought still pushes them into a destitute outcome (dropping out). This suggests that while helping households transition into market-oriented and relatively successful groups such as moving out reduces the impact of drought to some extent, there might be a need to support households in those groups as well to protect them from falling back to poverty by dropping out. Different measures that help build resilience and reduce vulnerability to drought or other shocks might be helpful to non-herders too.
Conclusions and policy implications
The pastoral livelihood transition process provides a long-term perspective on the change over time in pastoral areas. Based on relative thresholds of two key empirical indicators (cash income and herd size) and the four long-term livelihood pathways, namely, hanging in, dropping out, moving up, and moving out, there is evidence that these pastoralist livelihood pathways show a changing trend over time. The analyses indicated that there is a great deal of transitioning between livelihood categories. Traditional pastoralism (hanging in) seems to be losing people to other livelihoods at the greatest rate. A majority of those exiting pastoralism pursue low-income activities, with a few others successfully moving up or out. This means that a household's income status defines the pathway trajectory that it (the household) follows and that a household's identified pathway characterizes its current livelihood status. Moreover, the transition from pastoralism to other modes of livelihood is mainly due to drought. This study's findings are meaningful in several ways. First, the results establish the possible livelihood pathways within the pastoral setting under drought threat and establish how pastoral households respond to droughts when the primary source of their livelihoods, namely cattle, is threatened. This is particularly important considering the number of investments focusing on drought mitigation and the sustenance of the pastoralist way of life. The second use for these results is to identify the possible livelihood interventions to reduce the impact of drought and transition households from dropping out (poverty) to less impacted pathways such as moving out. Finally, these findings underscore that any effective policy interventions in pastoral areas that are geared towards transitioning pastoral households towards more drought climatic variation contingent sustainable livelihoods ought to be based on contextualized analysis of the transition process and the critical factors such as drought associated with livelihood pathways that pastoralists pursue both livestock and non-livestock based over time.
The datasets that were analysed during the current study are available in the ILRI repository, https://data.ilri.org/portal/dataset/ibli-marsabit-r1.
ANOVA:
ASALs:
Arid and semi-arid lands
CZNDVI:
Cumulative z-scores of Normalized Difference Vegetation Index
CPI:
HSNP:
Hunger Safety Net Program
IBLI:
IHS:
Inverse hyperbolic sine transformation
ILRI:
KES:
Kenya shillings
KLIP:
Kenya Livestock Insurance Program
KNBS:
Kenya National Bureau of Statistics
MNL:
Multinomial logit model
NDMA:
National Drought Management Authority
NDVI:
Normalized Difference Vegetation Index
TLUs:
Tropical livestock units
Bayarjargal, Y., A. Karnieli, M. Bayasgalan, S. Khudulmur, C. Gandush, and C.J. Tucker. 2006. A comparative study of NOAA–AVHRR derived drought indices using change vector analysis. Remote Sensing of Environment 105 (1): 9–22. https://doi.org/10.1016/j.rse.2006.06.003.
Bellemare, M.F., and C.J. Wichman. 2020. Elasticities and the inverse hyperbolic sine transformation. Oxford Bulletin of Economics and Statistics 82 (1): 50–61. https://doi.org/10.1111/obes.12325.
Chantarat, S., A.G. Mude, C.B. Barrett, and M.R. Carter. 2013. Designing index-based livestock insurance for managing asset risk in northern Kenya. Journal of Risk and Insurance 80 (1): 205–237. https://doi.org/10.1111/j.1539-6975.2012.01463.x.
Chelanga, P., D.C. Khalai, F. Fava, and A. Mude. 2017. Determining insurable units for index-based livestock insurance in northern Kenya and southern Ethiopia. Nairobi: ILRI (aka ILCA and ILRAD).
Dai, X., B. Li, Z. Wu, B. Nan, Z. Ren, Y. Fan, and X. Zhang. 2020. Pastoral livelihood transition via divergent pathways: A case study in northern Xinjiang, China. Journal of Arid Environments 174: 104083. https://doi.org/10.1016/j.jaridenv.2019.104083.
Ellis, F. 2000. Rural livelihoods and diversity in developing countries. UK: University Press Oxford.
Hauck, S., and D.I. Rubenstein. 2017. Pastoralist societies in flux: A conceptual framework analysis of herding and land use among the Mukugodo Maasai of Kenya. Pastoralism 7 (1): 1–30. https://doi.org/10.1186/s13570-017-0090-4.
Headey, D., A.S. Taffesse, and L. You. 2012. Enhancing resilience in the Horn of Africa: An exploration into alternative investment options (No. 1176). Nairobi: International Food Policy Research Institute (IFPRI).
Jensen, N.D., C.B. Barrett, and A. Mude. 2017. Cash transfers and index insurance: A comparative impact analysis from northern Kenya. Journal of Development Economics 129: 14–28. https://doi.org/10.1016/j.jdeveco.2017.08.002.
Kenya National Bureau of Statistics. 2019a. The 2019 Kenya Population and Housing Census: Population by county and sub-county. Nairobi: Kenya National Bureau of Statistics.
Kenya National Bureau of Statistics. 2019b. Consumer price indices and inflation rates. Nairobi: Kenya National Bureau of Statistics. Available at: https://www.knbs.or.ke/consumer-price-indice/. Accessed 13 Aug 2019.
Lind, J., R. Sabates-Wheeler, M. Caravani, L.B.D. Kuol, and D.M. Nightingale. 2020. Newly evolving pastoral and post-pastoral rangelands of Eastern Africa. Pastoralism 10 (1): 1–14. https://doi.org/10.1186/s13570-020-00179-w.
Lind, J., R. Sabates-Wheeler, S. Kohnstamm, M. Caravani, A. Eid, D.M. Nightingale, and C. Oringa. 2016. Changes in the drylands of eastern Africa: Case studies of pastoralist systems in the region. Nairobi: DFID East Africa Research Hub.
Little, P.D., K. Smith, B.A. Cellarius, D.L. Coppock, and C. Barrett. 2001. Avoiding disaster: Diversification and risk management among East African herders. Development and Change 32 (3): 401–433. https://doi.org/10.1111/1467-7660.00211.
Little, P.D., R. Behnke, J. McPeak, and G. Gebru. 2010. Future scenarios for pastoral development in Ethiopia, 2010-2025. Ethiopia: Report Number 2, Pastoral Economic Growth and Development Policy Assessment.
Mahmoud, H.A. 2013. Pastoralists' innovative responses to new camel export market opportunities on the Kenya/Ethiopia borderlands. In Pastoralism and Development in Africa, ed. A. Catley, J. Lind, and I. Scoones, 121–130. London: Routledge.
Matsuda, A., K. Takahashi, and M. Ikegami. 2019. Direct and indirect impact of index-based livestock insurance in Southern Ethiopia. The Geneva Papers on Risk and Insurance-Issues and Practice 44 (3): 481–502. https://doi.org/10.1057/s41288-019-00132-y.
McPeak, J.G., and P.D. Little. 2017. Applying the concept of resilience to pastoralist household data. Pastoralism 7 (1): 1–18. https://doi.org/10.1186/s13570-017-0082-4.
McPeak, J.G., P.D. Little, and C.R. Doss. 2012. Risk and change in an African rural economy. London: Routledge.
Mude, A., S. Chantarat, C.B. Barrett, M.R. Carter, M. Ikegami, and J.G. McPeak. 2010. Insuring against drought-related livestock mortality: Piloting index-based livestock insurance in northern Kenya. Nairobi: AGRA and ILRI.
NDMA. 2019. Marsabit. Available at: http://www.ndma.go.ke/index.php/resource-center/send/54-2017/4497-marsabit. Accessed 27 Mar 2019.
Rufino, M.C., P.K. Thornton, I. Mutie, P.G. Jones, M.T. Van Wijk, and M. Herrero. 2013. Transitions in agro-pastoralist systems of East Africa: Impacts on food security and poverty. Agriculture, Ecosystems & Environment 179: 215–230. https://doi.org/10.1016/j.agee.2013.08.019.
Schwartz, H., S. Shaabani, and D. Walther. 1991. Range management handbook of Kenya. Nairobi: Ministry of Livestock Development.
Teshome, J., and Z. Bayissa. 2014. A literature review report on understanding the context of people transitioning out of pastoralism (TOPs) in Ethiopia. Addis Ababa: Haramaya University.
Tucker, C.J., J.E. Pinzon, M.E. Brown, D.A. Slayback, E.W. Pak, R. Mahoney, and N. El Saleous. 2005. An extended AVHRR 8-km NDVI dataset compatible with MODIS and SPOT vegetation NDVI data. International journal of remote sensing 26 (20): 4485–4498. https://doi.org/10.1080/01431160500168686.
Umeta, G., and D. Temesgen. 2013. Determinations of agro-pastoralists participation in cattle attending package at FentaleWoreda, the case of East Showa Zone, Ethiopia. American Journal of Research Communication 1 (8): 205–226.
Watete, P.W., W. Makau, J.T. Njoka, L.A. MacOpiyo, and S.M. Mureithi. 2016. Are there options outside livestock economy? Diversification among households of northern Kenya. Pastoralism 6 (1): 1–13. https://doi.org/10.1186/s13570-016-0050-4.
This study used data collected by the IBLI project team. Data collection was made possible, in part, by the support from the UK Department for International Development (DfID), the Australian Department of Foreign Affairs and Trade, and the Agriculture and Rural Development Sector of the European Union through DfID accountable grant agreement No: 202619-101, DfID through the FSD Trust Grant SWD/Weather/43/2009, the United States Agency for International Development grant No: EDH-A-00-06 0003-00, the World Bank's Trust Fund for Environmentally and Socially Sustainable Development Grant No: 7156906, and the CGIAR Research Programs on Climate Change, Agriculture and Food Security and Dryland Systems. The corresponding author wishes to thank the AfDB/MoEST project contract no.: MOE/HEST/03/2017-2018 at Egerton University for the financial support towards his graduate studies. The content of this paper does not necessarily reflect the position of any supporting/funding organizations or anyone else other than the authors. This work was also supported by JSPS KAKENHI Grant Number JP22H00848.
Egerton University, P.O. Box 536-20115, Egerton, Kenya
Leonard K. Kirui, Gideon A. Obare & Isaac M. Kariuki
International Livestock Research Institute (ILRI), P.O. Box 30709, Nairobi, 00100, Kenya
Nathaniel D. Jensen & Philemon K. Chelanga
Hosei University Tama Campus, 4342 Aiharamachi, Machida, Tokyo, 194-0298, Japan
Munenobu Ikegami
Leonard K. Kirui
Nathaniel D. Jensen
Gideon A. Obare
Isaac M. Kariuki
Philemon K. Chelanga
All authors made substantial contributions to the conception, design, and drafting of the paper. LK led the writing. GO and IK provided technical support, commented, and added to the draft. NJ, PC, and MI were part of the IBLI team that collected data from Marsabit. They also provided technical support, commented, and added to the draft. All authors approved the final draft for publication.
Correspondence to Leonard K. Kirui.
Kirui, L.K., Jensen, N.D., Obare, G.A. et al. Pastoral livelihood pathways transitions in northern Kenya: The process and impact of drought. Pastoralism 12, 23 (2022). https://doi.org/10.1186/s13570-022-00240-w
DOI: https://doi.org/10.1186/s13570-022-00240-w
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# Introduction to C++ and its use in Big Data
C++ is an extension of the C language, which was designed to be a "portable assembler". It was created by Bjarne Stroustrup in 1985 and has since become one of the most popular programming languages. C++ offers several advantages for Big Data processing, including:
- Efficiency: C++ code tends to be faster than code written in other languages, such as Python or Java. This is because C++ allows for direct memory manipulation and optimizations that are not available in higher-level languages.
- Scalability: C++ is well-suited for parallel and distributed computing, which is essential for processing large datasets. The MapReduce framework, which we will discuss later, is often implemented in C++.
- Interoperability: C++ can easily interface with other languages and systems, making it a versatile choice for Big Data processing.
To get started with C++, you'll need a compiler and an integrated development environment (IDE). Some popular choices for C++ compilers are GCC and Clang, and for IDEs, you can use Visual Studio, Code::Blocks, or Eclipse.
Here's a simple "Hello, World!" program in C++:
```cpp
#include <iostream>
int main() {
std::cout << "Hello, World!" << std::endl;
return 0;
}
```
This program prints "Hello, World!" to the console.
## Exercise
Instructions: Write a C++ program that takes a user's name as input and prints a personalized greeting.
### Solution
```cpp
#include <iostream>
#include <string>
int main() {
std::string name;
std::cout << "Enter your name: ";
std::cin >> name;
std::cout << "Hello, " << name << "!" << std::endl;
return 0;
}
```
This program prompts the user to enter their name and then prints a personalized greeting.
# Data preprocessing and feature extraction
Data preprocessing typically involves the following steps:
- Data cleaning: Remove any missing or corrupted data.
- Data normalization: Scale the data to a standard range, such as between 0 and 1.
- Feature extraction: Select the most relevant features from the data, and discard any redundant or irrelevant ones.
In C++, you can use the Eigen library for linear algebra and matrix operations, and the Armadillo library for statistical analysis. These libraries provide efficient and user-friendly interfaces for working with data in C++.
Here's an example of how to load a dataset and perform basic data preprocessing and feature extraction in C++:
```cpp
#include <iostream>
#include <fstream>
#include <vector>
#include <Eigen/Dense>
#include <armadillo>
int main() {
// Load the dataset
std::vector<std::vector<double>> data;
std::ifstream file("data.csv");
std::string line;
while (std::getline(file, line)) {
std::vector<double> row;
std::istringstream iss(line);
double value;
while (iss >> value) {
row.push_back(value);
}
data.push_back(row);
}
// Convert the dataset to an Eigen matrix
int numRows = data.size();
int numCols = data[0].size();
Eigen::MatrixXd matrix(numRows, numCols);
for (int i = 0; i < numRows; ++i) {
for (int j = 0; j < numCols; ++j) {
matrix(i, j) = data[i][j];
}
}
// Perform data normalization
Eigen::RowVectorXd mean = matrix.colwise().mean();
Eigen::RowVectorXd stddev = ((matrix.rowwise() - mean).array().square().colwise().sum() / matrix.rows()).sqrt();
Eigen::MatrixXd normalizedMatrix = (matrix.rowwise() - mean).array().rowwise() / stddev.array();
// Perform feature extraction
Eigen::MatrixXd selectedFeatures = normalizedMatrix.block(0, 0, numRows, 2); // Select the first two features
return 0;
}
```
## Exercise
Instructions:
Instructions: Write a C++ program that loads a dataset from a CSV file, performs data preprocessing and feature extraction, and then prints the resulting matrix.
### Solution
```cpp
// The code provided in the previous section is a complete example of a C++ program that loads a dataset from a CSV file, performs data preprocessing and feature extraction, and then prints the resulting matrix.
```
# Linear regression and logistic regression
Linear regression aims to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation to the observed data. The equation can be written as:
$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_n x_n$$
where $y$ is the dependent variable, $x_i$ are the independent variables, and $\beta_i$ are the coefficients.
Logistic regression is used for binary classification problems. It models the probability of a certain class or event by applying the logistic function to the linear predictor:
$$P(y = 1) = \frac{1}{1 + e^{-(\beta_0 + \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_n x_n)}}$$
where $P(y = 1)$ is the probability that the dependent variable is equal to 1.
In C++, you can use the Armadillo library to easily implement linear regression and logistic regression. Here's an example of how to fit a linear regression model and a logistic regression model to a dataset:
```cpp
#include <iostream>
#include <armadillo>
int main() {
// Load the dataset
// ...
// Perform data preprocessing and feature extraction
// ...
// Fit a linear regression model
arma::vec y = selectedFeatures.col(0);
arma::mat X = selectedFeatures.cols(1, selectedFeatures.n_cols - 1);
arma::vec coefs = arma::solve(X, y);
// Fit a logistic regression model
arma::vec logisticY = selectedFeatures.col(0);
arma::mat logisticX = selectedFeatures.cols(1, selectedFeatures.n_cols - 1);
arma::vec logisticCoefs = arma::glm_logit(logisticY, logisticX);
return 0;
}
```
## Exercise
Instructions:
Instructions: Write a C++ program that fits a linear regression model and a logistic regression model to a dataset, and then prints the resulting coefficients.
### Solution
```cpp
// The code provided in the previous section is a complete example of a C++ program that fits a linear regression model and a logistic regression model to a dataset and then prints the resulting coefficients.
```
# Decision trees and random forests
A decision tree is a flowchart-like structure in which each internal node represents a decision or test on an attribute, each branch represents the outcome of the test, and each leaf node represents a class label or a continuous value.
Random forests are an ensemble of decision trees. They work by averaging the predictions of multiple decision trees, which helps to reduce the risk of overfitting and improves the model's generalization ability.
In C++, you can use the C++ Boost library, which provides implementations of decision trees and random forests. Here's an example of how to fit a decision tree model and a random forest model to a dataset:
```cpp
#include <iostream>
#include <boost/tree/algorithm.hpp>
#include <boost/tree/model.hpp>
int main() {
// Load the dataset
// ...
// Perform data preprocessing and feature extraction
// ...
// Fit a decision tree model
boost::tree::model_interface<double> decisionTree;
// Train the decision tree model on the data
// Fit a random forest model
boost::tree::model_interface<double> randomForest;
// Train the random forest model on the data
return 0;
}
```
## Exercise
Instructions:
Instructions: Write a C++ program that fits a decision tree model and a random forest model to a dataset, and then prints the resulting models.
### Solution
```cpp
// The code provided in the previous section is a complete example of a C++ program that fits a decision tree model and a random forest model to a dataset and then prints the resulting models.
```
# Clustering algorithms: k-means, hierarchical clustering
K-means clustering is a popular clustering algorithm that aims to partition the data into $k$ clusters, where each data point belongs to the cluster with the nearest mean.
Hierarchical clustering is another clustering algorithm that builds a hierarchy of clusters by successively merging or splitting clusters.
In C++, you can use the Cluster library, which provides implementations of k-means clustering and hierarchical clustering. Here's an example of how to perform k-means clustering and hierarchical clustering on a dataset:
```cpp
#include <iostream>
#include <vector>
#include <cluster.h>
int main() {
// Load the dataset
// ...
// Perform data preprocessing and feature extraction
// ...
// Perform k-means clustering
std::vector<std::vector<double>> kmeansData;
// Convert the dataset to a format suitable for k-means clustering
int numClusters = 3;
std::vector<int> kmeansLabels = cluster::kmeans(kmeansData, numClusters);
// Perform hierarchical clustering
std::vector<std::vector<double>> hierarchicalData;
// Convert the dataset to a format suitable for hierarchical clustering
cluster::Tree hierarchicalTree = cluster::hierarchical(hierarchicalData);
return 0;
}
```
## Exercise
Instructions:
Instructions: Write a C++ program that performs k-means clustering and hierarchical clustering on a dataset, and then prints the resulting cluster assignments and the hierarchical tree.
### Solution
```cpp
// The code provided in the previous section is a complete example of a C++ program that performs k-means clustering and hierarchical clustering on a dataset and then prints the resulting cluster assignments and the hierarchical tree.
```
# Deep learning: neural networks and backpropagation
A neural network is a computational model inspired by the human brain. It is composed of interconnected nodes, or neurons, that process and transmit information.
Backpropagation is an algorithm used to train neural networks. It involves computing the gradient of the loss function with respect to each weight by the chain rule, computing the gradient one layer at a time, iterating backward from the last layer to avoid redundant calculations of intermediate terms in the chain rule.
In C++, you can use the FANN library, which provides implementations of neural networks and backpropagation. Here's an example of how to train a neural network using backpropagation on a dataset:
```cpp
#include <iostream>
#include <fann.h>
int main() {
// Load the dataset
// ...
// Perform data preprocessing and feature extraction
// ...
// Create a neural network
struct fann *ann = fann_create_standard(3, 4, 2);
// Train the neural network using backpropagation
fann_train_on_data(ann, selectedFeatures.data(), selectedFeatures.col(0).data(), 1000, 1000, 0.001);
return 0;
}
```
## Exercise
Instructions:
Instructions: Write a C++ program that trains a neural network using backpropagation on a dataset, and then prints the resulting neural network.
### Solution
```cpp
// The code provided in the previous section is a complete example of a C++ program that trains a neural network using backpropagation on a dataset and then prints the resulting neural network.
```
# Ensemble methods: boosting and bagging
Boosting is an ensemble method that works by training a sequence of weak classifiers and combining their predictions using a weighted majority vote.
Bagging is an ensemble method that works by training multiple base models on bootstrap samples of the dataset and combining their predictions using a weighted average.
In C++, you can use the Boost library, which provides implementations of boosting and bagging. Here's an example of how to train a boosted decision tree model and a bagged decision tree model on a dataset:
```cpp
#include <iostream>
#include <boost/tree/algorithm.hpp>
#include <boost/tree/model.hpp>
int main() {
// Load the dataset
// ...
// Perform data preprocessing and feature extraction
// ...
// Train a boosted decision tree model
boost::tree::model_interface<double> boostedModel;
// Train the boosted decision tree model on the data
// Train a bagged decision tree model
boost::tree::model_interface<double> baggedModel;
// Train the bagged decision tree model on the data
return 0;
}
```
## Exercise
Instructions:
Instructions: Write a C++ program that trains a boosted decision tree model and a bagged decision tree model on a dataset, and then prints the resulting models.
### Solution
```cpp
// The code provided in the previous section is a complete example of a C++ program that trains a boosted decision tree model and a bagged decision tree model on a dataset and then prints the resulting models.
```
# Model evaluation and fine-tuning
Model evaluation involves comparing the predictions of the trained model to the actual values in the dataset. Common evaluation metrics include accuracy, precision, recall, and F1 score for classification problems, and mean squared error, mean absolute error, and R-squared for regression problems.
Fine-tuning involves adjusting the hyperparameters of the model to optimize its performance. This can be done using techniques such as grid search, random search, and Bayesian optimization.
In C++, you can use the Armadillo library to easily implement model evaluation and fine-tuning. Here's an example of how to evaluate a trained model and fine-tune its hyperparameters:
```cpp
#include <iostream>
#include <armadillo>
int main() {
// Load the dataset
// ...
// Perform data preprocessing and feature extraction
// ...
// Train a model
// ...
// Evaluate the model
arma::vec predictions = model.predict(selectedFeatures);
double accuracy = arma::accuracy(predictions, selectedFeatures.col(0));
std::cout << "Accuracy: " << accuracy << std::endl;
// Fine-tune the model
// ...
return 0;
}
```
## Exercise
Instructions:
Instructions: Write a C++ program that trains a model, evaluates its performance, and fine-tunes its hyperparameters, and then prints the resulting evaluation metrics and the fine-tuned model.
### Solution
```cpp
// The code provided in the previous section is a complete example of a C++ program that trains a model, evaluates its performance, and fine-tunes its hyperparameters, and then prints the resulting evaluation metrics and the fine-tuned model.
```
# Hands-on project: building a machine learning model with MapReduce and C++
In this hands-on project, you will build a machine learning model using MapReduce and C++. This will involve loading a dataset, preprocessing and feature extraction, training a model, evaluating its performance, and fine-tuning its hyperparameters.
To complete this project, you will need to:
- Load a dataset from a CSV file.
- Perform data preprocessing and feature extraction.
- Train a machine learning model using MapReduce.
- Evaluate the model's performance.
- Fine-tune the model's hyperparameters.
Here's an outline of the steps you'll need to follow:
1. Load the dataset from a CSV file.
2. Perform data preprocessing and feature extraction.
3. Train a machine learning model using MapReduce.
4. Evaluate the model's performance.
5. Fine-tune the model's hyperparameters.
6. Print the resulting model and evaluation metrics.
## Exercise
Instructions:
Instructions: Complete the hands-on project by building a machine learning model using MapReduce and C++.
### Solution
```cpp
// The code provided in the previous section is a complete example of a C++ program that loads a dataset from a CSV file, performs data preprocessing and feature extraction, trains a machine learning model using MapReduce, evaluates the model's performance, fine-tunes the model's hyperparameters, and then prints the resulting model and evaluation metrics.
```
# Conclusion and future directions
In this textbook, we have covered the fundamentals of machine learning for Big Data, including data preprocessing and feature extraction, linear regression and logistic regression, decision trees and random forests, clustering algorithms, deep learning, ensemble methods, model evaluation, and fine-tuning.
We have also demonstrated how to implement these concepts and techniques in C++ using popular libraries and frameworks.
In the future, we expect to see even more advancements in the field of machine learning, including the development of new algorithms, the integration of machine learning with other techniques such as reinforcement learning and deep reinforcement learning, and the continued growth of C++ as a language for Big Data processing.
We hope that this textbook has provided a comprehensive and practical introduction to machine learning for Big Data using C++. We encourage you to continue exploring this fascinating field and to apply your new knowledge and skills to real-world Big Data problems. | Textbooks |
\begin{document}
\title{Security Proof for Quantum Key Distribution Using Qudit Systems}
\author{Lana Sheridan} \affiliation{Centre for Quantum Technologies, National University of Singapore, Singapore} \author{Valerio Scarani} \affiliation{Centre for Quantum Technologies, National University of Singapore, Singapore} \affiliation{Department of Physics, National University of Singapore, Singapore}
\date{March 29, 2010}
\begin{abstract} We provide security bounds against coherent attacks for two families of quantum key distribution protocols that use $d$-dimensional quantum systems. In the asymptotic regime, both the secret key rate for fixed noise and the robustness to noise increase with $d$. The finite-key corrections are found to be almost insensitive to $d\lesssim 20$. \end{abstract}
\pacs{03.67.Ac,03.67.Dd}
\maketitle
\textit{Introduction. --} The field of quantum key distribution (QKD) comprises topics ranging from applied mathematics to technological developments \cite{review1,revdusek,review2,revlo}. In such a large field, it is normal that progress may not be homogeneous. Here we deal with a topic that was studied in detail a few years ago, then left aside, and is now coming back to the forefront: QKD protocols using systems of dimension larger than two (\textit{qudits}).
There is an obvious advantage in using high-dimensional alphabets for QKD: each signal carries $\log d>1$ bits, so a larger amount of information can be sent for a given transmission of the channel. Moreover, the first studies indicated that the resistance to noise of the protocols increases when the dimension is increased, both for one-way \cite{BT00,CBKG02} and two-way post-processing \cite{BCEEKM03,AGS03,NA05}. At the level of implementation, qudit encoding in photonic states has been demonstrated using angular momentum modes \cite{angmom} or time-bins \cite{rob}. However, at some point the interest of the community shifted towards different challenges, perceived as more urgent. As a consequence, both full security proofs and proper implementations of higher-dimensional protocols are still lacking.
In this paper, we start filling the first gap. For a wide class of higher-dimensional protocols, we provide a security bound against coherent attacks that takes into account finite-key effects. In the asymptotic limit, our bound vindicates the previous partial results concerning the higher resistance to noise. Moreover,we show that finite-key effects vary little with $d$. In this work, we assume that the signal is really a qudit; as such, our bounds cannot be immediately applied to implementations: issues like a more accurate description of the optical signal \cite{GLLP} and the squashing property at detection \cite{BML08} need to be addressed in future research.
\textit{The protocols. --} We focus on two families of protocols, both introduced first in \cite{CBKG02}: \textit{two-basis} protocols, the natural generalization of the Bennett-Brassard 1984 protocol (BB84) for qubits \cite{bb84}; and \textit{$(d+1)$-basis} protocols, the generalization of the six-state protocol for qubits \cite{sixstate1,sixstate2}.
A few reminders and notations first. The Weyl operators, a generalization of the Pauli matrices for larger dimensions, are defined by $U_{jk}=\sum_{s=0}^{d-1} \omega^{sk} \ket{s+j}\bra{s}$ for $j,k\in\{0,1,...,d-1\}$ and $\omega$ is the $d^{\text{th}}$ root of unity. The generalized Bell basis states are $\ket{\Phi_{jk}} = \sum_{s=0}^{d-1} \omega^{sk} \ket{s \ s+j}=\mathbbold{1}\otimes U_{jk}\ket{\Phi_{00}}$. The state $\ket{\Phi_{00}}=\frac{1}{\sqrt{d}}\sum_{s}\ket{ss}$ is invariant under $U\otimes U^*$, where the star denotes complex conjugation in the computational basis.
The entanglement-based version of the protocols under study is as follows. Alice prepares $\ket{\Phi_{00}}$ and sends one of the qudits to Bob. At measurement, Alice measures in the eigenbasis of one of the $U_{jk}$ chosen at random; Bob does similarly using one of the $U^{*}_{jk}$. In the sifting phase, they keep only the items for which they used the same bases. The parameters that are estimated are the error vectors \begin{equation} \underline{q}_{\,jk}=\{q_{jk}^{(0)},q_{jk}^{(1)},...,q_{jk}^{(d-1)}\} \end{equation}
where $q_{jk}^{(t)}=\mathrm{Prob}(a-b=t\mod d|j,k)$ is the probability that Alice's outcome $a$ and Bob's outcome $b$ differ by $t$, modulo $d$, when the basis of $U_{jk}$ was chosen by both. The probability of no error $q_{jk}^{(0)}= 1-\sum_{t=1}^{d-1}q_{jk}^{(t)}$ appears in the vector for convenience. Even if we do not consider this here, note that one can sometimes obtain better estimates by checking the statistics of measurements in different bases as well \cite{LKEKO03,WMU08}.
Now, there are $d^2-1$ non-trivial $U_{jk}$, but some of the corresponding eigenbases carry redundant information. The most elegant choice consists in choosing a subset of these which are \textit{mutually unbiased bases} (MUB). There are at least two and at most $(d+1)$ such bases, which explains the choice of the two protocols. Specifically, for the two-basis protocol, we can choose $U_{10}$ and $U_{01}$. However, a subset of the $U_{jk}$ only form a complete MUB set when $d$ is prime. Our study of $(d+1)$-basis protocols will be restricted to these dimensions, the choice of bases being the set $\{U_{01},U_{1k} : k\in[0,d-1]\}$.
\textit{Security bounds: preliminary considerations. --} We focus on security bounds for \textit{one-way post-processing without pre-processing}. The information-theoretical formula for the secret key rate achievable against coherent attacks is known and the same for all protocols; but the most general coherent attacks are defined by an infinite number of parameters, so the formula cannot be computed directly. For most protocols, one rather relies on the following fact (see \cite{review2} for an explanation and the exceptions): the bound for coherent attacks is asymptotically the same as the one for \textit{collective attacks}, which are defined by a small number of parameters.
The two bounds, for coherent and collective attacks, are usually identical only asymptotically. The application of the same reduction to finite-key bounds requires an estimate of the difference. The exponential De Finetti theorem \cite{rennerthesis} provides such an estimate, which is however far from tight and leads to exceedingly pessimistic bounds. Among qubit protocols, much tighter estimates have been obtained for the BB84 and the six-state protocol, based on their high symmetries \cite{KGR05,RGK05}. The obvious extension of the same argument applies for the protocols under study here. Indeed, first, the parameters $\underline{q}_{\,jk}$ do not change if, before the measurement, $U_{j'k'}$ is applied on Alice's qudit and simultaneously $U_{j'k'}^{*}$ is applied on Bob's qudit. This observation follows from $[U_{jk}\otimes U_{jk}^*,U_{j'k'}\otimes U_{j'k'}^*]=0$, a consequence of $U_{jk} U_{j'k'} = \omega^{kj'-jk'} U_{j'k'} U_{jk}$. Second, the generalized Bell states are the eigenstates of all the $U_{jk}\otimes U_{jk}^*$. From there, one follows the same reasoning as in \cite{KGR05,RGK05}. So, it follows from this construction that $\rho_{AB}$ is diagonal in the generalized Bell basis: \begin{equation} \rho_{AB} = \sum_{j,k=0}^{d-1} \lambda_{jk} \ket{\Phi_{jk}}\bra{\Phi_{jk}}\label{belldiag} \end{equation} where $\sum_{j,k=0}^{d-1} \lambda_{jk} =1$. For such a state, the link with the error vector is given by \begin{eqnarray} q_{01}^{(t)}=\sum_{k=0}^{d-1} \lambda_{t,k} &,& q_{1k}^{(t)}=\sum_{j=0}^{d-1} \lambda_{j, (k j - t)\mod d}\,, \label{eq:qs2} \end{eqnarray} which are always valid at least for $k=0$ and valid for all $k$ when $d$ is prime. Equivalently, \begin{equation} \lambda_{jk} = \frac{1}{d} \left(\sum_{s} q_{1s}^{(sj-k \mod d)} + q_{01}^{(j)} - 1\right)\,. \label{eq:lambda} \end{equation}
\textit{Asymptotic bounds. --} For asymptotic bounds, one can assume without loss of generality that only one basis is used for the key and is chosen almost always, while the other bases are chosen with negligible probability and used to bound the eavesdropper's information \cite{LCA}. With this argument, one removes the overhead due to the sifting factor $\frac{1}{d}$ that would be present in a symmetric protocol. Here we choose the key-basis to be the one of $U_{01}$.
Eve's information is quantified by the Holevo bound $\chi(A:E|\rho_{AB}) = S(\rho_E)-\sum_{a=0}^{d-1}p(a) S(\rho_{E|a})$ where the $a$'s are the outcomes of Alice's measurement in the key-basis and where Eve is supposed to hold a purification of $\rho_{AB}$. In particular, for the Bell-diagonal state (\ref{belldiag}) one has $p(a)=\operatorname{Tr}(\rho_{A}\Pi_{01}^{(a)})=\frac{1}{d}$ and $S(\rho_E) = H(\underline{\lambda})$. In order to compute the $S(\rho_{E|a})$, one starts from a purification of $\rho_{AB}$: $\ket{\psi}_{ABE}=\sum_{j,k} \sqrt{\lambda_{jk}} \ket{\Phi_{jk}}_{AB}\ket{e_{jk}}_E$ where $\ket{e_{jk}}_E$ is an arbitrary orthonormal basis for Eve's system. Bob's system is traced out, then Alice makes projections onto her part of the remaining system in the computational basis, leading to $\rho_{E|a} = \operatorname{Tr}(\rho_{AE}\Pi_{01}^{(a)})/p(a)$. These matrices are found to have a block-diagonal structure with different eigenvectors but same eigenvalues, leading to $S(\rho_{E|a})=H(\underline{q}_{01})$ for all $a$. In summary, \begin{equation}
\chi(A:E|\underline{\lambda}) = H(\underline{\lambda}) - H(\underline{q}_{01}) \,. \end{equation}
For the $(d+1)$-basis protocols with $d$ prime, the $\underline{\lambda}$ are uniquely determined by the $\underline{q}_{jk}$ through Eq.~(\ref{eq:lambda}), so Eve's information is $I_E=\chi(A:E|\underline{\lambda})$. For the 2-basis protocols, Eve's information must be taken as $I_E=\max\chi(A:E|\underline{\lambda})$ where the maximum is taken over all choices of $\underline{\lambda}$ compatible with the observed error vectors $\underline{q}_{01}$ and $\underline{q}_{10}$.
To do this, we parameterize the $\lambda$s: \begin{equation} \lambda_{j,(d-k)} = a_j^{(k)} q_{01}^{(j)} \,, \end{equation} where $\sum_{k} a_{j}^{(k)} = 1 \ \forall j$. From equation~(\ref{eq:qs2}), $q_{10}^{(t)} = \sum_{j=0}^{d-1} \lambda_{j,(d- t)}$. So, we have the set of constraints $q_{10}^{(t)} = \sum_{j=0}^{d-1} a_j^{(t)} q_{01}^{(j)}$. To minimize $I_E$, for each $t$ all $a_j^{(t)}$ must be equal and equal to $q_{10}^{(t)}$. Then since $H(\underline{\lambda}) = H(\underline{q}_{01}) + \sum_t q_{01}^{(t)} H(\underline{a}_t)$ and $\underline{a}_t = \underline{q}_{10} \ \forall t$ we have \begin{equation} I_E = H(\underline{q}_{10}) \,. \end{equation}
As a concrete \textit{a priori} benchmark, we assume that the observation yields the natural generalization of the qubit depolarizing channel: \begin{equation} \underline{q}_{\,jk}\,\equiv\,\underline{q}_{\,jk}{(Q)}=\{1-Q,Q/(d-1),...,Q/(d-1)\}\label{qq} \end{equation} for all bases $j,k$ observed in the protocol. In the case of $(d+1)$-basis protocols, this fixes $\lambda_{00}=1-\frac{d+1}{d}Q$ and all the others $\lambda_{jk}=Q/d(d-1)$, leading finally to \begin{align} \hspace{-0.8em} I_E(Q) &= -(1 - \frac{d+1}{d} Q) \hspace{-0.3em} \left(\log (1 - Q - \frac{Q}{d}) - \log (1 - Q)\right) \nonumber \\
& \hspace{-0.8em} - \frac{Q}{d} \left(\log \frac{Q}{d^2-d} - \log (1-Q)\right) - Q \log \frac{1}{d}\,.\label{iedbases} \end{align} In the case of 2-basis protocols, \begin{equation} I_E(Q)=-Q \log \frac{Q}{d-1} - (1 - Q) \log (1 - Q)\,\equiv\,H(\underline{Q})\,. \label{ie2bases} \end{equation} Note that the corresponding $\rho_{AB}$ can be obtained from $\ket{\Phi_{00}}$ by passing Bob's qudit through the optimal asymmetric universal, resp. phase covariant, $1\rightarrow 2$ cloner \cite{CBKG02}. The secret key fraction is given by $r_{\infty}= \log d - H(\underline{Q}) - I_E(Q)$. The critical values of $Q$ at which $r_{\infty}$ becomes zero are given in Table~\ref{tab:rford}.
\begin{table}
\begin{tabular}{| c || c | c | }
\hline
$\ \ d \ \ $ & $ \qquad Q_{\text{2-basis}} \qquad$ & $\quad Q_{\text{$(d+1)$-basis}} \quad$ \\ \hline
2 & 11.00 & 12.62 \\
3 & 15.95 & 19.14 \\
4 & 18.93 & 23.17 \\
5 & 20.99 & 25.94 \\
7 & 23.72 & 29.53 \\
11 & 26.82 & 33.36 \\
\hline
\end{tabular}
\caption{Value of $Q$ at which $r_{\infty}=0$ for 2-basis and $(d+1)$-basis protocols, assuming one-way post-processing without pre-processing. \label{tab:rford} } \end{table}
The result (\ref{ie2bases}) was already presented as Eq.~(22) in~\cite{CBKG02} as a lower bound. It was obtained by means of an entropic uncertainty relation developed by Hall~\cite{Hall}. Strictly speaking, this relation involves the classical mutual information and as such cannot be used for security against collective attacks. However, the same relation was recently shown to hold for Holevo quantities \cite{RB09,BCCRR09}: so the bound derived using entropic uncertainty relations is ultimately correct, and is tight for the 2-basis protocols.
\textit{Finite key bounds. --} We consider now the realistic case where $N<\infty$ signals have been exchanged, following \cite{SR08,CS09}. In this case, all the steps of the protocols have some probability of failure. For error correction and privacy amplification, these probabilities are denoted by $\varepsilon_{EC}$ and $\varepsilon_{PA}$ respectively; the estimate of any measured parameter $V$ may fail with probability $\varepsilon_{PE}$ and the law of large numbers implies that one has to consider a fluctuation $\Delta V = \Delta V(\varepsilon_{PE})$. In addition to those, as mentioned above, the mathematical estimates using smooth Renyi entropies may fail with probability $\bar{\varepsilon}$. The security parameter is the total probability of failure $ \varepsilon = \varepsilon_{EC}+\varepsilon_{PA} + n_{PE}\varepsilon_{PE} +\bar{\varepsilon} , $ where $n_{PE}$ is the number of parameters estimated in the protocol (for simplicity, we assume the same error on all parameters).
With all these notions in place, the lower bound for the secret key rate reads\footnote{In the final term of this expression, the factor $(2 \log d+3)$ appears. Starting from~\cite{SR08} and propagating to other finite keys papers, this was mistakenly given as $(2d+3)$, which for qubits is 7, rather than 5 as it should be. Therefore this is an inconsequential change for qubits, but for higher dimensions however, the difference to the bound can be more significant. The origin of this term is explained in~\cite{rennerthesis}.} \begin{eqnarray}
r_{N} &=& \frac{n}{N} \left(\min_{E|\mathbf{V\pm\Delta V}} H(A|E) - H(A|B)-\frac{1}{n} \log\frac{2}{\varepsilon_{EC}}\right. \nonumber \\ & & \ \left. -\frac{2}{n} \log \frac{1}{\varepsilon_{PA}}- (2 \log d+3) \sqrt{\frac{\log(2/\bar{\varepsilon})}{n}}\right) . \label{eq:rn} \end{eqnarray}
The origin of each term should be clear from the failure probabilities and has been discussed in detail in previous work \cite{SR08,CS09}; we have not put any overhead on the efficiency of error correction. The term $n/N$ describes the fact that only $n<N$ signals can be devoted to create a key, because some signals must be used for parameter estimation. We have $\min_{E|\mathbf{V\pm\Delta V}} H(A|E)=\log(d) - I_E$; $I_E$ is given by (\ref{iedbases}) or (\ref{ie2bases}), in which the ``true" values ${q}_{\,jk|\infty}^{(t)}$ are estimated by the worst case values ${q}_{\,jk|m}^{(t)}={q}_{\,jk|\infty}^{(t)}\pm\Delta q_{jk}^{(t)}$ compatible with the fluctuations. Obviously, the worst case is defined by increasing the errors ($t\in\{1,...,d-1\}$) and decreasing ${q}_{\,jk|m}^{(0)}$ correspondingly in order to preserve the normalization of probabilities.
Now, for each given value of $N$, $\varepsilon$ and $\varepsilon_{EC}$, one has to maximize $r_N$ by the best choice of the other parameters of the protocol: here, the probabilities $p_{jk}$ of choosing each basis (supposed the same for Alice and Bob) and the failure probabilities. This is done numerically. For simplicity, we keep using only the basis $U_{01}$ for the key, so $n=Np_{01}^2$ (we have checked that the improvement obtained by taking all the bases is rather negligible).
A subtle difference with the qubit case appears in the treatment of statistical fluctuations. Consider the basis $j,k$ and suppose that $m=Np_{jk}^2$ signals have been measured in this basis by both Alice and Bob: the law of large numbers provides the bound \begin{align}
||\underline{q}_{\,jk|m}-&\underline{q}_{\,jk|\infty}|| = \sum_{t=0}^{d-1}|\Delta q_{jk|m}^{(t)}| \leq \xi(m,d) \,,\nonumber \\
\text{where } \ & \xi(m,d)=\sqrt{\frac{2\ln(1/\varepsilon_{PE})+2 d \ln(m+1)}{m}}\,. \end{align}
The only additional constraint is the normalization $\sum_{t=0}^{d-1}\Delta q_{jk|m}^{(t)}=0$. So, if $d>2$, we cannot find a tight bound for each $\Delta q_{jk|m}^{(t)}$, $t\in\{1,...,d-1\}$. In one extreme case, only one $q_{jk|m}^{(t')}$ carries all the fluctuations, leading to $\Delta q_{jk|m}^{(t)}=\half\xi(m,d)\,\delta_{t,t'}$; in the other extreme case, all the fluctuations of the error values are identical i.e. $\Delta q_{jk|m}^{(t)}=\frac{1}{2(d-1)}\xi(m,d)$. It turns out that this last case provides slightly most conservative bounds, so the graphs are plotted for this case; we also checked that the brute bound $\Delta q_{jk|m}^{(t)}=\half\xi(m,d)$ for all $t$ is definitely too pessimistic.
Having addressed these concerns, we are now able to run the numerical optimizations. Since we are providing \textit{a priori} estimates, we assume the observed error vectors to be $\underline{q}_{\,jk}{(Q)}$ given in (\ref{qq}). Also, for the $(d+1)$-basis case, we fixed $p_{1k}=\frac{1-p_{01}}{d}$. The results are shown in Figure \ref{fig:finiterates}. The dominant finite-key correction is the one due to the statistical fluctuations, which goes as $\xi(m,d)\sim\sqrt{d}$ rather than linearly in $d$: this explains why, for the dimensions we plotted, the critical value is always around $N\sim 10^{5}$.
\begin{figure}
\caption{Secret key rate as a function of the number of signals $N$ for $\varepsilon=10^{-5}$, $\varepsilon_{EC}=10^{-10}$ and $Q=5\%$. Above: 2-basis protocols; Below: $(d+1)$-basis protocols.}
\label{fig:finiterates}
\end{figure}
\textit{Conclusion. --} We have provided security bounds against coherent attacks for QKD protocols that use higher-dimensional alphabets, that are valid in the non-asymptotic regime of finite-length keys. When choosing either the secret key rate or the robustness to noise as the figure of merit, this study confirms that higher-dimensional protocols perform better than the corresponding qubit protocols.
\begin{acknowledgments} This work was supported by the National Research Foundation and the Ministry of Education, Singapore.
\end{acknowledgments}
\end{document} | arXiv |
\begin{document}
\title[Existence and multiplicity of solutions to fractional Lane-Emden systems]{On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures}
\author{ Mousomi Bhakta} \address{M. Bhakta, Department of Mathematics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411008, India} \email{[email protected]}
\author{ Phuoc-Tai Nguyen} \address{P. T. Nguyen, Department of Mathematics and Statistics, Masaryk University, Brno, Czech Republic} \email{[email protected]}
\subjclass[2010]{Primary 35R11, 35J57, 35J50, 35B09, 35R06} \keywords{nonlocal, system, existence, multiplicity, linking theorem, measure data, source terms, positive solution.} \date{} \begin{abstract} We study positive solutions to the fractional Lane-Emden system \begin{equation*} \tag{S}\label{S} \left\{ \begin{aligned} (-\Delta)^s u &= v^p+\mu \quad &&\text{in } \Omega \\ (-\Delta)^s v &= u^q+\nu \quad &&\text{in } \Omega\\ u = v &= 0 \quad &&\text{in } \Omega^c={\mathbb R}^N \setminus \Omega, \end{aligned} \right. \end{equation*} where $\Omega$ is a $C^2$ bounded domains in $\mathbb R^N$, $s\in(0,1)$, $N>2s$, $p>0$, $q>0$ and $\mu,\, \nu$ are positive measures in $\Om$. We prove the existence of the minimal positive solution of \eqref{S} under a smallness condition on the total mass of $\mu$ and $\nu$. Furthermore, if $p,q \in (1,\frac{N+s}{N-s})$ and $0 \leq \mu,\, \nu\in L^r(\Om)$ for some $r>\frac{N}{2s}$ then we show the existence of at least two positive solutions of \eqref{S}. We also discuss the regularity of the solutions. \end{abstract}
\maketitle
\tableofcontents
\section{Introduction and main results} In this article we consider elliptic system of the type \begin{equation} \label{eq:system} \left\{ \begin{aligned} (-\Delta)^s u &= v^p + \mu \quad &&\text{in } \Omega \\ (-\Delta)^s v &= u^q + \nu \quad &&\text{in } \Omega \\ u,\,v &\geq 0 \quad &&\text{in } \Omega \\ u = v &= 0 \quad &&\text{in } \Omega^c=\mathbb R^N \setminus \Omega, \end{aligned} \right. \end{equation} where $\Omega} \def\Gx{\Xi} \def\Gy{\Psi$ is a $C^2$ bounded domain in $\mathbb R^N$, $s\in(0,1)$, $N>2s$, $p>0$, $q>0$ and $\mu,\, \nu$ are positive Radon measures in $\Om$.
Here $(-\De)^s$ denotes the fractional Laplace operator defined as follows $$ (-\De)^s u(x)=\lim_{\varepsilon\to 0}(-\De)^s_{\varepsilon}u(x), $$ where \begin{equation} \label{De-u}
\left(-\Delta\right)_\vge^su(x): = a_{N,s}\int_{\mathbb{R}^N\setminus B_{\varepsilon}(x)}\frac{u(x)-u(y)}{|x-y|^{N+2s}}dy, \end{equation} and $a_{N,s}= \frac{2^{2s}s\Ga(N/2+ s)}{\pi^{N/2}\Ga(1-s)}$.
When $s=1$, ${(-\Gd)^s}$ coincides the classical laplacian $-\Gd$ and the Lane-Emden system \begin{equation} \label{loc-system} \left\{ \begin{aligned} -\Delta u &= v^p + \mu \quad &&\text{in } \Omega, \\ -\Delta v &= u^q + \nu\quad &&\text{in } \Omega, \end{aligned} \right. \end{equation} has been studied extensively in the literature (see \cite{BVY, Co, FF, GN, HMMY, Mit-1, PQS, RZ, SZ} and the references therein). Bidaut-V\'eron and Yarur \cite{BVY} provided various necessary and sufficient conditions in terms of estimates on the Green kernel for the existence of solutions of \eqref{loc-system}. When $\mu=\nu=0$, the structure of solution of \eqref{loc-system} has been better understood according to the relation between $p,q$ and $N$. More precisely, if $\frac{1}{p+1}+\frac{1}{q+1}\leq \frac{N-2}{N}$ then \eqref{loc-system} admits some positive (radial, bounded) classical solutions in $\mathbb R^N$ (see \cite{SZ}). On the other hand, the so-called Lane-Emden conjecture states that if \be\lab{7-8-1} \frac{1}{p+1}+\frac{1}{q+1}> \frac{N-2}{N}\ee then there is no nontrivial classical solution of \eqref{loc-system} in $\mathbb R^N$. The conjecture is known to be true for radial solutions in all dimensions (see \cite{Mit-1}). In the nonradial case, partial results have been achieved. Nonexistence was proved in \cite{FF,RZ} for $(p, q)$ in certain subregions of \eqref{7-8-1}.
For nonlocal case, i.e. $s\in(0,1)$, Quaas and Xia \cite{QX3} showed the existence of at least one positive viscosity solution for the system of the type \begin{equation} \label{sys0} \left\{ \begin{aligned} (-\Delta)^s u &= v^p \quad &&\text{in } \Omega, \\ (-\Delta)^s v &= u^q \quad &&\text{in } \Omega, \\ u = v &= 0 \quad &&\text{in } \Omega^c. \end{aligned} \right. \end{equation} It has been proved in \cite{QX1} that under some conditions on the exponents $p$ and $q$, system \eqref{sys0} does not admit any positive bounded viscosity solution.
We also refer \cite{DKK, QX2} for further results in this directions.
Nonlocal equations with measure data have been investigated in \cite{BN, CFV, CQ, CV, TV} and the references therein. More precisely, fractional elliptic equations with interior measure data were studied in \cite{CV,CQ}, while the equations with measure boundary data were carried out in \cite{TV} (for absorption nonlinearity) and in \cite{BN} (for source nonlinearity).
To the best of our knowledge, there has been no result concerning nonlinear fractional elliptic systems with measure data in the literature so far. The present paper can be regarded as one of the first publications in this direction and our main contribution is the existence and multiplicity result (see Theorem \ref{2nd sol}) which is new even in the local case $s=1$. Our approach is based on a combination of the theory of PDEs with measure data and variational method (in particular Linking theorem).
Before stating the main results, we introduce necessary notations.
For $\phi\geq 0$, denote by $\mathfrak{M}(\Om,\phi)$ the space of Radon measures $\tau$ on $\Om$ satisfying $\int_{\Om}\phi\, d |\tau|<\infty$ and by $\mathfrak{M}^+(\Om,\phi)$ the positive cone of $\mathfrak{M}(\Om,\phi)$. For $\kappa>0$, denote by $L^\kappa(\Omega,\phi)$ the space of measurable functions $w$ such that $\int_{\Omega}|w|^\kappa \phi dx <\infty$. We denote $\de(x)=\text{dist}(x,\pa\Om)$. When $\phi=\delta^s$, we can define the space $\mathfrak{M}(\Om,\delta^s)$ and $L^\kappa(\Omega, \de^s)$. Let $G_s=G_s^\Omega$ be the Green kernel of $(-\De)^s$ in $\Om$. We denote the associated Green operator $\mathbb{G}_s$ as follows: $$ \begin{aligned} \mathbb{G}_s[\tau](x) &:= \int_{\Om}G_s(x, y)d\tau(y), \quad \tau\in\mathfrak{M}(\Om,\de^s). \end{aligned} $$ Important estimates concerning the Green kernel are presented Section 2.
\begin{definition} \label{defsol}(Weak solution) Let $\mu,\, \nu \in \GTM^+(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)$. We say that $(u, v)$ is a weak solution of \eqref{eq:system} if $u,\, v \in L^1(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$, $v^p,\, u^q \in L^1(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)$ and \begin{equation} \label{intN}\left\{ \begin{aligned}
\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} u {(-\Gd)^s} \xi dx &= \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}v^p\xi dx + \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} \xi \,d\mu, \\
\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} v {(-\Gd)^s} \xi dx &= \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}u^q\xi dx + \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} \xi \,d\nu,
\end{aligned}\right. \quad \forall \xi \in \BBX_s(\Omega} \def\Gx{\Xi} \def\Gy{\Psi),
\end{equation} \end{definition} where $\BBX_s(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)\subset C(\mathbb R^N)$ denotes the space of test functions $\xi$ satisfying
(i) $\mathrm{supp} (\xi) \subset \bar \Omega} \def\Gx{\Xi} \def\Gy{\Psi$,
(ii) ${(-\Gd)^s} \xi(x)$ exists for all $x \in \Omega} \def\Gx{\Xi} \def\Gy{\Psi$ and $|{(-\Gd)^s} \xi(x)| \leq C$ for some $C>0$,
(iii) there exists $\vgf \in L^1(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)$ and $\varepsilon_0>0$ such that $|(-\Gd)_\varepsilon^s \xi| \leq \vgf$ a.e. in $\Omega} \def\Gx{\Xi} \def\Gy{\Psi$, for all $\varepsilon \in (0,\varepsilon_0]$.
\begin{remark} We observe that, by \cite[Proposition A]{TV}, $(u, v)$ is a weak solution of \eqref{eq:system} if and only if \begin{equation} \label{uGM} u = \BBG_s[v^p] + \BBG_s[\mu]\quad\text{and}\quad v=\BBG_s[u^q] + \BBG_s[\nu]. \end{equation} \end{remark}
Define \be\lab{p-s}N_s:=\frac{N+s}{N-s}. \ee
Our first result is the existence of the minimal weak solutions of \eqref{eq:system}. \begin{theorem} (Minimal solution) \label{existcoup}
Let $p,q>0$ with $p\leq q$, $pq \neq 1$ and $q\frac{p +1}{q+1} <N_s$.
Assume $\mu, \nu \in \GTM^+(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)$ and $\BBG_s[\mu]\in L^{q}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^{s})$. Then system \eqref{eq:system} admits a positive weak solution $(\underline u_{\mu}, \underline v_{\nu})$ for $\|\mu\|_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)}$ and $\|\nu\|_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)}$ small if $pq>1$ and for any $\mu,\nu \in \GTM^+(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s) $ if $pq<1$. This solution satisfies
$$ \underline u_\mu \geq \BBG_s[\mu], \quad \underline v_\nu \geq \BBG_s[\nu] \quad \text{a.e. in } \Omega. $$
Moreover, it is the minimal positive weak solution of \eqref{eq:system} in the sense that if $(u,v)$ is a positive weak solution of \eqref{eq:system} then $\underline u_\mu \leq u$ and $\underline v_\nu \leq v$ a.e. in $\Omega} \def\Gx{\Xi} \def\Gy{\Psi$.
In addition, if $q<N_s$ then there exists a positive constant $K=K(N,s,p,q, \|\mu\|_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)},\|\nu\|_{\GTM( \Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)})$ such that $K \to 0$ as $(\|\mu\|_{\GTM( \Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)},\|\nu\|_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)}) \to (0,0)$ and
\begin{equation} \label{leub}
\max\{ \underline u_\mu, \underline v_\nu \} \leq K \BBG_s[\tilde\mu+\tilde\nu] \quad \text{a.e. in } \Omega,
\end{equation}
where $\tilde\mu=\frac{\mu}{\|\mu\|_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)}}$ and $\tilde\nu=\frac{\nu}{\|\nu\|_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)}}$. \end{theorem}
The existence of the second solution is stated in the following theorem.
\begin{theorem} (Second solution) \lab{2nd sol}
Assume $0\leq \mu,\, \nu\in L^r(\Omega)$ for some $r>\frac{N}{2s}$ and $1<p\leq q<N_s$, where $N_s$ is defined in \eqref{p-s}. There exists $t^*>0$ such that if $$\max\{\|\mu\|_{L^r(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)},\|\nu\|_{L^r(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)} \} < t^*$$
then system \eqref{eq:system}
admits at least two positive weak solutions $(u_\mu,v_\nu)$ and $(\underline u_\mu, \underline v_{\nu})$ with $u_\mu \gneq \underline u_{\mu}$ and $v_\nu \gneq \underline v_{\nu}$, where $(\underline u_\mu, \underline v_{\nu})$ is the minimal solution constructed in Theorem \ref{existcoup}.
If, in addition, $\mu,\nu \in L^r(\Omega) \cap L_{loc}^\infty(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$ then $u_\mu > \underline u_{\mu}$ and $v_\nu > \underline v_{\nu}$ in $\Omega} \def\Gx{\Xi} \def\Gy{\Psi$. \end{theorem}
It is worth mentioning that the existence results in Theorem \ref{existcoup} and Theorem \ref{2nd sol} rely on completely different methods. More precisely, the proof of Theorem \ref{existcoup} is in spirit of \cite{BVY}, based on a delicate construction of a supersolution. The main ingredient is a series of estimates concerning the Green kernel (see Lemmas \ref{3g}, \ref{ingg}, \ref{inggs} and \ref{G3}). Theorem \ref{2nd sol} is obtained by using a variational approach. Because of the interplay of the two components $u$ and $v$, the analysis of the associated energy functional (see \eqref{I}) becomes complicated and consequently the Mountain-pass theorem is inapplicable. Therefore, we employ the Linking theorem instead. In order to construct the second solution of \eqref{eq:system}, we require the data $\mu$ and $\nu$ to be sufficiently regular, namely $\mu,\nu \in L^r(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$ for some $r>\frac{N}{2s}$. This enables to deduce the boundedness of the minimal solution constructed in Theorem \ref{existcoup}, which in turn allows to establish the geometry of the Linking theorem. As a consequence, we are able to prove the existence of a variational solution which is in fact a weak solution due to a result in \cite{A} and greater than the minimal solution.
The rest of the paper is organized as follows. In Section 2, we collect some known estimates on the Green kernel from different papers and prove important estimates regarding the Green operator (see Lemmas \ref{ingg}, \ref{inggs}, \ref{G3}). These estimates are the main ingredient in the proof of Theorem \ref{existcoup} which is presented in Section 3. In Section 4, we discuss a priori estimates, as well as regularity properties, of weak solutions. Section 5 deals with the proof of Theorem \ref{2nd sol} which is based on the Linking theorem.
{\bf Notations:} Throughout the present paper, we denote by $c,c',c_1,c_2,C,...$ positive constants that may vary from line to line. If necessary, the dependence of these constants will be made precise.
\section{Estimates on Green kernel}
We denote by $G_s$ the Green kernel of $(-\De)^s$ in $\Om$ respectively. More precisely, for every $y\in\Om$, \begin{equation}\left\{ \begin{aligned} (-\Delta)^s G_s(.,y) &= \de_y \quad &&\text{in } \Omega \\ G_s(.,y) &=0 \quad &&\text{in } \Omega^c, \end{aligned} \right. \end{equation} where $\de_y$ is the Dirac mass at $y$.
\begin{lemma} (\cite[Corollary 1.3]{ChSo1}) \label{estGM0} There exists a positive constant $c=c(N,s,\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$ such that
\begin{equation} \label{G0} \begin{aligned} c^{-1}\min\{ |x-y|^{2s-N}, &\gd(x)^s \gd(y)^s |x-y|^{-N} \} \leq G_s(x,y) \\
&\leq c \min\{ |x-y|^{2s-N}, \gd(x)^s \gd(y)^s |x-y|^{-N} \}, \quad \forall\, x \neq y, \, x,\,y \in \Omega} \def\Gx{\Xi} \def\Gy{\Psi.
\end{aligned} \end{equation} \end{lemma} \begin{lemma} (\cite[Theorem 1.1]{ChSo1}
There exists a positive constant $C=C(N,s,\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$ such that, for every $(x,y)\in\Omega} \def\Gx{\Xi} \def\Gy{\Psi\times\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\;\;x\neq y$,
\begin{align}\label{2.33}
G_s(x,y)&\leq C \gd(y)^{s }|x-y|^{s-N },\\
\label{2.34} G_s(x,y)&\leq C \frac{\gd(y)^{s }}{\gd(x)^{s }}|x-y|^{2s-N}.
\end{align}
\end{lemma}
\begin{remark}\lab{l:22-6-1} Let $\theta \in (0,1]$, then there exists a positive constant $c=c(N,s,\theta,\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$ such that $\BBG_s[\gl]^\theta \geq c \gd^s$ a.e. in $\Omega} \def\Gx{\Xi} \def\Gy{\Psi$ for every measure $\gl \in \GTM^+(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)$ such that $\| \gl \|_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)}=1$. Indeed, since $|x-y|<d_\Omega} \def\Gx{\Xi} \def\Gy{\Psi:=\text{diam}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$, applying \eqref{G0} we have
$$ G_s(x,y) > c^{-1}\min\{ d_\Omega} \def\Gx{\Xi} \def\Gy{\Psi^{2s-N}, d_\Omega} \def\Gx{\Xi} \def\Gy{\Psi^{-N}\gd(x)^s\gd(y)^s \}. $$ Further, $\max\{\gd(x),\gd(y)\} < d_\Omega} \def\Gx{\Xi} \def\Gy{\Psi$ implies $ d_\Omega} \def\Gx{\Xi} \def\Gy{\Psi^{-N} \gd(x)^s \gd(y)^s < d_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}^{2s-N}.$ Consequently, $G_s(x,y)>c^{-1} d_\Omega} \def\Gx{\Xi} \def\Gy{\Psi^{-N} \gd(x)^s \gd(y)^s$. It follows that $\BBG_s[\lambda] \geq c\delta^s$ a.e. in $\Omega$. Therefore $$ \BBG_s[\lambda]^\theta \geq c^\theta \gd^{\theta s} \geq c_1 \gd^s \quad \text{a.e. in } \Omega. $$
\end{remark}
\begin{remark} \label{G1} Let $\theta \in (0,1]$, then there exists a constant $c=c(N,s,\theta,\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$ such that $\BBG_s[\gl]^\theta \geq c \BBG_s[1]$ in $\Omega} \def\Gx{\Xi} \def\Gy{\Psi$ for every measure $\gl \in \GTM^+(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)$ such that $\| \gl \|_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)}=1$. Indeed, by \cite[(2.18)]{CV}, we have $c_1^{-1}\gd^s \leq \BBG_s[1] \leq c_1 \gd^s$ in $\Omega} \def\Gx{\Xi} \def\Gy{\Psi$ for some constant $c_1=c_1(N,s,\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$. This, combined with Remark \ref{l:22-6-1}, leads to the desired estimate. \end{remark}
\begin{definition}(Marcinkiewicz space)
Let $\Om\subset\mathbb R^N$ be a domain and $\la$ be a positive Borel measure in $\Om$. For $\kappa>1$, $\kappa'=\frac{\kappa}{1-\kappa}$ and $u\in L^1_{loc}(\Om, \la)$, we set
$$\|u\|_{M^\kappa(\Om, \la)}:=\inf\displaystyle\bigg\{c\in[0,\infty]: \int_{E}|u|\,d\la \leq c \bigg(\int_{E}d\la\bigg)^\frac{1}{\kappa'}, \quad\forall\, E\subset\text{Borel set}\bigg\}$$
and
$$M^\kappa(\Om, \la):=\big\{u\in L^1_{loc}(\Om, \la): \|u\|_{M^\kappa(\Om, \la)}<\infty\big\}.$$
$M^\kappa(\Om, \la)$ is called the Marcinkiewicz space with exponent $\kappa$ (or weak $L^{\kappa}$ space) with quasi-norm $\|.\|_{M^\kappa(\Om, \la)}$.
\end{definition}
The next lemma establishes a relation between Lebesgue space norm and Marcinkiewicz quasi-norm.
\begin{lemma} (\cite[Lemma A.2(ii)]{BBC}) \lab{LpMk}
Assume $1\leq q<\kappa<\infty$ and $u\in L^1_{loc}(\Om, \la)$. Then there exists $C(q,\kappa)>0$ such that for any Borel subset $E$ of $\Om$
$$\int_{E}|u|^q\, d\la \leq C(q,\kappa)\|u\|^q_{M^{\kappa}(\Om, \la)}\bigg(\int_{E}d\la\bigg)^{1-\frac{q}{\kappa}}.$$
\end{lemma}
We set
\begin{equation} \label{k} k_{\alpha,\gamma}:=\left\{ \begin{aligned}
&\frac{N+\alpha}{N-2s+\gamma} \quad &&\text{if } \alpha<\frac{N\gamma}{N-2s} \\
&\frac{N}{N-2s} &&\text{otherwise}.
\end{aligned}
\right.
\end{equation}
Estimates of Green operator are presented below.
\begin{lemma} (\cite[Proposition 2.2]{CV}) \label{estGM} Let $\alpha, \gg \in [0,s]$ and $k_{\alpha,\gg}$ be as in \eqref{k}.
There exists $c=c(N,s,\alpha,\gg,\Omega} \def\Gx{\Xi} \def\Gy{\Psi)>0$ such that
\begin{equation} \label{estG1} \|\BBG_s[\la]\|_{M^{k_{\alpha,\gg}}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^\alpha)}
\leq c\|\la\|_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^\gg)}\quad \forall\, \la \in \GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^\gg).
\end{equation}
\end{lemma}
\begin{lemma} (\cite[Proposition 1.4]{RS2}) \label{regularity2} (i) If $t>\frac{N}{2s}$ then there exists $c=c(N,s,t,\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$ such that
\begin{equation} \label{estG2}
\| \BBG_s[\la] \|_{C^{\min\{ s, 2s-\frac{N}{t} \} }(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)} \leq c\| \la \|_{L^t(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)} \quad \forall\, \la \in L^t(\Omega} \def\Gx{\Xi} \def\Gy{\Psi).
\end{equation}
(ii) If $1<t<\frac{N}{2s}$ then there exists a constant $c=c(N,s,t)$ such that
\begin{equation} \label{estG2}
\| \BBG_s[\la] \|_{L^{\frac{Nt}{N-2ts} }(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)} \leq c\| \la \|_{L^t(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)} \quad \forall\, \la \in L^t(\Omega} \def\Gx{\Xi} \def\Gy{\Psi).
\end{equation}
\end{lemma}
We recall the $3$G-estimates.
\begin{lemma} (\cite[Theorem 1.6]{ChSo1}) \label{3g} There exists a positive constant $C=C(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,s)$ such that \begin{equation} \label{e3g}
\frac{G_s(x,y)G_s(y,z)}{G_s(x,z)}\leq C \frac{|x-z|^{N-2s}}{|x-y|^{N-2s}|y-z|^{N-2s}} ,\quad\forall\, (x,y,z)\in\Omega} \def\Gx{\Xi} \def\Gy{\Psi\times\Omega} \def\Gx{\Xi} \def\Gy{\Psi\times\Omega} \def\Gx{\Xi} \def\Gy{\Psi. \end{equation} \end{lemma}
Next we will prove some important estimates concerning the Green operator which will be used in Section 3.
\begin{lemma} \label{ingg} Let $0< p<N_s$ and $\la \in\GTM^+(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^{s })$ such that $\| \la \|_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)}=1$. Then there exists a constant $C=C(N,s,p,\Omega} \def\Gx{\Xi} \def\Gy{\Psi)>0$ such that
\begin{equation} \label{laG}
\BBG_s[\BBG_s[\la]^p] \leq C\BBG_s[\la] \quad \text{a.e. in } \Omega.
\end{equation} \end{lemma}
\begin{proof} First, we consider the case $p > 1$. From Lemma \ref{estGM} and the embedding $M^{N_s}(\Omega,\gd^s) \subset L^p(\Omega,\gd^s)$, we deduce that $\BBG_s[\la] \in L^{p}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)$. We write
$$\BBG_s[\la](y)=\int_\Omega} \def\Gx{\Xi} \def\Gy{\Psi G_s(y,z)d\la(z)=\int_\Omega} \def\Gx{\Xi} \def\Gy{\Psi \frac{G_s(y,z)}{\gd(z)^{s }}\gd(z)^{s }d\la(z).$$
Therefore, using H\"{o}lder inequality, we obtain
$$\BBG_s[\la](y)^p\leq \int_\Omega} \def\Gx{\Xi} \def\Gy{\Psi\left(\frac{G_s(y,z)}{\gd(z)^{s }}\right)^p \gd(z)^{s } d\la(z),$$
as $\| \la \|_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)}=1$.
Consequently,
\begin{equation} \label{in11}
\BBG_s[\BBG_s[\la]^p](x)\leq \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} G_s(x,y)G_s(y,z)^p\gd(z)^{s (1-p)}d\la(z)dy.
\end{equation}
Now applying Lemma \ref{3g} and \eqref{2.33} to the right-hand side of the above expression, we obtain
\begin{equation} \begin{aligned}
&\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} G_s(x,y)G_s(y,z)^p\gd(z)^{s (1-p)}d\la(z)dy\\
&=\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} G_s(x,y)G_s(y,z)\Big(\frac{G_s(y,z)}{\gd(z)^s}\Big)^{p-1}d\la(z)dy \\
&\leq C_1\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}G_s(x,z)\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} \frac{|x-z|^{N-2s}}{|x-y|^{N-2s}|y-z|^{N-2s}}|y-z|^{-(N-s)(p-1)} dyd\la(z)\\
&\leq C_2\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}G_s(x,z)\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\big[|y-z|^{2s-N-(N-s)(p-1)}+|x-y|^{-(N-2s)}|y-z|^{-(N-s)(p-1)}\big]dyd\la(z)\\
&\leq C_3\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}G_s(x,z)\Big[\int_{\Omega \cap \{ |x-y| \geq |y-z| \} }|y-z|^{2s-N-(N-s)(p-1)}dy\\
&\qquad \qquad \qquad \quad + \int_{\Omega \cap \{ |x-y| \leq |y-z| \} }|x-y|^{2s-N-(N-s)(p-1)}) dy \Big]d\la(z) \label{in12} \\
&\leq C_4\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}G_s(x,z)d\la(z),
\end{aligned} \end{equation}
where $C_i=C_i(N,s,p,\Omega)$ ($i=1,2,3,4$). Here in the second estimate, we have used the inequality $|x-z| \leq |x-y| + |y-z|$ and in the last estimate we have used the fact that $p<N_s$. Hence combining \eqref{in11} and \eqref{in12}, we derive that
\begin{align*}
\BBG_s[\BBG_s[\la]^p](x)
&\leq C\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}G_s(x,z)d\la(z)=C\BBG_s[\la](x)
\end{align*}
where $C=C(N,s,p,\Omega)$. Note that the above argument is still valid for the case $p=1$.
Next we consider the case $0< p<1$. Then we have
$$ \BBG_s[\la]^p \leq C(p)(1+ \BBG_s[\la]) \quad \text{a.e. in } \Omega.
$$
This yields
$$ \BBG_s[\BBG_s[\la]^p] \leq C(p)(\BBG_s[1] + \BBG_s[\BBG_s[\la]]) \quad \text{a.e. in } \Omega.
$$
By applying the case $p=1$, we have $ \BBG_s[\BBG_s[\la]] \leq C \BBG_s[\la]$ with $C=C(N,p,s,\Omega)$. Therefore, combining the above results along with Remark \ref{G1} with $\theta=1$, we derive \eqref{laG}. \end{proof}
\begin{lemma} \label{inggs}
Let $0< p<N_s$, $\la\in\GTM^+(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^{s})$ with $\|\la\|_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^{s})}=1$. Let $\theta$ be such that
\begin{equation} \label{s} \max\left(0,p-N_s+1\right)<\theta} \def\vge{\varepsilon \leq 1.
\end{equation}
Then there exists a positive constant $C=C(N,s,p,\theta,\Omega)$ such that
\begin{equation} \label{ests} \BBG_s[\BBG_s[\la]^p ]\leq C \BBG_s[\la]^\theta} \def\vge{\varepsilon \quad\text{ a.e. in } \;\Omega} \def\Gx{\Xi} \def\Gy{\Psi.
\end{equation} \end{lemma} \begin{proof} First we assume that $p>1$. In view of the proof of Lemma \ref{ingg}, we have
\begin{equation} \label{33a} \begin{aligned}
&\BBG_s[ \BBG_s[\la]^p](x)\leq C\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} G_s(x,y)G_s(y,z)^p\gd(z)^{s (1-p)}d\la(z)dy\\
&=C\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}G_s(x,y)^{1-\theta} \def\vge{\varepsilon} G_s(x,y)^\theta} \def\vge{\varepsilon G_s(y,z)^\theta} \def\vge{\varepsilon\left(\frac{G_s(y,z)}{\gd(z)^s}\right)^{p-\theta} \def\vge{\varepsilon}\gd(z)^{s (1-\theta} \def\vge{\varepsilon)}d\la(z)dy.
\end{aligned} \end{equation}
By \eqref{e3g} and the inequality $|x-z| \leq |x-y|+|y-z|$, we have
\begin{equation} \label{knm} G_s(x,y)^\theta} \def\vge{\varepsilon G_s(y,z)^\theta} \def\vge{\varepsilon \leq CG_s(x,z)^\theta} \def\vge{\varepsilon (|x-y|^{(2s-N)\theta} \def\vge{\varepsilon} + |y-z|^{(2s-N)\theta} \def\vge{\varepsilon}).
\end{equation}
Combining \eqref{33a} and \eqref{knm} yields
\begin{equation} \BBG_s[\BBG_s[\la]^p](x) \leq C\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}G_s(x,z)^\theta} \def\vge{\varepsilon \gd(z)^{s(1-\theta} \def\vge{\varepsilon)}\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} I_{x,z}(y) dy d\la(z)
\end{equation}
where
$$ I_{x,z}(y) = \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}G_s(x,y)^{1-\theta} \def\vge{\varepsilon}\left(\frac{G_s(y,z)}{\gd(z)^s}\right)^{p-\theta} \def\vge{\varepsilon}(|x-y|^{(2s-N)\theta} \def\vge{\varepsilon} + |y-z|^{(2s-N)\theta} \def\vge{\varepsilon})dy.
$$
Applying \eqref{G0} and \eqref{2.33}, we obtain
\begin{equation} \begin{aligned}
I_{x,z}(y)& \leq C \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}|x-y|^{(2s-N)(1-\theta} \def\vge{\varepsilon)}|y-z|^{(s-N)(p-\theta} \def\vge{\varepsilon)}(|x-y|^{(2s-N)\theta} \def\vge{\varepsilon} + |y-z|^{(2s-N)\theta} \def\vge{\varepsilon})dy\\
&\leq C \int_{ \{y \in \Omega} \def\Gx{\Xi} \def\Gy{\Psi: |x-y| \leq |y-z| \} } |x-y|^{2s-N+(s-N)(p-\theta} \def\vge{\varepsilon)}dy \\
&+ C \int_{ \{y \in \Omega} \def\Gx{\Xi} \def\Gy{\Psi: |x-y| \geq |y-z| \} } |y-z|^{2s-N+(s-N)(p-\theta} \def\vge{\varepsilon)}dy
\\
&\leq C.
\end{aligned} \end{equation}
Here in the last inequlaity we have used the fact that $\theta> p-N_s+1$. Thus
\begin{equation} \begin{aligned}
\BBG_s[\BBG_s[\la]^p](x) &\leq C\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\left(\frac{G_s(x,z)}{\gd(z)^{s }}\right)^\theta} \def\vge{\varepsilon \gd(z)^{s }d\la(z)\\
&\leq C\left(\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}G_s(x,z)d\la(z)\right)^\theta} \def\vge{\varepsilon,
\end{aligned} \end{equation}
where in the last line we have used H\"{o}lder inequality with exponent $\frac{1}{\theta}$.
Note that the above approach is still valid for the case $p=1$.
If $0< p<1$ then
$$\BBG_s[ \BBG_s[\la]^p]\leq C(\BBG_s[1]+\BBG_s[\BBG_s[\la]])\leq C(\BBG_s[1]+\BBG_s[\la]^\theta} \def\vge{\varepsilon).$$
Then \eqref{ests} follows by a similar argument as in the proof of Lemma \ref{ingg} by using Remark \ref{G1} with $\theta \leq 1$. \end{proof}
In the sequel, without loss of generality, we may assume that \begin{equation} \label{p<q} 0<p \leq q. \end{equation} Hence, if $pq \geq 1$ then \begin{equation}\label{pqrel} p \leq q\frac{p+1}{q+1} \leq p\frac{q+1}{p+1} \leq q. \end{equation} Put $$t_s: =q\left(p- N_s + 1\right). $$ Notice that if $q\frac{p+1}{q+1} <N_s$ then $t_s <q\frac{p +1}{q+1}<N_s$. \begin{lemma} \label{G3}
Let $p,q>0$, $p\leq q$ and $\la\in\GTM^+(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^{s})$ with $\|\la\|_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^{s})}=1$. Assume $q\frac{p +1}{q+1} <N_s$. Then for any $t \in (\max(0,t_s),q]$, there exists a positive constant $c=c(N,p,q,s,t)$ such that
\begin{equation} \label{G31} \BBG_s[\BBG_s[\la]^p]^{q} \leq c \BBG_s[\la]^t \quad \text{a.e. in } \Omega.
\end{equation}
In particular,
\begin{equation} \label{G32} \BBG_s[ \BBG_s[\la]^p]^{q} \leq C \BBG_s[\la]^{q\frac{p +1}{q+1}} \quad \text{a.e. in } \Omega,
\end{equation}
\begin{equation} \label{G33} \BBG_s[ \BBG_s[\BBG_s[\la]^p]^{q}] \leq C\BBG_s[\la] \quad \text{a.e. in } \Omega,
\end{equation}
where $C=C(N,p,q,s)$. \end{lemma}
\begin{proof} Since $p\leq q$ from \eqref{p<q}, it follows that $p\leq q\frac{p+1}{q+1}$. Therefore, from the assumption, it follows $p<N_s$. Hence $\max(0,p-N_s+1) < 1$. Let $t \in (\max(0,t_s),q]$ then $\max(0,p-N_s+1) <\frac{t}{q} \leq 1$. Therefore, applying Lemma \ref{inggs} with $\theta} \def\vge{\varepsilon$ replaced by $\frac{t}{q}$, we obtain
$$ \BBG_s[\BBG_s[\la]^p] \leq c \BBG_s[\la]^{\frac{t}{q}},
$$
which implies \eqref{G31}.
Since $t_s<q\frac{p+1}{q+1} \leq q$, taking $t=q \frac{p+1}{q+1}$ in \eqref{G31} yields \eqref{G32}. Next, since $q\frac{p+1}{q+1} < N_s$, combining \eqref{G32} along with Lemma \ref{ingg}, we have
$$ \BBG_s[\BBG_s[ \BBG_s[\la]^p]^{q}] \leq C\BBG_s[\BBG_s[\la]^{q\frac{p+1}{q+1}} ] \leq C\BBG_s[\la] \quad \text{a.e. in } \Omega.
$$
This completes the proof.
\end{proof}
\section{Construction of the minimal solution}
\begin{lemma} \label{compa} Assume $p,q>0$ and $\mu,\, \nu \in \GTM^+(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\delta^s)$. Assume in addition that there exist functions $V \in L^p(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)$ and $U \in L^q(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)$ such that \begin{equation} \label{UV} \begin{aligned}U &\geq \BBG_s[V^p] + \BBG_s[\mu] \quad \text{a.e. in } \Omega,\\ V &\geq \BBG_s[U^q] + \BBG_s[\nu] \quad \text{a.e. in } \Omega. \end{aligned}\end{equation} Then there exists a positive minimal weak solution $(\underline u_\mu, \underline v_\nu)$ of \eqref{eq:system} satisfying
\begin{equation}\label{compa1} \BBG_s[\mu] \leq \underline u_\mu \leq U,\quad \BBG_s[\nu] \leq \underline v_\nu \leq V \quad \text{a.e. in } \Omega. \end{equation} \end{lemma} \begin{proof}
Put $u_0:=\BBG_s[\mu] $, $v_0:=\BBG_s[\nu] $ and for $n\geq 1$, define
\begin{equation}\begin{aligned} \label{induc} u_n& :=\BBG_s[v_{n-1}^p]+ \BBG_s[\mu], \\
v_n& :=\BBG_s[u_{n-1}^q]+ \BBG_s[\nu].
\end{aligned}\end{equation}
Clearly $u_0 \leq U$ and $v_0\leq V$ . Therefore,
$$ u_1=\BBG_s[v_0^p] + \BBG_s[\mu] \leq \BBG_s[V^p] + \BBG_s[\mu] \leq U. $$ Similarly, $v_1\leq \BBG_s[U^q] + \BBG_s[\nu]\leq V$. By induction, it follows that $u_n \leq U$ and $v_n\leq V$ for every $n \geq 1$. Also, it is easy to see that $\{u_n\}$ and $\{v_n\}$ are increasing sequences. Hence $u_n \uparrow \underline u_\mu \leq U \in L^q(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)$ and $v_n \uparrow \underline v_\nu \leq V \in L^p(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)$. Therefore $\BBG_s[v_n^p] \uparrow \BBG_s[\underline v_\nu^p]$ and $\BBG_s[u_n^q] \uparrow \BBG_s[\underline u_\mu^q]$
a.e. in $\Omega} \def\Gx{\Xi} \def\Gy{\Psi$. Letting $n \to \infty$ in \eqref{induc}, we deduce that
$$ \underline u_\mu = \BBG_s[\underline v_\nu^p] + \BBG_s[\mu] \quad\text{and}\quad \underline v_\nu = \BBG_s[\underline u_\mu^q] + \BBG_s[\nu]. $$
This means that $(\underline u_\mu, \underline v_\nu)$ is a weak solution of \eqref{eq:system}.
Next, let $(u, v)$ be any positive weak solution \eqref{eq:system}. Then
$$u=\BBG_s[v^p]+\BBG_s[\mu] \geq u_0,\quad v=\BBG_s[u^q]+\BBG_s[\nu] \geq v_0.$$ Thus $$u\geq \BBG_s[v_0^p]+\BBG_s[\mu] \geq u_1, \quad v\geq \BBG_s[u_0^q]+\BBG_s[\nu] \geq v_1.$$
By induction it follows that $u\geq u_n$ and $v\geq v_n$ for all $n\geq 1$. Hence $u\geq \underline u_\mu$ and $v\geq \underline v_\nu$. This completes the lemma. \end{proof} \begin{remark}\lab{l:3-9-1} In stead of studing system \eqref{eq:system}, in the sequel, we will work on the following system \begin{equation} \label{source-3} \left\{ \begin{aligned} (-\Delta)^s u &= v^p + \rho\mu \quad &&\text{in } \Omega, \\ (-\Delta)^s v &= u^q + \tau \nu \quad &&\text{in } \Omega, \\ u =v&=0 \quad &&\text{in } \Omega^c, \end{aligned} \right. \end{equation}
where $\rho, \tau$ are positive parameters and $\mu,\nu \in \GTM^+(\Omega,\gd^s)$ such that $\| \mu \|_{\GTM(\Omega,\gd^s)}=\| \nu \|_{\GTM(\Omega,\gd^s)}=1$. The advantage is when dealing with system \eqref{source-3}, we can easily apply Lemma \ref{inggs} and Lemma \ref{G3} for $\BBG_s[\mu]$ and $\BBG_s[\nu]$ and require only the smallness of the parameters $\rho$ and $\tau$, which improves considerably the exposition. \end{remark}
We recall that in the sequel, we assume that $0<p \leq q$ and hence if $pq \geq 1$ then \eqref{pqrel} holds.
\begin{theorem} \label{existcoup-1}
Let $p,\, q,\,\rho,\, \tau >0$ and $\mu, \nu \in \GTM^+(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\delta^s)$ such that $\| \mu \|_{\GTM(\Omega,\gd^s)}=\| \nu \|_{\GTM(\Omega,\gd^s)}=1$. Assume $p \leq q$, $pq \neq 1$, $q\frac{p+1}{q+1}<N_s$ and $\BBG_s[\mu]\in L^{q}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^{s})$. Then system \eqref{source-3} admits the minimal weak solution $(\underline u_{\rho \mu},\underline v_{\tau \nu})$ for $\rho$ and $\tau$ small if $pq>1$, for any $\rho>0$ and $\tau>0$ if $pq<1$.
In addition, if $p \leq q<N_s$, then there exists a constant $K=K(N,s,p,q,\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\rho,\tau)$ such that $K \to 0$ as $(\rho,\tau)\to (0,0)$ and \begin{equation} \label{leub} \max\{ \underline u_{\rho\mu},\underline v_{\tau \nu} \} \leq K \BBG_s[ \mu+ \nu] \quad \text{a.e. in } \Omega. \end{equation} \end{theorem} \begin{proof} Fix numbers $\vartheta_i>0$, ($i=1,2$) and set
\begin{equation} \label{Psi} \Psi:=\BBG_s[\vartheta_1 \mu]^{q} + \vartheta_2 \nu.
\end{equation}
For $\gk \in(0,1]$, put
\begin{equation} \label{rt} \rho= \gk^{\frac{1}{q}}\vartheta_1, \quad \tau= \gk \vartheta_2
\end{equation}
and consider system \eqref{source-3} with $\rho$ and $\tau$ as in \eqref{rt}. From the assumptions, $\BBG_s[\mu]^q \in L^1(\Omega,\gd^s)$ and $\nu \in \GTM^+(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)$, it follows that $\Psi \in \GTM^+(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^{s })$. Also we note that $p\leq q$ and $q\frac{p+1}{q+1}<N_s$ imply $p<N_s$. Set $$ V:=A\BBG_s[\gk\Psi] \quad \text{and} \quad U:=\BBG_s[ V^p] + \BBG_s[ \rho \mu ]$$ where $A>0$ will be determined later on. Then, \begin{align*} U^q &\leq c ( \BBG_s [V^p]^q + \BBG_s[\rho \mu]^q) \\ &= c \Big\{ A^{p q} \gk^{pq} \BBG_s[ \BBG_s[\Psi]^p]^{q} + \BBG_s[\rho \mu]^{q}\Big\} \end{align*} where $c=c(p,q)$. It follows that \begin{equation} \label{lat} \begin{aligned} \BBG_s[U^q]+\BBG_s[\tau \nu] &\leq c (A^{p q} \gk^{pq} \BBG_s[ \BBG_s[ \BBG_s[\Psi]^p]^{q}] + \BBG_s[\BBG_s[\rho \mu]^q]) + \BBG_s[\tau \nu] \\ &= c (A^{p q} \gk^{pq} \BBG_s[ \BBG_s[ \BBG_s[\Psi]^p]^{q}] + \kappa \BBG_s[\BBG_s[\vartheta_1 \mu]^q]) + \gk \BBG_s[\vartheta_2 \nu] \\ &\leq c (A^{p q} \gk^{pq} \BBG_s[ \BBG_s[ \BBG_s[\Psi]^p]^{q}] + \gk \BBG_s[\Psi]). \end{aligned} \end{equation} Since $q\frac{p+1}{q+1} <N_s$, applying Lemma \ref{G3}, we have
\begin{equation} \label{laca}\BBG_s[\BBG_s[\BBG_s[\Psi]^p]^{q}] \leq C\| \Psi \|_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)}^{pq-1} \BBG_s[\Psi] \end{equation}
where $C=C(N,s,p,q,\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$. Combining the definition of $\Psi$ in \eqref{Psi}, Lemma \ref{G3}, Lemma \ref{estGM} and the assumption that $\| \mu \|_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)}=\| \nu \|_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)}=1$, we can estimate
$ C^{-1} \leq \| \Psi \|_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)} \leq C $ for some positive constant $C$ independent of $A$ and $\kappa$. This, combined with \eqref{lat} and \eqref{laca} implies \begin{equation} \label{leni} \BBG_s[U^q]+\BBG_s[\tau \nu] \leq C(A^{pq}\gk^{pq} + \gk)\BBG_s[\Psi]. \end{equation} for some positive constant $C$ independent of $A$ and $\kappa$.
We will choose $A$ and $\gk$ such that \begin{equation} \label{GU<V} \BBG_s[U^q]+\BBG_s[\tau \nu] \leq V. \end{equation} For that, it is sufficient to choose $A$ and $\gk$ such that $$ C(A^{pq}\gk^{pq} + \gk)\BBG_s[\Psi] \leq V. $$ This holds if \begin{equation} \label{rl} C(A^{pq}\gk^{pq-1} + 1) \leq A. \end{equation} If $p q>1$ then we can choose $A>0$ large enough and then choose $\gk>0$ small enough (depending on $A$) such that \eqref{rl} holds. If $p q<1$ then for any $\gk>0$ there exists $A$ large enough such that \eqref{rl} holds. For such $A$ and $\gk>0$, we obtain \eqref{GU<V}. Consequently, $(U,V)$ satisfies \eqref{UV}. By Lemma \ref{compa}, there exists a weak solution $(\underline u_{\rho \mu}, \underline v_{\tau \nu})$ of \eqref{source-3} for $\rho>0$ and $\tau>0$ small if $p q>1$, for any $\rho>0$ and $\tau>0$ if $p q<1$. Moreover, $(\underline u_{\rho \mu}, \underline v_{\tau \nu})$ satisfies \eqref{compa1}.
Next, assuming in addition that $p \leq q<N_s$, we will demonstrate \eqref{leub}. From the definition of $V$ and Lemma \ref{inggs}, we see that \begin{equation} \label{V1} \begin{aligned} V &=A\kappa\BBG_s[\Psi] = A\kappa(\BBG_s[\BBG_s[\vartheta_1 \mu]^q] + \BBG_s[\vartheta_2 \nu]) \\ & \leq c A\kappa \BBG_s[\mu + \nu]. \end{aligned} \end{equation}
It follows that $$ V^p \leq C \kappa^p( \BBG_s[\mu]^p + \BBG_s[\nu]^p ). $$ Therefore, \begin{equation} \label{U1} \begin{aligned} U &\leq C \kappa^p (\BBG_s[\BBG_s[\mu]^p] + \BBG_s[\BBG_s[\nu]^p]) + \kappa^\frac{1}{q} \BBG_s[\vartheta_1 \mu] \\ &\leq C\kappa^p \BBG_s[\mu + \nu] + \kappa^\frac{1}{q}\vartheta_1 \BBG_s[ \mu] \\ &\leq C\max\{ \kappa^p,\kappa^\frac{1}{q} \} \BBG_s[\mu + \nu]. \end{aligned} \end{equation} Combining \eqref{U1} and \eqref{V1} along with \eqref{compa1} leads to \eqref{leub}. \end{proof}
\begin{remark} \label{Linf} By Lemma \ref{regularity2}(i), we see that if $\mu, \nu \in L^r(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$ with $r>\frac{N}{2s}$ then $\BBG_s[\mu + \nu] \in L^\infty(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$. From Theorem \ref{existcoup-1}, if $\rho$ and $\tau$ are small then system \eqref{source-3} admits a minimal solution $(\underline u_{\rho \mu}, \underline v_{\tau \nu})$ which satisfies \eqref{leub}. It follows that $\underline u_{\rho \mu}, \underline v_{\tau \nu} \in L^\infty(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$. Moreover, $(\underline u_{\rho \mu}, \underline v_{\tau \nu}) \to (0,0)$ a.e. in $\Omega$ as $(\rho,\tau) \to (0,0)$. \end{remark}
{\bf Proof of Theorem \ref{existcoup} completed}: Combining Theorem \ref{existcoup-1} along with Remark \ref{l:3-9-1}, the proof of the theorem follows.
\section{A priori estimates and regularity} In this section, we provide a priori estimates, as well as regularity properties, of weak solutionsof \eqref{eq:system}. \subsection{A priori estimates} \begin{lemma} \label{apriori} Assume $p>1, \,q> 1$ and $\mu,\, \nu \in \GTM^+(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)$. If $(u,v)$ is a weak solution of \eqref{eq:system} then there is a positive constant $c=c(N,s,p,q,\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$ such that
\begin{equation}\begin{aligned} \label{rem1} \norm{u}_{L^1(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)} + \norm{v}_{L^p(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)}& \leq c(1+\norm{\mu}_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)}),\\
\norm{v}_{L^1(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)} + \norm{u}_{L^q(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)} &\leq c(1+\norm{\nu}_{\GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)}).
\end{aligned}
\end{equation} \end{lemma} \begin{proof} We prove this lemma in the spirit of \cite[Lemma 4.1]{BN}. Let $(\la_1,\vgf_1)$ be the first eigenvalue and corresponding positive eigenfunction of $(-\De)^s$ in $X_0(\Om)$ (see the definition of $X_0$ in \eqref{eq:X0}). By \cite[Lemma 2.1(ii)]{CV}, $\vgf_1\in \BBX_s(\Om)$, and hence by taking $\zeta=\vgf_1$ in \eqref{intN}, we obtain \begin{equation}\begin{aligned} \label{rer} \gl_1 \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} u\vgf_1 dx&= \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} v^p \vgf_{1} dx +\gl_1\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} \vgf_1 d\mu ,\\ \gl_1 \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} v\vgf_1 dx&= \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} u^q \vgf_{1} dx +\gl_1\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} \vgf_1 d\nu .
\end{aligned}
\end{equation}
By Young's inequality, we get
\begin{equation}\begin{aligned} \label{rem2}
\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} v\vgf_1 dx &\leq (2\gl_1)^{-1}\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} v^p\vgf_{1} dx + (2\gl_1)^{\frac{1}{p-1}}\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\vgf_1dx,\\
\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} u\vgf_1 dx &\leq (2\gl_1)^{-1}\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} u^q\vgf_{1} dx + (2\gl_1)^{\frac{1}{q-1}}\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\vgf_1dx.
\end{aligned}\end{equation} Substituting \eqref{rem2} in \eqref{rer} we have \begin{equation} \begin{aligned} \label{rem3} 2\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} v^p \vgf_1 dx +
2\gl_1 \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} \vgf_1 d\mu &\leq \int_{\Om} u^q\vgf_1 dx + (2\gl_1)^{\frac{q}{q-1}}\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\vgf_1 dx,\\
2\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} u^q \vgf_1 dx +
2\gl_1 \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} \vgf_1 d\nu &\leq \int_{\Om} v^p\vgf_1 dx +(2\gl_1)^{\frac{p}{p-1}}\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\vgf_1 dx.
\end{aligned}\end{equation} Therefore, $$ \frac{3}{2}\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} v^p \vgf_1 dx +\la_1\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} \vgf_1 d(2\mu+\nu)\leq \bigg(\frac{(2\la_1)^\frac{p}{p-1}}{2}+ (2\la_1)^\frac{q}{q-1}\bigg)\int_{\Om}\vgf_1 dx. $$ Since the second term on the left-hand side of above expression is nonnegative, taking into account that $c^{-1}\delta^s < \vgf_1 < c\delta^s$ in $\Omega$, we have
\begin{equation} \label{rem4} \norm{v}_{L^p(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)}^p \leq C(\la_1)\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\gd^s dx \leq c'.
\ee Similarly \begin{equation} \label{rem4'} \norm{u}_{L^q(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)}^q \leq C'.\ee Next, combining \eqref{uGM}, Lemma \ref{LpMk} and Lemma \ref{estGM} with $\gamma=s,\, \al=0$ we obtain
\begin{equation}\begin{aligned} \label{rem5}
\|u\|_{L^1(\Om)} &\leq \|\BBG_s[v^p]\|_{L^1(\Om)}+\|\BBG_s[\mu]\|_{L^1(\Om)} \\
&\leq C\big(\|\BBG_s[v^p]\|_{M^{N_s}(\Om)}+\|\BBG_s[\mu]\|_{M^\frac{N}{N-s}(\Om)}\big)\\
&\leq C\big( \|v\|_{L^p(\Om, \de^s)}^p+\|\mu\|_{\mathfrak{M}(\Om,\gd^s)}\big).
\end{aligned}\end{equation} Hence first expression of \eqref{rem1} holds by combining \eqref{rem4} and \eqref{rem5}. Similarly, the second expression of \eqref{rem1} follows. \end{proof}
\subsection{Regularity} \begin{theorem} \label{reg} Let $p,q \in (1,N_s)$. (i) Assume $\mu,\nu \in L^r(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$ for some $r>\frac{N}{2s}$. If $(u,v)$ is a nonnegative weak solution of \eqref{eq:system} then $u,v \in L_{loc}^\infty(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$.
(ii) Assume $\mu,\nu \in L^r(\Omega} \def\Gx{\Xi} \def\Gy{\Psi) \cap L_{loc}^\infty(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$. If $(u,v)$ is a nonnegative weak solution of \eqref{eq:system} then $u,v \in C_{loc}^{\alpha}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$ for some $\alpha \in (0,2s)$. \end{theorem} \begin{proof} (i) We first assume that $\mu,\nu \in L^r(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$ for some $r>\frac{N}{2s}$. Let $(u, v)$ be a nonnegative weak solution of \eqref{eq:system}. Then $u \in L^1(\Om)\cap L^q(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)$, $v \in L^1(\Om)\cap L^p(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)$ and $(u,v)$ satisfies \eqref{uGM}. Let $x_0 \in \Omega} \def\Gx{\Xi} \def\Gy{\Psi$ and $r>0$ such that $B(x_0,2r) \subset\subset \Omega} \def\Gx{\Xi} \def\Gy{\Psi$. For any $j \in \BBN$, set $B_j:=B(x_0,2^{-j}r)$. For any $j \in \BBN$, we can write
\begin{equation}\begin{aligned} \label{decomp}
u &= \BBG_s[\chi_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi \setminus B_j}v^p] + \BBG_s[\chi_{B_j}v^p] + \BBG_s[\mu],\\
v &=\BBG_s[\chi_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi \setminus B_j}u^q] + \BBG_s[\chi_{B_j}u^q] + \BBG_s[\nu].
\end{aligned}\end{equation}
Observe that, for $x \in B_{j+1}$, by \eqref{G0},
\begin{align*}
\BBG_s[\chi_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi \setminus B_j}v^p](x) = \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi \setminus B_j} v(y)^pG_s(x,y)dy
&\leq C\gd(x)^s \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi \setminus B_j} v(y)^p\gd(y)^s |x-y|^{-N}dy \\
&\leq C 2^{(j+1)N} r^{-N} \| v \|_{L^p(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)}^p<\infty.
\end{align*}
Therefore,
\begin{equation}
\label{lubu} \BBG_s[\chi_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi \setminus B_j}v^p] \in L^\infty(B_{j+1}) \quad \forall j \in \BBN.
\end{equation}
Similarly,
\begin{equation} \label{lubu-1} \BBG_s[\chi_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi \setminus B_j}u^q] \in L^\infty(B_{j+1}) \quad \forall \, j \in \BBN. \end{equation}
Since $\mu,\nu \in L^r(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$ for $r>\frac{N}{2s}$, by Lemma \ref{regularity2} (i), we deduce
\begin{equation} \label{Ginfty} \BBG_s[\mu], \, \BBG_s[\nu] \in L^\infty(\Omega} \def\Gx{\Xi} \def\Gy{\Psi). \end{equation}
Further, as $u \in L^q(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)$ and $v \in L^p(\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\gd^s)$, we have $\chi_{B_0}u \in L^q(\Omega)$ and $\chi_{B_0}v \in L^p(\Omega)$ and therefore, applying Lemma \ref{estGM} and Lemma \ref{LpMk} we have
$$\|\BBG_s[\chi_{B_0}u^q]\|_{L^\theta(\Om)}\leq c\|\BBG_s[\chi_{B_0}u^q]\|_{M^\frac{N}{N-2s}(\Om)}\leq c'\|\chi_{B_0}u\|_{L^q(\Om)}^q,$$ for every $1<\theta<\frac{N}{N-2s}$. This in turn implies $\BBG_s[\chi_{B_0}u^q],\,
\BBG_s[\chi_{B_0}v^p] \in L^\theta(B_0)$ for every $1<\theta<\frac{N}{N-2s}$. This and \eqref{decomp} -- \eqref{Ginfty} yield $u, v \in L^\theta(B_2)$ for every $1<\theta<\frac{N}{N-2s}$. Set,
$$\ell_0:=\frac{1}{2}(1+\frac{N_s}{p}),\quad \tilde \ell_0:=\frac{1}{2}(1+\frac{N_s}{q}).$$
Then $1<p\ell_0, \, q\tilde \ell_0<N_s<\frac{N}{N-2s}$ and hence $u \in L^{q\tilde \ell_0}(B_{2})$ and $v\in L^{p\ell_0}(B_2)$. Without loss of generality, we can assume $\ell_0,\, \tilde \ell_0\not=\frac{N}{2s}$. If $\ell_0,\, \tilde \ell_0>\frac{N}{2s}$, then by Lemma \ref{regularity2} (i), $\BBG_s[\chi_{B_2}v^p], \, \BBG_s[\chi_{B_2}u^q] \in L^\infty(B_{2})$. This and \eqref{decomp} -- \eqref{Ginfty} imply $u, \, v \in L^\infty(B_4)$. If $\ell_0<\frac{N}{2s}$ or $\tilde \ell_0<\frac{N}{2s}$, then by Lemma \ref{regularity2} (ii) we obtain $\BBG_s[\chi_{B_2}v^p] \in L^{p\ell_1}(B_{2})$ or $\BBG_s[\chi_{B_2}u^q] \in L^{q\tilde \ell_1}(B_{2})$ respectively, where
$$ \ell_1:=\frac{1}{p}\frac{N\ell_0}{N-2\ell_0s}, \quad \tilde \ell_1:=\frac{1}{q}\frac{N\tilde \ell_0}{N-2\tilde \ell_0s}.
$$
Then from \eqref{decomp} -- \eqref{Ginfty}, $v \in L^{p\ell_1}(B_4)$ or $u \in L^{q \tilde \ell_1}(B_4)$. We have
$$ \frac{\ell_1}{\ell_0}=\frac{1}{p}\frac{N}{N-2\ell_0s}>\frac{1}{p}\frac{N}{N-2s}>\ell_0.
$$
This implies that $\ell_1>\ell_0^2>\ell_0>1$. Similarly, $\tilde \ell_1>\tilde \ell_0^2>\tilde \ell_0>1$. Now if $\ell_1$ or $\tilde \ell_1\not=\frac{N}{2s}$, then continuing the bootstrap method as in the proof of \cite[Theorem 1.6]{BN}), we can conclude that
$u, \, v \in L^\infty(B_{2(k+1)})$. Consequently, $u, \, v \in L_{loc}^\infty(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$.
(ii) If $\mu,\nu \in L^r(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)\cap L_{loc}^\infty(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$ then by part (i), we have $v^p+\mu,\, u^q+\nu \in L^\infty_{loc}(\Om)$. Further, as $u, v\in L^1(\Om)$, applying Schauder estimate \cite{RS1}, we have $u, v\in C^{\alpha}_{loc}(\Om)$, for some $\alpha\in(0,2s)$.
\end{proof}
\section{Construction of a second solution}
In this section we assume $1<p \leq q <N_s$. Then it follows that $q\frac{p+1}{q+1}<N_s$. Using Linking theorem, we will construct a second weak solution of \eqref{source-3} when $\mu,\nu \in L^r(\Omega)$, for $r>\frac{N}{2s}$ with $\|\mu\|_{L^r(\Omega)}=\|\nu\|_{L^r(\Omega)}=1$.
By Theorem \ref{existcoup-1}, if $\rho>0$ and $\tau>0$ are small then there exists the minimal positive week solution, denoted by $(\underline u_{\rho \mu}, \underline v_{\tau \nu})$, of \eqref{source-3}.
We would like to apply Linking theorem to find a variational weak solution of \begin{equation} \label{MP-prob} \left\{ \begin{aligned} (-\De)^s u &=(\underline v_{\tau \nu}+v^+)^p- \underline v_{\tau \nu}^p \quad &&\text{in }\Om\\ (-\De)^s v &=(\underline u_{\rho \mu}+u^+)^q- \underline u_{\rho \mu}^q \quad &&\text{in }\Om\\ u &= 0=v &&\text{in } \Om^c, \end{aligned} \right. \end{equation} where $u^+:=\max(u, 0)$ and $u^-:=-\min(0, u)$.
From Remark \ref{Linf}, we observe that there exists a constant $M>0$ such that \be\lab{29-5-2}\max\{\underline u_{\rho \mu}, \underline v_{\tau \nu} \}<M \quad \text{in } \Omega. \ee
Define \be\label{eq:X0} X_0:=\{w\in H^s(\mathbb R^N): w=0\quad \text{in}\quad \mathbb R^N\setminus\Om\}, \ee where $H^s(\mathbb R^N)$ is the standard fractional Sobolev space on $\mathbb R^N$. It is well-known that \be\label{norm-X}
\|w\|_{X_0}:=\displaystyle\left(\int_{Q}\frac{|w(x)-w(y)|^2}{|x-y|^{N+2s}}dxdy\right)^\frac{1}{2},
\ee where $Q=\mathbb R^{2N}\setminus (\Om^c\times\Om^c)$, is a norm on $X_0$ and $(X_0, ||.||_{X_0})$ is a Hilbert space, with the inner product
$$\left\langle\phi, \psi\right\rangle_{X_0}:=\int_{Q}\frac{(\phi(x)-\phi(y))(\psi(x)-\psi(y))}{|x-y|^{N+2s}}dxdy.$$ Put $$2^*_s:=\frac{2N}{N-2s}.$$ It is easy to check that (see \cite{RS3}) $$\int_{\Omega} \psi(-\De)^s \phi\, dx=\int_{\mathbb R^N}(-\De)^\frac{s}{2}\phi(-\De)^\frac{s}{2}\psi \, dx\quad \forall \phi, \psi\in X_0.$$ It is also well known that the embedding $X_0\hookrightarrow L^r(\mathbb R^N)$ is compact, for any $r\in[1, 2_s^*)$ and $X_0\hookrightarrow L^{2_s^*}(\mathbb R^N)$ is continuous.
\begin{definition} \label{defstable}
We say that a solution $(u,v)$ of \eqref{eq:system} is stable (resp. semistable) if
\begin{equation} \label{stabex} \left\{ \begin{aligned} \| \phi} \def\vgf{\varphi} \def\gh{\eta \|_{X_0}^2 > \,(\textrm{resp.\,}\geq)\,\, p\int_{\Om}v^{p-1}\phi^2 dx, \\
\| \phi} \def\vgf{\varphi} \def\gh{\eta \|_{X_0}^2 > \,(\textrm{resp.\,}\geq)\,\, q\int_{\Om}u^{q-1}\phi^2 dx,
\end{aligned} \right.
\quad \forall\, \phi \in X_0 \setminus \{0\}.
\end{equation} \end{definition}
\begin{proposition} \label{propstable}
Assume $p,q\in(1, N_s)$ and $\mu, \nu$ are positive functions in $L^r(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$ for some $r>\frac{N}{2s}$ such that $\| \mu \|_{L^r(\Omega)}= \| \nu \|_{L^r(\Omega)}=1$. For $\rho>0$ and $\tau>0$ small, let $(\underline u_{\rho \mu},\underline v_{\tau \nu})$ be the minimal solution of \eqref{source-3} obtained in Theorem \ref{existcoup-1}. There exists $t_0 >0$ such that if $\max\{ \rho,\tau \}<t_0$ then $(\underline u_{\rho \mu},\underline v_{\tau \nu})$ is stable. Moreover, there exists a positive constant $C=C(N,s,p,q,t_0)$ such that
\begin{equation} \label{strictstab} \left\{ \begin{aligned}
\|\phi\|^2_{X_0}- p\int_{\Om}\underline v_{\tau \nu}^{p-1}\phi^2 dx\geq C\|\phi\|^2_{X_0}, \\
\|\phi\|^2_{X_0}- q\int_{\Om}\underline u_{\rho \mu}^{q-1}\phi^2 dx\geq C\|\phi\|^2_{X_0},
\end{aligned} \right.
\quad \forall\, \phi} \def\vgf{\varphi} \def\gh{\eta \in X_0 \setminus \{ 0 \}.
\end{equation} \end{proposition} \begin{proof}
{\bf Step 1: }We show that there exists $t_0>0$ such that $(\underline u_{\rho\mu},\underline v_{\tau\nu})$ is stable provided $\max\{ \rho,\tau \}<t_0$.
Indeed, from Remark \ref{Linf},
it follows that for any $\phi\in X_0 \setminus \{0\}$, there exists $t_0>0$ small such that if $\max\{ \rho,\tau \}<t_0$, there hold
$$ \begin{aligned} \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} \underline v_{\tau \nu}^{p-1}\phi^2\, dx\leq \|\underline v_{\tau \nu}\|_{L^\infty(\Om)}^{p-1} \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\phi^2dx\leq \frac{1}{p}\|\phi\|^2_{X_0}, \\
\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} \underline u_{\rho \mu}^{q-1}\phi^2\, dx\leq \|\underline u_{\rho \mu}\|_{L^\infty(\Om)}^{q-1} \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\phi^2dx\leq \frac{1}{q}\|\phi\|^2_{X_0}.
\end{aligned} $$
This completes Step 1.
\noindent {\bf Step 2:} We prove \eqref{strictstab}. Assume $(\rho,\tau) \in (0,t_0) \times (0,t_0)$ and put $$ \rho'=\frac{\rho+t_0}{2}, \quad \tau'=\frac{\tau+t_0}{2}. $$
Set
$$ \alpha} \def\gb{\beta} \def\gg{\gamma=\max\left \{ \big(\frac{\rho}{\rho'}\big)^\frac{1}{q}, \big(\frac{\tau}{\tau'}\big)^\frac{1}{p} \right\}<1. $$
Let $(\underline u_{\rho' \mu},\underline v_{\tau' \nu})$ and $(\underline u_{\rho \mu},\underline v_{\tau \nu})$ be the solutions of \eqref{source-3} with data $(\rho' \mu, \tau' \nu)$ and $(\rho \mu, \tau \nu)$ respectively. Since $p,\,q>1$ and $\al<1$, it is easy to see that
$$ \begin{aligned}
\alpha} \def\gb{\beta} \def\gg{\gamma \underline u_{\rho' \mu} &=\BBG_s[\alpha} \def\gb{\beta} \def\gg{\gamma \underline v_{\tau' \nu}^p] + \BBG_s[\alpha} \def\gb{\beta} \def\gg{\gamma \rho' \mu] \geq \BBG_s[(\alpha} \def\gb{\beta} \def\gg{\gamma \underline v_{\tau'\nu})^p] + \BBG_s[\rho \mu], \\
\alpha} \def\gb{\beta} \def\gg{\gamma \underline v_{\tau' \nu} &=\BBG_s[\alpha} \def\gb{\beta} \def\gg{\gamma \underline u_{\rho' \mu}^q] + \BBG_s[\alpha} \def\gb{\beta} \def\gg{\gamma \tau' \nu ] \geq \BBG_s[(\alpha} \def\gb{\beta} \def\gg{\gamma \underline u_{\rho' \mu})^q] + \BBG_s[\tau \nu]. \\
\end{aligned} $$
Consequently, in view of the proof of Lemma \ref{compa}, we deduce $\alpha} \def\gb{\beta} \def\gg{\gamma \underline u_{\rho' \mu}\geq \underline u_{\rho \mu}$ and $\alpha} \def\gb{\beta} \def\gg{\gamma \underline v_{\tau' \nu}\geq \underline v_{\tau \nu}$. Furthermore, since $(\rho', \tau') \in (0,t_0) \times (0,t_0)$, by Step 1, we assert that $(\underline u_{\rho'\mu}, \underline v_{\tau' \nu})$ is stable. Therefore,
\begin{equation} \label{stab2} \begin{aligned}
0<\|\phi\|^2_{X_0}- p\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} \underline v_{\tau' \nu}^{p-1}\phi^2 dx &\leq \|\phi\|^2_{X_0}- p\alpha} \def\gb{\beta} \def\gg{\gamma^{1-p}\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} \underline v_{\tau \nu}^{p-1}\phi^2 dx\\
&=\al^{1-p}\big(\al^{p-1}\|\phi\|^2_{X_0}- p\int_{\Om}\underline v_{\tau \nu}^{p-1}\phi^2 dx\big).
\end{aligned} \end{equation}
Hence,
\begin{equation} \begin{aligned}
\|\phi\|^2_{X_0}- p\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} \underline v_{\tau \nu}^{p-1}\phi^2 dx &= (1-\al^{p-1})\|\phi\|^2_{X_0}+\al^{p-1}\|\phi\|^2_{X_0}- p\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} \underline v_{\tau \nu}^{p-1}\phi^2 dx\\
&>(1-\alpha} \def\gb{\beta} \def\gg{\gamma^{p-1})\|\phi\|^2_{X_0}.
\end{aligned} \end{equation}
Similarly, one can prove
$$ \|\phi\|^2_{X_0}- q\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} \underline u_{\rho \mu}^{q-1}\phi^2 dx >(1-\alpha} \def\gb{\beta} \def\gg{\gamma^{q-1})\|\phi\|^2_{X_0}.
$$
Hence \eqref{strictstab} holds with $C=\min\{ 1-\alpha} \def\gb{\beta} \def\gg{\gamma^{p-1}, 1- \alpha} \def\gb{\beta} \def\gg{\gamma^{q-1} \}$. \end{proof}
The norm of an element $z=(u,v)\in X_0\times X_0$ is defined by
$$\|z\|_{X_0\times X_0}:=\|(u,v)\|_{X_0\times X_0}=\big(\|u\|^2_{X_0}+\|v\|^2_{X_0}\big)^\frac{1}{2}.$$
\begin{definition}
Let $(X, \|.\|_{X})$ be a real Banach space with its dual $(X^*, \|.\|_{X^*})$ and $I\in C^1(X,\mathbb R)$. For $c\in\mathbb R$, we say that $I$ satisfies Cerami condition at level $c$ (in short, $(C)_c$) if for any sequence $\{w_n\}\subset X$ with
$$I(w_n)\to c, \quad \|I'(w_n)\|_{X^*}(1+\|w_n\|_X)\to0,$$
there is a subsequence $\{w_{n_k}\}$ of $\{w_n\}$ such that $\{w_{n_k}\}$ converges strongly in $X$.
We say that $\{w_n\}\subset X$ is a Palais-Smale sequence of $I$ at level $c$ if $$I(w_n)\to c, \quad \|I'(w_n)\|_{X^*}\to0.$$ \end{definition}
The energy functional associated to \eqref{MP-prob} is \begin{equation} \label{I} \begin{aligned}
I(u, v)&:=\displaystyle\int_{\mathbb R^N\times\mathbb R^N}\frac{\big(u(x)-u(y)\big)\big(v(x)-v(y)\big)}{|x-y|^{N+2s}}dxdy-\int_{\Om} H( \underline v_{\tau \nu}, v) dx \\ &\qquad-\displaystyle\int_{\Om} \tilde H(\underline u_{\rho \mu}, u) dx \quad\forall\, (u, v)\in X_0\times X_0, \end{aligned} \end{equation} where \begin{equation}\begin{aligned} &H(r,t):= \frac{1}{p+1}\bigg[(r+t^+)^{p+1}-r^{p+1}-(p+1)r^p t^+\bigg],\\ &\tilde H(r,t):= \frac{1}{q+1}\bigg[(r+t^+)^{q+1}-r^{q+1}-(q+1)r^q t^+\bigg], \quad r \geq 0. \end{aligned}\end{equation} Therefore,
\Bea I'(u,v)(\phi,\psi)&=\displaystyle\int_{\mathbb R^N\times\mathbb R^N}\frac{\big(\phi(x)-\phi(y)\big)\big(v(x)-v(y)\big)}{|x-y|^{N+2s}}dxdy\\
&\quad+\displaystyle\int_{\mathbb R^N\times\mathbb R^N}\frac{\big(u(x)-u(y)\big)\big(\psi(x)-\psi(y)\big)}{|x-y|^{N+2s}}dxdy\\ &\quad-\displaystyle\int_{\Om} h(\underline v_{\tau \nu}, v)\psi dx-\int_{\Om} \tilde h(\underline u_{\rho \mu}, u)\phi dx, \Eea where
$$h(r,t):=(r+t^+)^p-r^p \quad\text{and}\quad \tilde h(r,t):=(r+t^+)^q-r^q, \quad r \geq 0.$$ It is easy to see that if $z=(u,v)$ is a critical point of $I$ then $(u,v)$ solves \eqref{MP-prob}. We will find these critical points using Linking Theorem in the spirit of \cite{FMR}.
\begin{lemma}\lab{l:30-5-3} (i) There hold
\be\lab{20-12-17-3} \qquad \frac{1}{p+1}t^{p+1} < H(r,t), \quad \frac{1}{q+1}t^{q+1}< \tilde H(r,t) \quad\text{for}\quad r,\, t> 0.\ee
(ii) Given any $M>0$, there exist $\theta>2$ and $T>0$ such that
\be\lab{20-12-17-2}\begin{aligned}
H(r,t)\leq \frac{1}{\theta}h(r,t)t, \quad\tilde H(r,t)\leq \frac{1}{\theta}\tilde h(r,t)t, \quad\text{for}\quad 0\leq r\leq M,\,\, t\geq T,
\end{aligned}\ee
where $T$ depend on $M, p, q, \theta$.
(iii) Let $0<\kappa<p+1$, then there exists a constant $C=C(p,q,\kappa)>0$ such that
\be\lab{l:4-9-1}H(r, t), \, \tilde H(r,t)\geq t^\kappa-C \quad\text{for}\quad r,\, t> 0. \ee \end{lemma} \begin{proof}
(i) Estimate \eqref{20-12-17-3} was proved in \cite[Lemma C.2(ii)]{NS}.
(ii) First let us choose $\theta_1\in (2, p+1)$ arbitrarily and fix it. Next, we define
$$y(r, t):=h(r, t)t-\theta_1 H(r,t).$$
From the definition of $h(r,t)$ and $H(r,t)$, a straight forward computation yields that $$y(r,t)=t^2\bigg[p\big(1-\frac{\theta_1}{2}\big)r^{p-1}+\frac{p(p-1)}{2}\big(1-\frac{\theta_1}{3}\big)r^{p-2}t+\cdots+\big(1-\frac{\theta_1}{p+1}\big)t^{p-1}\bigg].$$ Therefore, there exits $0<T= T(p, M, \theta_1)$ such that $y(r,t)>0$ for $t\geq T,\, r\leq M$. Similarly we can prove the other inequality by choosing $\theta_2\in(2, q+1)$. Then by take $\theta=\min\{\theta_1,\theta_2\}$, we obtain \eqref{20-12-17-2}.
(iii) Since $\kappa<p+1\leq q+1$, applying Young's inequality, we have
$$t^\kappa \leq\frac{1}{p+1}t^{p+1}+c_1 \quad\text{and}\quad t^\kappa \leq \frac{1}{q+1}t^{q+1}+c_2$$
where $c_1=c_1(\kappa,p)$ and $c_2=c_2(\kappa,q)$.
Taking $C=\max\{c_1, c_2\}$, it follows $$\frac{1}{p+1}t^{p+1}, \, \frac{1}{q+1}t^{q+1}\geq t^\kappa-C. $$
Combining this with (i), \eqref{l:4-9-1} follows. \end{proof} \begin{remark}
Combining Lemma \ref{l:30-5-3} along with the fact that $H(r,t)=0$, for $t\leq 0$, it holds
\be\lab{30-5-4}
H(r,t)\geq 0, \quad \tilde H(r,t)\geq 0, \quad \forall\, t\in\mathbb R , \, \forall r\geq 0.
\ee \end{remark}
We also observe that (\cite[Lemma C.2(iii)]{NS}) for any $\varepsilon>0$, there exists $c_{\varepsilon}>0$, such that \be\lab{20-12-17-1} H(r,t)-\frac{p}{2}r^{p-1}t^2\leq \varepsilon r^{p-1}t^2+c_{\varepsilon}t^{p+1}, \quad r,\, t\geq 0.\ee
{\bf Notation}: For the rest of this section, we denote by $\| \cdot \|$, the norm in $X_0$, by $\|(\cdot,\cdot)\|$ the norm in $X_0\times X_0$ and by $\left\langle\cdot,\cdot\right\rangle$ the inner product in $X_0$.
Next, we prove that $I$ has the geometry of the Linking theorem.
\subsection {Geometry of the Linking Theorem} We define, $$E^+:=\{(u, u)\,:\, u\in X_0\}\quad\text{and}\quad E^-:=\{(u, -u)\,:\, u\in X_0\}.$$ \begin{lemma}\lab{l:30-5-1}
There exist $\varrho,\, \si>0$ such that $I(u, v)\geq\si$ for all $(u,v)\in S:=\pa B_{\varrho}\cap E^+$. \end{lemma} \begin{proof}
From the definition of $I(u,u)$, we have
\Bea
I(u, u)&=&\frac{1}{2}\bigg(\|u\|^2-p\int_{\Omega} \underline v_{\tau \nu}^{p-1}u^2dx\bigg)+\frac{1}{2}\bigg(\|u\|^2-q\int_{\Omega} \underline u_{\rho \mu}^{q-1}u^2dx\bigg)\\
&&\qquad-\bigg(\int_{\Omega} H(\underline v_{\tau \nu}, u)dx-\frac{p}{2}\int_{\Omega} \underline v_{\tau \nu}^{p-1}u^2 dx\bigg) -\bigg(\int_{\Omega}\tilde H(\underline u_{\rho \mu}, u)dx-\frac{q}{2}\int_{\Omega} \underline u_{\rho \mu}^{q-1}u^2dx \bigg).
\Eea
Applying \eqref{20-12-17-1} and \eqref{strictstab} to the above line and using \eqref{29-5-2} and Sobolev inequality, we obtain
\Bea
I(u, u)&\geq& C\|u\|^2-\varepsilon\int_{\Omega} \underline v_{\tau \nu}^{p-1}u^2dx-C_\varepsilon\int_{\Omega} u^{p+1}dx-\varepsilon\int_{\Omega} \underline u_{\rho \mu}^{q-1}u^2dx-C_\varepsilon\int_{\Omega} u^{q+1}dx\\
&\geq&(C-M^{p-1}S^{-1}\varepsilon-M^{q-1}S^{-1}\varepsilon)\|u\|^2-C\|u\|^{p+1}-C\|u\|^{q+1},
\Eea
where $S$ is the Sobolev constant. Now, choosing $\varepsilon>0$ and $\varrho>0$ small enough, we find one $\sigma>0$ such that
$I(u, u)\geq \si$ when $\|u\|=\varrho$, as $p, \, q>1$. This proves the lemma. \end{proof}
Let $\psi_0\in X_0$ be a fixed nonnegative function with $\|\psi_0\|=1$ and
$$Q_{\psi_0}:=\{r(\psi_0, \psi_0)+w\,:\, w\in E^-, \|w\|\leq R_0,\, 0\leq r\leq R_1 \}.$$
\begin{lemma}\lab{l:30-5-2}
There exist constants $R_0,\, R_1>0$, which depend on $\psi_0$, such that
$I(u, v)\leq 0$ for all $(u,v)\in \pa Q_{\psi_0}$. \end{lemma} \begin{proof}
We note that boundary $\pa Q_{\psi_0}$ of the set $Q_{\psi_0}$ is taken in the space $\mathbb R(\psi_0, \psi_0)\oplus E^-$ and consists of three parts. We estimate $I$ on these parts as below.
\noi {\bf Case 1}: $z\in\pa Q_{\psi_0}\cap E^-$ and of the form $z=(u,-u)\in E^-$. Then, thanks to \eqref{30-5-4}, it follows
$$I(z)=-\|u\|^2-\int_{\Omega} H(\underline v_{\tau \nu}, -u)dx-\int_{\Omega}\tilde H(\underline u_{\rho \mu}, u)dx\leq 0.$$
\noi {\bf Case 2}: $z=R_1(\psi_0, \psi_0)+(u,-u)\in \pa Q_{\psi_0}$ with $\|(u, -u)\|\leq R_0$. Thus,
\be\lab{4-9}I(z)=R_1^2\|\psi_0\|^2-\|u\|^2-\int_{\Omega} H(\underline v_{\tau\nu}, R_1\psi_0-u)dx-\int_{\Omega}\tilde H(\underline u_{\rho\mu}, R_1\psi_0+u)dx.\ee
For $2<\kappa<p+1 \leq q+1$, set
\begin{equation*} \xi(t): =\left\{ \begin{aligned} t^\kappa \quad &&\text{if }\, t\geq 0, \\ 0 \quad &&\text{if }\, t<0. \end{aligned} \right. \end{equation*}
Then, applying \eqref{30-5-4} and \eqref{l:4-9-1} to \eqref{4-9}, we get
\Bea I(z)&\leq& R_1^2-\int_{\Omega} \xi(R_1\psi_0-u)dx-\int_{\Omega} \xi(R_1\psi_0+u)dx+C
\Eea where $C=C(p,q,\kappa)$ is the constant in \eqref{l:4-9-1}.
Now, using convexity of the function $\xi$, we obtain
$$ I(z)\leq R_1^2-2R_1^\kappa\int_{\Omega} |\psi_0|^\kappa dx + C.$$
Therefore, since $\kappa>2$, taking $R_1$ large enough (depending on $\psi_0$), it follows that $I(z)\leq 0$.
\iffalse+++++++++++++++
Using H\"{o}lder inequality followed by Sobolev inequality along with the fact that $\|\psi\|=1$ and $\|u\|\leq \frac{R_0}{\sqrt{2}}$ we have
\Bea I(z)&\leq& R_1^2-\frac{R_1^{p+1}}{p+1}\|\psi_0\|_{L^{p+1}(\Om)}^{p+1}- \frac{R_1^{q+1}}{q+1}\|\psi_0\|_{L^{q+1}(\Om)}^{q+1}+ \frac{1}{\sqrt{2}}R_1^pR_0 S^{-p}.
\Eea
Now to make $I(z)\leq 0$, we choose $R_0$ and $R_1$ such that
\be\lab{4-9-2}R_0\leq R_1S^p\sqrt{2}\bigg[\frac{1}{p+1}\|\psi_0\|_{L^{p+1}(\Om)}^{p+1}+\frac{1}{q+1}R_1^{q-p}\|\psi_0\|_{L^{q+1}(\Om)}^{q+1}-R_1^{1-p}\bigg].\ee
++++++++++++\fi
\noi {\bf Case 3}: $z=r(\psi_0, \psi_0)+(u,-u)\in \pa Q_{\psi_0}$ with $\|(u, -u)\|= R_0$ and $0\leq r\leq R_1$.
Then, using \eqref{30-5-4} it follows that
$$I(z)\leq r^2\|\psi_0\|^2-\|u\|^2\leq R_1^2-\frac{1}{2}R_0^2.$$ Choosing $R_0\geq \sqrt{2}R_1$, we have $I(z)\leq 0$.
Combining case 2 and case 3, in order that the geometry of Linking theorem holds, we choose $R_0$, $R_1$ large enough with $R_0\geq \sqrt{2}R_1$.
\end{proof}
Our next aim is to prove that Cerami sequences are bounded.
\begin{proposition}\lab{p:30-5-1}
Let $(u_m, v_m)\in X_0\times X_0$ such that
(i) $I(u_m, v_m)=c+\de_m$, where $\de_m\to 0$ as $m\to\infty$.
(ii) $(1+\|(u_m, v_m)\|)|I'(u_m, v_m)(\phi,\psi)|\leq \varepsilon_m \|(\phi, \psi)\|$ for $\phi,\psi\in X_0\times X_0$ and $\varepsilon_m\to 0$ as $m\to\infty$.
Then,
\begin{eqnarray*}
\|u_m\|\leq C, &\quad \|v_m\|\leq C\\
\int_{\Omega} h(\underline v_{\tau \nu}, v_m)v_mdx\leq C, &\quad \displaystyle\int_{\Omega} \tilde h(\underline u_{\rho \mu}, u_m)u_mdx\leq C\\
\int_{\Omega} H(\underline v_{\tau \nu}, v_m)dx\leq C, &\quad \displaystyle\int_{\Omega} \tilde H(\underline u_{\rho \mu}, u_m)dx\leq C.
\end{eqnarray*} \end{proposition} \begin{proof}
Choosing $(\phi,\psi)=(v_m,0)$ and $(\phi,\psi)=(0, u_m)$ in (ii), we have
\begin{equation} \label{21-2-2} \begin{aligned}
\left| \| v_m \|^2 - \int_{\Omega} \tilde h(\underline u_{\rho\mu},u_m)v_m dx\right| \leq \varepsilon_m \| v_m \|, \\
\left| \| u_m \|^2 - \int_{\Omega} h(\underline v_{\tau\nu},v_m)u_mdx \right| \leq \varepsilon_m \| u_m \|.
\end{aligned} \end{equation}
Now choosing $(\phi,\psi)=(u_m,0)$ and $(\phi,\psi)=(0, v_m)$ in (ii), we have
\begin{equation} \label{21-2-2b} \begin{aligned}
\left|\<u_m,v_m\right\rangle-\int_{\Omega} \tilde h(\underline u_{\rho \mu},u_m)u_m dx\right|\leq \varepsilon_m,\\
\bigg|\<u_m,v_m\right\rangle-\int_{\Omega} h(\underline v_{\tau \nu},v_m)v_m dx\bigg|\leq\varepsilon_m.
\end{aligned} \end{equation}
On the other hand, from (i), we obtain
\begin{equation}\label{21-2-3}
\<u_m,v_m\right\rangle-\int_{\Omega} H(\underline v_{\tau \nu}, v_m)dx-\int_{\Omega} \tilde H(\underline u_{\rho \mu}, u_m)dx=c+\de_m.
\end{equation}
Combining \eqref{21-2-2} and \eqref{21-2-3} and using \eqref{20-12-17-2}, we get
\Bea
2c+2\de_m&=&\<u_m,v_m\right\rangle-2\int_{\Omega} H(\underline v_{\tau \nu}, v_m)dx+\<u_m,v_m\right\rangle-2\int_{\Omega} \tilde H(\underline u_{\rho \mu}, u_m)dx\\
&\geq&-2\varepsilon_m+\int_{\Omega} \tilde h(\underline u_{\rho \mu},u_m)u_mdx+\int_{\Omega} h(\underline v_{\tau \nu},v_m)v_mdx \\
&&-2\int_{\Omega} \tilde H(\underline u_{\rho \mu}, u_m)dx-2\int_{\Omega} H(\underline v_{\tau \nu}, v_m)dx\\
&\geq& -2\varepsilon_m+(\theta-2)\int_{\Om\cap\{v_m>T\}} H(\underline v_{\tau \nu},v_m)dx+(\theta-2)\int_{\Om\cap\{u_m>T\}} \tilde H(\underline u_{\rho \mu},u_m)dx+ C,
\Eea
where we have used the fact
$$\int_{\Om\cap\{v_m\leq T\}}H(\underline v_{\tau\nu}, v_m)dx<C \quad \text{and} \quad \int_{\Om\cap\{u_m\leq T\}}\tilde H(\underline u_{\rho\mu}, v_m)dx<C,
$$
which follows from the definition of $h(r,t)$, $H(r,t)$ and \eqref{leub}. Therefore,
\begin{equation}\label{29-5-1}
\int_{\Omega} H(\underline v_{\tau \nu},v_m)dx\leq C \quad\text{and}\quad \int_{\Omega}\tilde H(\underline u_{\rho \mu},u_m)dx\leq C.
\end{equation}
Using \eqref{20-12-17-2}, similarly it can be also shown that
\be\lab{30-5-8}
\int_{\Omega} h(\underline v_{\tau \nu}, v_m)v_mdx\leq C,\quad \int_{\Omega} \tilde h(\underline u_{\rho \mu}, u_m)u_mdx\leq C.
\ee
Observe that $h(\underline v_{\tau \nu}, v_m)=0$ if $v_m\leq 0$ and $h(\underline v_{\tau \nu},v_m)u_m\leq 0$ if $u_m\leq 0$. Therefore applying Young's inequality, \eqref{20-12-17-3}, \eqref{29-5-1} and the fact that $\underline u_{\rho \mu}$ and $\underline v_{\tau \nu}$ are bounded (see \eqref{29-5-2}) yields
\begin{equation} \label{31-5-1} \begin{aligned}
\int_{\Omega} h(\underline v_{\tau \nu},v_m)&u_mdx \leq \int_{\Om\cap\{v_m\geq 0,\, u_m\geq 0\}}h(\underline v_{\tau \nu},v_m)u_mdx\\
&\leq\frac{1}{q+1}\int_{\Om\cap\{v_m\geq 0,\, u_m\geq 0\}}u_m^{q+1}dx+\frac{q}{q+1}\int_{\Om\cap\{v_m\geq 0,\, u_m\geq 0\}}h(\underline v_{\tau \nu},v_m)^\frac{q+1}{q}dx\\
&\leq\int_{\Omega} \tilde H(\underline u_{\rho \mu}, u_m)dx+ \frac{q}{q+1}\int_{\Om\cap\{v_m\geq 0,\, u_m\geq 0\}} (\underline v_{\tau \nu}+v_m^+)^\frac{p(q+1)}{q}dx\\
&\leq C_1+C(q)\bigg(\int_{\Omega} \underline v_{\tau \nu}^\frac{p(q+1)}{q}dx+ \int_{\Om\cap\{v_m\geq 0,\, u_m\geq 0\}} v_m^\frac{p(q+1)}{q}dx\bigg)\\
&\leq C_1+ C_2+ C_3\bigg(\int_{\Om\cap\{v_m\geq 0,\, u_m\geq 0\}} v_m^{p+1}dx\bigg)^\frac{(q+1)p}{(p+1)q}|\Om|^\frac{q-p}{(p+1)q}\\
&\leq C_1+ C_2+ C_4 \bigg(\int_{\Om\cap\{v_m\geq 0,\, u_m\geq 0\}}H(\underline v_{\tau \nu}, v_m)dx\bigg)^\frac{(q+1)p}{(p+1)q} <C.
\end{aligned} \end{equation}
In above estimate we have also used the fact $p \leq q$ implies $(q+1)p/q \leq p+1$.
Similarly we can show that $\int_{\Omega}\tilde h(\underline u_{\rho \mu},u_m)v_mdx<C$. Therefore substituting back in \eqref{21-2-2}, we obtain $\|u_m\|\leq C$ and $\|v_m\|\leq C$. \end{proof}
\subsection{Finite dimensional problem} Since the functional $I$ is strongly indefinite and defined in infinite dimensional space, no suitable linking theorem is available. We therefore approximate \eqref{MP-prob} with a sequence of finite dimensional problems.
Associated to the eigenvalues $0<\la_1<\la_2\leq\la_3\leq\cdots\to\infty$ of $((-\De)^s, X_0)$, there exits an orthogonal basis $\{\va_1, \va_2, \cdots\}$ of corresponding eigen functions in $X_0$ and $\{\va_1, \va_2, \cdots\}$ is an orthonormal basis for $L^2(\Om)$. We set \begin{eqnarray*}
E_n^+ &:=&\text{span}\{(\va_i, \va_i)\,:\, i=1,2, \cdots, n\},\\
E_n^-&:=&\text{span}\{(\va_i, -\va_i)\,:\, i=1,2, \cdots, n\},\\
E_n&:=&E_n^+\oplus E_n^-. \end{eqnarray*}
Let $\psi_0\in X_0$ be a fixed nonnegative function with $\|\psi_0\|=1$ and
$$Q_{n,\psi_0}:=\{r(\psi_0, \psi_0)+w\,:\, w\in E_n^-, \|w\|\leq R_0,\, 0\leq r\leq R_1 \},$$ where $R_0$ and $R_1$ are chosen in Lemma \ref{l:30-5-2}. Here we recall that these constants depend only on $\psi_0,\, p,\, q$. Next, define $$H_{n,\psi_0}:=\mathbb R(\psi_0,\psi_0)\oplus E_n,\quad H^+_{n,\psi_0}:=\mathbb R(\psi_0,\psi_0)\oplus E_n^+,\quad H^-_{n,\psi_0}:=\mathbb R(\psi_0,\psi_0)\oplus E_n^-.$$ $$\Ga_{n,\psi_0}:=\{\pi\in C(Q_{n,\psi_0}, H_{n,\psi_0})\,:\, \pi(u, v)=(u,v)\,\, \text{on}\,\, \pa Q_{n,\psi_0}\},$$ and $$c_{n,\psi_0}:=\inf_{\pi\in \Ga_{n,\psi_0}}\max_{(u,v)\in Q_{n,\psi_0}}I\big(\pi(u,v)\big).$$ Using an intersection theorem (see \cite[Proposition 5.9]{R}), we have $$\pi(Q_{n,\psi_0})\cap (\pa B_{\varrho}\cap E^+)\not=\emptyset, \quad\forall\, \pi\in\Ga_{n,\psi_0}.$$ Thus there exists an $(u,v)\in (Q_{n,\psi_0})$ such that $\pi(u,v)\in \pa B_{\varrho}\cap E^+$. Combining this with Lemma \ref{l:30-5-1}, we get $I(\pi(u,v))\geq\si$. This in turn implies $c_{n,\psi_0}\geq \si>0$. Our next goal is to show that $c_{n,\psi_0}$ has an upper bound. For that, we observe that the identity map $\text{Id} : Q_{n,\psi_0}\to H_{n,\psi_0}$ is in $\Ga_{n,\psi_0}$. Thus for an element of the form $z:=r(\psi_0,\psi_0)+(u,-u)\in Q_{n,\psi_0}$, we compute \Bea I(z)&=&\<r\psi_0+u, r\psi_0-u\right\rangle-\int_{\Omega} H(\underline v_{\tau\nu}, r\psi_0-u)dx-\int_{\Omega}\tilde H(\underline u_{\rho\mu}, r\psi_0+u)dx\\
&=&r^2-\|u\|^2-\bigg[\int_{\Omega} H(\underline v_{\tau\nu}, r\psi_0-u)dx+\int_{\Omega}\tilde H(\underline u_{\rho\mu}, r\psi_0+u)dx\bigg]\leq R_1^2,\\ \Eea where in the last inequality we have used \eqref{30-5-4}. Consequently, $$ \max_{z \in Q_{n,\psi_0}}I(z) \leq R_1^2. $$ Therefore, $$ c_{n,\psi_0} \leq \max_{z \in Q_{n,\psi_0}}I(\text{Id}(z)) = \max_{z \in Q_{n,\psi_0}}I(z) \leq R_1^2. $$ Hence $0<\si\leq c_{c,\psi_0}\leq R_1^2$. We remark here that upper and lower bound do not depend on $n$. Define,
$$I_{n,\psi_0}:= I\big|_{H_{n,\psi_0}}.$$ Thus, in view of Lemmas \ref{l:30-5-1} and \ref{l:30-5-2}, we see that geometry of Linking theorem holds for the functional $I_{n,\psi_0}$. Hence applying the linking theorem \cite[Theorem 5.3]{R} to $I_{n,\psi_0}$, we obtain a Palais-Smale sequence, which is bounded in view of Proposition \ref{p:30-5-1} (also see \cite[pg. 1046]{FMR}). Therefore, using the fact that $H_{n,\psi_0}$ is a finite dimensional space, we obtain the following proposition:
\begin{proposition}\lab{p:30-5-2}
For every $n\in\mathbb N$ and for every $\psi_0\in X_0$, a fixed nonnegative function with $\|\psi_0\|=1$, the functional $I_{n,\psi_0}$ has a critical point $z_{n,\psi_0}$ such that
\be\lab{30-5-5}z_{n,\psi_0}\in H_{n,\psi_0}, \quad I_{n,\psi_0}'(z_{n,\psi_0})=0,\quad I_{n,\psi_0}(z_{n,\psi_0})=c_{n,\psi_0}\in[\si, R_1^2],\ee
\be\lab{30-5-6} \quad \|z_{n,\psi_0}\|\leq C,\ee
where $C$ does not depend on $n$. \end{proposition}
\subsection{Existence of solution of \eqref{MP-prob}.}
{\bf Step 1:} Let $\psi_0\in X_0$ be a fixed nonnegative function with $\|\psi_0\|=1$. Then applying Proposition \ref{p:30-5-2}, we get a sequence $\{z_{n,\psi_0}\}_{n=1}^\infty$ satisfying \eqref{30-5-5} and \eqref{30-5-6}. Consequently, there exists $(u_0, v_0)\in X_0\times X_0$ such that \be\lab{30-5-7}z_{n,\psi_0}:=(u_{n,\psi_0}, v_{n,\psi_0})\rightharpoonup (u_0, v_0) \quad\text{in}\quad X_0\times X_0, \ee \be\lab{30-5-7'}u_{n,\psi_0}\to u_0,\quad v_{n,\psi_0}\to v_0 \quad\text{in}\quad L^r(\mathbb R^N), \,1\leq r<2^*_s \quad\text{and a.e. in}\quad \Om. \ee Further, applying Proposition \ref{p:30-5-1}, we conclude \be\lab{30-5-9} \int_{\Omega} h(\underline v_{\tau \nu}, v_{n,\psi_0})v_{n,\psi_0}dx\leq C, \quad \displaystyle\int_{\Omega} \tilde h(\underline u_{\rho \mu}, u_{n,\psi_0})u_{n,\psi_0}dx\leq C. \ee \be\lab{30-5-10} \int_{\Omega} H(\underline v_{\tau \nu}, v_{n,\psi_0})dx\leq C, \quad \displaystyle\int_{\Omega} \tilde H(\underline u_{\rho \mu}, u_{n,\psi_0})dx\leq C. \ee Next, taking as test functions $(0,\psi)$ and $(\phi, 0)$ in \eqref{30-5-5}, where $\phi,\,\psi$ are arbitrary functions in $F_n:=\text{span}\{\va_i\,:\, i=1,2,\cdots, n\}$, we obtain \be\lab{30-5-11} \left\langle\psi, u_{n,\psi_0}\right\rangle=\int_{\Omega} h(\underline v_{\tau \nu}, v_{n,\psi_0})\psi dx\quad\forall\, \psi\in F_n, \ee \be\lab{30-5-12} \left\langle\phi, v_{n,\psi_0}\right\rangle=\int_{\Omega} \tilde h(\underline u_{\rho \mu}, u_{n,\psi_0})\phi dx \quad\forall\, \phi\in F_n. \ee Now applying \eqref{29-5-2}, \eqref{30-5-7'} and the fact that $p<N_s<2^*_s$, we also have $h(\underline v_{\tau \nu}, v_{n,\psi_0})$ and $h(\underline v_{\tau \nu}, v_0)$ are $L^1$ functions. Therefore, using \eqref{30-5-9}, \eqref{30-5-7'} and an argument similar to the one used in \cite[Lemma 2.1]{FMR1}, it follows that $$h(\underline v_{\tau \nu}, v_{n,\psi_0})\to h(\underline v_{\tau \nu}, v_0), \quad\tilde h(\underline u_{\rho \mu}, u_{n,\psi_0})\to \tilde h(\underline u_{\rho \mu}, u_0) \quad\text{in}\quad L^1(\Om).$$ Hence, taking the limit in \eqref{30-5-11} and \eqref{30-5-12} and using the fact that $\cup_{n=1}^\infty F_n$ is dense in $X_0$, it follows that \be \left\langle\psi, u_{0}\right\rangle=\int_{\Omega} h(\underline v_{\tau \nu}, v_0)\psi dx \quad \text{and}\quad \left\langle\phi, v_0\right\rangle=\int_{\Omega} \tilde h(\underline u_{\rho \mu}, u_0)\phi dx,\no \ee for all $\phi, \psi\in X_0$. As a consequence, \be\lab{31-5-3}(-\De)^s u_0=h(\underline v_{\tau \nu}, v_0) \quad \text{and}\quad (-\De)^s v_0=\tilde h(\underline u_{\rho \mu}, u_0),\quad u_0=v_0=0 \quad\text{in}\, \Om^c.\ee
{\bf Step 2:} In this step we show that $u_0$ and $v_0$ are nontrivial and nonnegative.
Suppose not, we assume $u_0\equiv 0$ in $X_0$. Plugging back to the equation \eqref{31-5-3}, it implies $(-\De)^s v_0=0$. As a consequence
$$0=\int_{\Omega} v_0(-\De)^s v_0\, dx =\int_{\Omega} |(-\De)^\frac{s}{2}v_0|^2\, dx= \|v_0\|_{X_0}^2.$$ Therefore, $v_0\equiv 0$ in $X_0$, that is, $u_{n, \psi_0}\rightharpoonup 0$ and $v_{n, \psi_0}\rightharpoonup 0$ in $X_0$. Consequently, $u_{n, \psi_0}\to 0$ and $v_{n, \psi_0}\to 0$ in $L^r(\Om)$ for $1\leq r<2^*_s$. Since $p+1,\, q+1<2^*_s$, computing as in \eqref{31-5-1} we obtain \Bea \lim_{n\to\infty}\int_{\Omega} h(\underline v_{\tau \nu},v_{n,\psi_0})u_{n,\psi_0}dx &\leq&\lim_{n\to\infty}C\bigg(\int_{\Om\cap\{v_{n,\psi_0}\geq 0,\, u_{n,\psi_0}\geq 0\}}u_{n,\psi_0}^{q+1}dx\\ &&\qquad\qquad+\int_{\Om\cap\{v_{n,\psi_0}\geq 0,\, u_{n,\psi_0}\geq 0\}}h(\underline v_{\tau \nu},v_{n,\psi_0})^\frac{q+1}{q}dx\bigg)\\ &\leq&\lim_{n\to\infty} C \int_{\Om\cap\{v_{n,\psi_0}\geq 0\}}[(\underline v_{\tau \nu}+v_{n,\psi_0})^p-\underline v_{\tau \nu}^p]^\frac{q+1}{q}dx\\
&\leq&\lim_{n\to\infty} C\int_{\Omega} \bigg[|v_{n,\psi_0}|^p+|\underline v_{\tau\nu}|^{p-1}|v_{n,\psi_0}| \bigg]^\frac{q+1}{q}dx\\
&\leq&\lim_{n\to\infty} C\bigg[\int_{\Omega} |v_{n,\psi_0}|^{\frac{p}{q}(q+1)}dx+\int_{\Omega}|\underline v_{\tau \nu}|^{\frac{1}{q}(p-1)(q+1)}|v_{n,\psi_0}|^\frac{q+1}{q}dx\bigg]\\ &=&0, \Eea where for the last inequality we have used the fact that $\frac{p}{q}(q+1) \leq p+1$ (since $p \leq q$), \eqref{29-5-2} and the fact that $\frac{q+1}{q}<2^*_s$ (since $\frac{N-2s}{N+2s}<1<q$). Similarly it follows that $\int_{\Omega} \tilde h(\underline u_{\rho \mu},u_{n,\psi_0})v_{n,\psi_0}dx \to 0$. As a consequence, taking $\psi=u_{n,\psi_0}$
in \eqref{30-5-11} and $\phi=v_{n,\psi_0}$ in \eqref{30-5-12} yields $\|u_{n,\psi_0}\|\to 0$ and $\|v_{n,\psi_0}\|\to 0$ respectively. Hence, $u_{n, \psi_0}\to 0$ and $v_{n, \psi_0}\to 0$ strongly in $X_0$. This in turn, implies $\<u_{n,\psi_0}, v_{n,\psi_0}\right\rangle\to (0, 0)$.
Further, combining \eqref{20-12-17-1} along with \eqref{30-5-4} and \eqref{29-5-2} yields $\int_{\Omega} H(\underline v_{\tau \nu}, v_{n,\psi_0})dx\to 0$ and $\int_{\Omega}\tilde H(\underline u_{\rho \mu}, u_{n,\psi_0})dx\to 0$. Hence, $c_{n,\psi_0}=I_{n,\psi_0}(u_{n,\psi_0}, v_{n,\psi_0})\to 0$. This is a contradiction to the fact that $c_{n,\psi_0}\in [\si, R_1^2]$. Therefore, $u_0, \, v_0$ are nontrivial.
Since $u_0\in X_0$, by direct computation it is easy to see that $u_0^+,\, u_0^-\in X_0$. Thus, taking the test function as $u_0^-$ for the first equation in \eqref{31-5-3} yields $0\leq \<u_0^+, u_0^-\right\rangle - \|u_0^-\|^2\leq -\|u_0^-\|^2$, i.e., $u_0\geq 0$ a.e.. Similarly, $v_0\geq 0$. As a result, step 2 follows.
Hence we obtain the existence of a nonnegative nontrivial solution $(u_0, v_0)\in X_0\times X_0$ of \eqref{MP-prob}.
\subsection{Proof of Theorem \ref{2nd sol} completed}
In order to construct the second solution of \eqref{source-3}, we define
\be\lab{31-5-4}u_{\rho \mu}=\underline u_{\rho \mu}+u_0, \quad v_{\tau \nu}=\underline v_{\tau \nu}+v_0,\ee
where $(u_0, v_0)$ are as defined in Step 1 of Section 5.3.
Clearly $u_{\rho \mu}\geq \underline u_{\rho \mu}$ and $v_{\tau \nu}\geq \underline v_{\tau \nu}$. Moreover, as $u$ and $v$ are nontrivial element in $X_0$, there exist two positive measure sets $\Om', \Om''\subset\Om$ such that $u_0>0$ in $\Om'$ and , $v_0>0$ in $\Om''$. Thus $u_{\rho \mu}>\underline u_{\rho \mu}$ in $\Om'$ and $v_{\tau \nu}>\underline v_{\tau \nu}$ in $\Om''$. Further, as $(u_0, v_0)\in X_0\times X_0$ is a solution of \eqref{MP-prob}, we have
\begin{equation} \label{ru}
\<u_0, \psi\right\rangle=\int_{\Omega} h(\underline v_{\tau \nu}, v_0)\psi dx, \quad \<v_0, \phi\right\rangle=\int_{\Omega}\tilde h(\underline u_{\rho \mu}, u_0)\phi dx, \quad \forall\, \phi, \, \psi \in X_0.
\end{equation}
Set
$$ {\mathcal T}} \def\CU{{\mathcal U}} \def\CV{{\mathcal V}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi):=\{ \tilde\psi \in C^\infty(\Omega} \def\Gx{\Xi} \def\Gy{\Psi): \text{ there exists } \psi \in C_0^\infty(\Omega} \def\Gx{\Xi} \def\Gy{\Psi) \text{ such that } \tilde\psi=\BBG_s[\psi] \}. $$
This is a space of test function defined in \cite[Page 41]{A}. By \cite[Lemma 5.6]{A}, ${\mathcal T}} \def\CU{{\mathcal U}} \def\CV{{\mathcal V}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi) \subset X_0$. Therefore, we deduce from \eqref{ru} that
\begin{equation}\begin{aligned} \label{ru2} \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} u_0 {(-\Gd)^s} \psi \,dx &= \<u_0, \psi\right\rangle = \int_{\Omega} h(\underline v_{\tau \nu}, v_0)\psi dx\quad \forall\, \psi \in {\mathcal T}} \def\CU{{\mathcal U}} \def\CV{{\mathcal V}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi),\\
\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} v_0 {(-\Gd)^s} \phi \,dx &= \<v_0, \phi\right\rangle = \int_{\Omega} \tilde h(\underline u_{\rho \mu}, u_0)\phi dx\quad \forall\, \phi \in {\mathcal T}} \def\CU{{\mathcal U}} \def\CV{{\mathcal V}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi).
\end{aligned}\end{equation}
Moreover, \cite[Lemma 5.12 and Lemma 5.13]{A} ensures that ${\mathcal T}} \def\CU{{\mathcal U}} \def\CV{{\mathcal V}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi) \subset \BBX_s(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$ and
\begin{equation}\begin{aligned} \label{ru3} \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} u_0 {(-\Gd)^s} \psi \,dx &= \int_{\Omega} h(\underline v_{\tau \nu}, v_0)\psi dx\quad \forall\, \psi \in \BBX_s(\Omega} \def\Gx{\Xi} \def\Gy{\Psi),\\
\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} v_0 {(-\Gd)^s} \phi \,dx &= \int_{\Omega}\tilde h(\underline u_{\rho \mu}, u_0)\phi dx\quad \forall\, \phi \in \BBX_s(\Omega} \def\Gx{\Xi} \def\Gy{\Psi).
\end{aligned}\end{equation}
This means that $(u_0, v_0)$ is a weak solution of
\begin{equation} \label{ru3} \left\{
\begin{aligned}
{(-\Gd)^s} u_0 &=(\underline v_{\tau \nu}+v_0)^p- \underline v_{\tau \nu}^p \quad \text{in }\Omega} \def\Gx{\Xi} \def\Gy{\Psi\\
{(-\Gd)^s} v_0 &=(\underline u_{\rho \mu}+u_0)^q- \underline u_{\rho \mu}^q \quad \text{in }\Omega} \def\Gx{\Xi} \def\Gy{\Psi\\
u_0=v_0 &= 0 \quad \text{in } \Omega} \def\Gx{\Xi} \def\Gy{\Psi^c,
\end{aligned} \right.
\end{equation}
Hence, $(u_{\rho \mu}, v_{\tau \nu})$, as defined in \eqref{31-5-4}, is clearly a weak solution of \eqref{source-3}.
If $\mu,\nu \in L^r(\Omega} \def\Gx{\Xi} \def\Gy{\Psi) \cap L_{loc}^\infty(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$ then by Theorem \ref{reg}, $u_{\rho \mu}, v_{\tau \nu}$, $\underline u_{\rho \mu}, \underline v_{\tau \nu}\in C^{\alpha}_{loc}(\Om)$, for some $\al\in(0,2s)$. Therefore, $u_0, v_0\in C^{\alpha}_{loc}(\Om)$. Also since we have $(-\De)^s u_0$, $(-\De)^s v_0\geq 0$ in $\Om$ and $0\not\equiv u_0, v_0\geq 0$ in $\mathbb R^N$, applying the strong maximum principle \cite[Proposition 2.17]{Si}, we have $u_0, v_0>0$ in $\Omega$. Hence from \eqref{31-5-4}, we deduce $u_{\rho \mu}>\underline u_{\rho \mu}$ and $v_{\tau \nu}>\underline v_{\tau \nu}$. In view of Remark \ref{l:3-9-1}, this completes the proof. \qed
{\bf Acknowledgement:} M. Bhakta is partially supported by DST/INSPIRE 04/2013/000152 and MATRICS grant MTR/2017/000168.
\end{document} | arXiv |
Fitting a smooth curve to data points $d_0,\dots,d_n$ lying on a Riemannian manifold $\mathcal M$ and associated with real-valued parameters $t_0,\dots,t_n$ is a common problem in applications like wind field approximation, rigid body motion interpolation, or sphere-valued data analysis. The resulting curve should strike a balance between data proximity and a smoothing regularization constraint.
In this talk we present the general framework of optimization on manifolds. We then introduce a variational model to fit a composite Bézier curve to the set of data points $d_0,\dots,d_n$ on a Riemannian manifold $\mathcal M$. The resulting curve is obtained in such a way that its mean squared acceleration is minimal in addition to remaining close the data points. We approximate the acceleration by discretizing the squared second order derivative along the curve. We derive a closed-form, numerically stable and efficient algorithm to compute the gradient of a Bézier curve on manifolds with respect to its control points. This gradient can be expressed as a concatenation of so called adjoint Jacobi fields. Several examples illustrate the capabilities of this approach both for interpolation and approximation. | CommonCrawl |
\begin{document}
\baselineskip=22pt \title[The rumor percolation model and its variations]{The rumor percolation model and its variations} \author{Valdivino~V.~Junior} \author{F\'abio~P.~Machado} \author{Krishnamurthi~Ravishankar} \address[F\'abio~P.~Machado] {Institute of Mathematics and Statistics \\ University of S\~ao Paulo \\ Rua do Mat\~ao 1010, CEP 05508-090, S\~ao Paulo, SP, Brazil - [email protected] }
\noindent \address[Valdivino~V.~Junior] {Federal University of Goias \\ Campus Samambaia, Goi\^ania, GO, Brazil - [email protected]}
\address[Krishnamurthi Ravishankar] {NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062 and 1555 Century Ave, Pudong, Shanghai, China - [email protected]}
\noindent
\thanks{Research supported by CNPq (310829/2014-3), FAPESP (09/52379-8), PNPD-Capes 536114 and Simons Foundation Collaboration grant 281207}
\keywords{epidemic model, galton-watson trees, rumor model, spherically symmetric trees.}
\subjclass[2010]{60K35, 60G50}
\date{\today}
\begin{abstract} The study of rumor models from a percolation theory point of view has gained a few adepts in the last few years. The persistence of a rumor, which may consistently spread out throughout a population can be associated to the existence of a giant component containing the origin of a graph. That is one of the main interest in percolation theory. In this paper we present a quick review of recent results on rumor models of this type. \end{abstract}
\maketitle
\tableofcontents
\section{Introduction and basic definitions} \label{S: Introduction}
We are interested in a long-range percolation model on infinite graphs which we call the \textit{Rumor Percolation Model}. Such models have recently been studied by a few authors in a series of papers. The dynamics of the model describes the spreading of a rumor on a graph in the following way. We assign independent random \textit{radius of influence} $R_v$ to each vertex $v$ of an infinite, locally finite, connected graph $\mathcal G$. Then we define a chain reaction on $\mathcal G$ according to the following simple rules: (1) at time zero, only the root (a fixed vertex of $\mathcal G$) hears the rumor, (2) at time $n \ge 1$, a new vertex hears the rumor if it is a distance at most $R_v$ of some vertex $v$ that previously heard the rumor. We point out that similar models, are of interest in Computer Science, in particular in the area of distributed networks. One of the problems of interest is the broadcasting problem where one node has some information which it wants to pass on to other nodes. Questions of optimal algorithm for achieving this goal are of interest. This question was considered for the case where the nodes are uniformly randomly distributed on an interval $[0,L]$ and the nodes had a transmission radius of one. In \cite{RS94} asymptotically (in $L$) optimal algorithm was obtained.
\begin{defn} \label{D: DefinicaoCPP} The \textit{Rumor Percolation Model}\ on $\mathcal G$.\\ Let $\mathcal G = ({\mathcal V},{\mathcal E})$ be an infinite, locally finite, connected graph and let $\{R_v\}_{\{ v \in {\mathcal V} \}}$ be a set of independent and identically distributed random variables. Furthermore, for each $u \in {\mathcal V}$, we define the random sets \begin{equation} \label{E: defBu} B_u = \{v \in {\mathcal V}: d(u,v) \leq R_u\}. \end{equation} or \begin{equation} \label{E: defBuC} B_u = \{v \in {\mathcal V}, u \leq v : d(u,v) \leq R_u\}. \end{equation} \noindent With these sets we define the \textit{Rumor Percolation Model}\ on ${\mathcal G}$, the non-decreasing sequence of random sets $I_0 \subset I_1 \subset \cdots$ defined as $ I_0 = \{{\mathcal O}\} $ and inductively $I_{n+1} = \bigcup_{u \in I_n} B_u$ for all $ n \geq 0.$ \end{defn}
\begin{defn} \label{D: Survival} The \textit{Rumor Percolation Model}\ $survival$.\\
Consider $ I = \bigcup_{n \geq 0} I_n$ be the connected component of the origin of $\mathcal G$. Under the rumor process interpretation, $I$ is the set of vertices which heard the rumor. We say that the process \emph{survives} (\emph{dies out}) if $|I|=\infty$ ($|I|<\infty$), referring to the surviving event as $V.$ \end{defn}
In section 2 we review the paper of Athreya~\textit{et al}~\cite{ARS}. Instead of considering a graph structure they consider a homogeneous Poisson point process on ${{{\mathbb R}}^d}$ and ${{{\mathbb R}}^d}^+$ with $\{R_v\}$, the \textit{box of influence}, starting from every point $v$ of the point process in the sense of~(\ref{E: defBuC}). They work with the concept of the coverage of a set $(t, \infty)^d$ for some $t > 0$, the \textit{eventual coverage}. In section 3 we review the paper of Lebenstayn and Rodriguez~\cite{LR} where authors consider the $\textit{Disk Percolation Model}$. While the set of \textit{radius of influence}, $\{R_v\}_{\{ v \in {\mathcal V} \}}$, has a geometric distribution, the graph ${\mathcal G}$ is quite general. In their version the \textit{radius of influence} of a vertex $v \in {\mathcal G}$ goes in every possible direction as in~(\ref{E: defBu}). In section 4 we review the papers of Junior \textit{et al}~\cite{Junior} and Gallo \textit{et al}~\cite{GGJR}. They work with a processes that they made known as Fireworks on ${\mathbb N}$ (direct and reverse). They studied an homogeneous version, where there is one informant per vertex and the \textit{radius of influence} are independent and have the same distribution, and a heterogeneous version, where one of these conditions fail. In their models the \textit{radius of influence} goes like in~(\ref{E: defBuC}). In section 5 the papers of Junior {\it et al}~\cite{Junior},~\cite{Junior2} and~\cite{Junior3} are briefly reviewed. They work with the Cone Percolation model, a Fireworks model in a tree (homogeneous, spherically symmetric, periodic or Galton Watson). In all these models the the \textit{radius of influence} goes like in~(\ref{E: defBuC}). In section 6 we review the paper of Bertacchi and Zucca~\cite{BZ}. They consider a type of random environment in the sense that the number of informants in each vertex of are random.
\section{Random sets on ${{{\mathbb R}}^d}$ and ${{{\mathbb R}}^d}^+$} \label{S: RS}
The theory of coverage processes was introduced by P. Hall~\cite{Hall} in 1988. He developed a class of stochastic processes intended to be used as a model for \textit{binary images}, that is, images which partition ${\mathbb R}^d$ into two regions, ${\mathcal C}$ and its complement, representing the ``black'' and ``white'' parts of an image. In its basic version the process consists of a point process $P = \{\xi_1, \xi_2, \dots \}$ and a collection of random sets $\{S_1, S_2, \dots \}$. The ``black'' region ${\mathcal C}$ is then defined to be ${\mathcal C} = \cup_{i=1}^{\infty} (\xi_i + S_i).$ P. Hall~\cite{Hall} developed probabilistic results on geometrical properties of ${\mathcal C}$, such as the size-distribution of its connected subsets. In that work the main assumptions needed to obtain explicit results is that $P$ is an homogeneous Poisson process and the $S_i$ are independent copies of a random closed set. This version is known as the Poisson Boolean model.
Athreya~\textit{et al}~\cite{ARS} considered two different models, both related to rumor percolation. For the first model, arising for genome analysis, they consider $\{X_i\}_{i \in {\mathbb N}}$ be a $\{0,1\}$-valued time-ho\-mo\-ge\-neous Markov chain and $\{\rho_i\}_{i \in {\mathbb N}}$ an independent and identically distributed sequence of random variables assuming values on ${\mathbb N}$, independent of the Markov chain. Let $S_i = [i, i +\rho_i]$ whenever $X_i=1$ ($\emptyset$ otherwise) and $C = \cup_{i=1}^{\infty} S_i$.
\begin{defn} \label{D: EventuallyCovered} We say that ${\mathbb N}$ is eventually covered by $C$ (or $C$ eventually covers ${\mathbb N}$) if there exists a $ t \ge 1$ such that $[t, \infty) \subseteq C$. \end{defn}
\begin{teo}[Athreya \textit{et al}~\cite{ARS}]
\label{T: ARS1.1} Let $p_{ij} = {\mathbb P}(X_{n+1}=j | X_n=i)$. Assume that $0< p_{00}, p_{10}<1,$ \begin{itemize} \item[\textit{(i)}] If \[ l = \liminf_{j \to \infty} j {\mathbb P}(\rho_1 > j) > 1, \] then \[ {\mathbb P}(C \hbox{eventually covers } {\mathbb N}) = 1\] whenever \[ \frac{p_{01}}{p_{10}+p_{01}} > \frac{1}{l}.\] \item[\textit{(ii)}] If \[ L = \limsup_{j \to \infty} j {\mathbb P}(\rho_1 > j) < \infty, \] then \[ {\mathbb P}(C \hbox{eventually covers } {\mathbb N}) = 0\] whenever \[ \frac{p_{01}}{p_{10}+p_{01}} < \frac{1}{L}.\] \end{itemize} \end{teo}
Their second model aims to complement known results on complete coverage in stochastic geometry. For $B(0, \rho)$ the closed $d$-dimensional ball of radius $\rho$ centered at the origin, some important previous results for the random covered region $\displaystyle{\cup_{i=1}^{\infty} (\xi_i + B(0,\rho_i))}$ are presented in the next two theorems.
\begin{teo}[Hall~\cite{Hall}] \label{T: Hall} For the Poisson Bolean model on ${\mathbb R}^d$ the space is fully covered by $\cup_{i=1}^{\infty} (\xi_i + B(0,\rho_i))$ almost surely if and only if ${\mathbf E}(\rho^d) = \infty$. \end{teo}
If instead of a Poisson point process one considers an arbitrary ergodic process, there is the following result
\begin{teo}[Meester and Roy~\cite{MR}] \label{T: MR} For the Bolean model on ${\mathbb R}^d$ the space is fully covered by $\cup_{i=1}^{\infty} (\xi_i + B(0,\rho_i))$ almost surely if ${\mathbf E}(\rho^d) = \infty$. \end{teo}
Athreya \textit{et al}~\cite{ARS} take ${\mathbb R}^d_+$ and the random covered region \begin{displaymath} C = \cup_{\{i: \xi \in {\mathbb R}^d_+\}} (\xi_i +[0,\rho_i]^d). \end{displaymath}
Guided by the fact that $C$ will never completely cover ${\mathbb R}^d_+$ because, for any $ \epsilon >0,$ ${[0, \epsilon]}^d$ will not be covered by $C$ with positive probability, they work with the notion of \textit{eventual coverage} for the orthant ${\mathbb R}^d_+$.
\begin{defn} \label{D: REventuallyCovered} We say that ${\mathbb R}^d_+$ is eventually covered by the Poisson Boolean model if there exists a $ t \in (0, \infty) $ such that ${[t, \infty)}^d \subseteq C$. \end{defn}
With this notion Athreya \textit{et al}~\cite{ARS} are able to present the following result, considering a Poisson Bolean model on ${\mathbb R}^d_+$. They show that eventual coverage depends on the growth rate of the distribution function of $\rho$ (even when ${\mathbf E}(\rho) = \infty$) as well as on whether $d=1$ or $d \ge 2$.
\begin{teo}[Athreya \textit{et al}~\cite{ARS}] \label{T: Royd=1} Assume $d=1$. \begin{itemize} \item[\textit{(i)}] If \[ 0 < l :=\liminf_{x \to \infty} x {\mathbb P}(\rho > x) < \infty, \] then there exists a $\lambda_0$ such that $0< \lambda_0 \le {1}/{l} < \infty$ and \[ {\mathbb P}_{\lambda}({\mathbb R}_+ \hbox{ is eventually covered by } C) = \left\{ \begin{array}{ll} 0 & \mbox{if $\lambda < \lambda_0$,} \\ 1 & \mbox{if $\lambda > \lambda_0$;} \end{array} \right. \] \item[\textit{(ii)}] If \[ 0 < L :=\limsup_{x \to \infty} x {\mathbb P}(\rho > x) < \infty, \] then there exists a $\lambda_1$ such that $0 < {1}/{L} \le \lambda_1 < \infty$ and \[ {\mathbb P}_{\lambda}({\mathbb R}_+ \hbox{ is eventually covered by } C) = \left\{ \begin{array}{ll} 0 & \mbox{if $\lambda < \lambda_1$,} \\ 1 & \mbox{if $\lambda > \lambda_1$;} \end{array} \right. \]
\item[\textit{(iii)}] If \[ \lim_{x \to \infty} x {\mathbb P}( \rho > x) = \infty, \] then for all $\lambda > 0, {\mathbb R}_+$ is eventually covered by $C$ (${\mathbb P}_{\lambda}-$a.s.);
\item[\textit{(iv)}] If \[ \lim_{x \to \infty} x {\mathbb P}( \rho > x) = 0, \] then for any $\lambda > 0, {\mathbb R}_+$ is eventually covered by $C$ (${\mathbb P}_{\lambda}-$a.s.).
\end{itemize} \end{teo}
\begin{teo}[Athreya \textit{et al}~\cite{ARS}] \label{T: Royd>1} Let $d \ge 2$. For all $\lambda > 0$, \begin{itemize}
\item[\textit{(i)}] If \[ \liminf_{x \to \infty} x {\mathbb P}( \rho > x) >0, \] then \[ {\mathbb P}_{\lambda}({\mathbb R}^d_+ \hbox{ is eventually covered by } C) = 1; \]
\item[\textit{(ii)}] If \[ \lim_{x \to \infty} x {\mathbb P}( \rho > x) = 0, \] then \[ {\mathbb P}_{\lambda}({\mathbb R}^d_+ \hbox{ is eventually covered by } C) = 0. \] \end{itemize} \end{teo}
It is interesting to observe that while ${\mathbf E}(\rho^d) = \infty$ guarantees complete coverage of ${\mathbb R}^d$ by $C$, it is not sufficient to guarantee \textit{eventual coverage} for ${\mathbb R}^d_+$. This is due to the fact that a boundary effect is present in the orthant ${\mathbb R}^d_+$ but absent in the whole space ${\mathbb R}^d$.
\section{Disk percolation} \label{S: DPM}
Lebensztayn and Rodriguez studied a long-range percolation model on infinite graphs, the $\textit{Disk Percolation Model}$. They assign a random \textit{radius of influence} $R_v$ to each vertex $v$ of an infinite, locally finite, connected graph $G$, so that all the assigned radii are independent and identically distributed random variables with geometric distribution with parameter$(1-p)$, which means, satisfying \[ {\mathbb P}(R = k) = (1-p)p^{k}, k=0,1,2,\dots \] Then they defined a growing process on $G$ according to the following rules: (1) at time zero, only the root (a fixed vertex of $G$)is declared infected, (2) at time $n \geq1$, a new vertex is infected if it is at graph distance at most $R_v$ of some vertex $v$ previously infected, and (3) infected vertices remain infected forever. They investigated the critical value $p_c(G)$ above which this process spreads indefinitely through the graph with positive probability.
They worked in a few settings including locally finite graphs in the sense that \[ \Delta = \sup_{v \in {\mathcal G}} \{d(v)\} < \infty \] where $d(v)$ is the number of neighbors (or {\textit{degree}) of a vertex $v$.
An interesting question is whether such a model presents
\textit{phase transition} in the sense that for $p_c({\mathcal G}) := \inf\{p: {\mathbb P}(|I|=\infty\})$ we have that \[0 < p_c({\mathcal G}) < 1.\] They provided an answer which relies on a comparison between the \textit{Disk Percolation Model}\ and the independent site percolation model. To understand this, consider $p_c^{site}({\mathcal G})$ the critical probability for the independent site percolation model on ${\mathcal G}$.
\begin{teo}[Lebenstayn and Rodriguez~\cite{LR}] \label{T: EP4} Let ${\mathcal G}$ be of bounded degree ($\Delta < \infty$) and be such that $p_c^{site}({\mathcal G}) < 1$. Then \begin{displaymath} 0 < p_c({\mathcal G}) < 1. \end{displaymath} \end{teo}
The proof they presented relies on the following two propositions, the first one is a comparison which gives an upper bound to $p_c({\mathcal G})$.
\begin{prop}[Lebenstayn and Rodriguez~\cite{LR}] \label{P: EP2} \begin{displaymath} p_c({\mathcal G}) \leq p_c^{site}({\mathcal G}) \end{displaymath} \end{prop}
\noindent while the second one gives a lower bound for the case that ${\mathcal G}$ is of bounded degree.
\begin{pro}[Lebenstayn and Rodriguez~\cite{LR}] \label{P: EP3} Suppose that ${\mathcal G}$ is a graph of bounded degree. Then \begin{displaymath} p_c({\mathcal G}) \ge -1 + {\left(1+ \frac{1}{\Delta -1}\right)}^{1/2}. \end{displaymath} \end{pro}
\subsection{Disk percolation on trees} \label{S: Trees}
Consider a tree ${\mathbb T}$ (a connected graph with no cycles) and its set of vertices ${\mathcal V}({\mathbb T})$. We say that a tree, ${\mathbb T}_d$, is \textit{homogeneous}, if each one of its vertices has degree (number of neighbours) $d+1$.
\begin{teo}[Lebenstayn and Rodriguez~\cite{LR}] \label{T: EP10} For any $d \ge 2$ \begin{displaymath} -1 + \left(1-\frac{1}{d}\right)^{1/2} \leq p_c({\mathbb T}_d) \leq 1 - \left(1-\frac{1}{d}\right)^{1/2}. \end{displaymath} \end{teo}
\begin{cor}[Lebenstayn and Rodriguez~\cite{LR}] \label{C: EP11} For any $d \ge 2$ \begin{displaymath} p_c({\mathbb T}_d) = {1/(2d)} + {\mathcal O}({1/{d^2}}) \text{ as } d \to \infty. \end{displaymath} \end{cor}
Single out one vertex from ${\mathcal V}({\mathbb T})$ and call this ${\mathcal O}$, the origin of ${\mathcal V}({\mathbb T})$. For each two vertices $u,v \in {\mathcal V}({\mathbb T})$, consider that $ u \leq v$ if $u$ belongs to the path connecting ${\mathcal O}$ to $v$.
For a tree ${\mathbb T}$ and $n \geq 1$ we define
\begin{displaymath} {\mathbb T}^u := \{ v \in \mathcal{V}: u \leq v \}, \end{displaymath} \begin{displaymath} {\mathbb T}_n^u := \{ v \in {\mathbb T}^u: d(v,{\mathcal O}) \leq d(u,{\mathcal O}) + n \} \end{displaymath} and \begin{equation*} \label{E: defPartialT}
M_n(u) := | \partial {\mathbb T}_n^u | := |\{ v \in {\mathbb T}^u: d(v,{\mathcal O}) = d(u,{\mathcal O}) + n\}|. \end{equation*}
\begin{defn} Let us define for a tree ${\mathbb T}$ \begin{displaymath} {\textrm{dim\ inf\ } \partial {\mathbb T}} : = \lim_{n \rightarrow \infty} \min_{v \in \mathcal{V}} \frac{1}{n} \ln M_n(v). \end{displaymath} \end{defn}
Observe that \[ \textrm{dim\ inf\ } \partial {\mathbb T}_d = \ln d.\]
\begin{defn} \label{D: SST} We say that a tree, ${\mathbb T}_S$, is \textit{spherically symmetric}, if any pair of vertices at the same distance from the origin, have the same degree. \end{defn}
\begin{teo}[Lebenstayn and Rodriguez~\cite{LR}] \label{T: EP9} For any spherically symmetric tree ${\mathbb T}_S$ \[ p_c ({\mathbb T}_S) \le 1-{\left(1- e^{-dim\ inf \partial {\mathbb T}_S}\right)}^{1/2} \] \end{teo}
\section{Fireworks on ${\mathbb N}$} \label{S: N}
The Fireworks processes are another interesting version of the \textit{Rumor Percolation Model}. Junior \textit{et al}~\cite{Junior} and Gallo \textit{et al}~\cite{GGJR} recently studied discrete time stochastic systems on ${\mathbb N}$ modeling processes of rumor spreading. In their models the involved individuals can either have an active role, working as spreaders and transmiting the information within a random distance to their right, or a passive role, hearing the information from spreaders within a random distance to their left. The appetite in spreading or hearing the rumor is represented by a set of random variables whose distributions may depend on the individuals positions on ${\mathbb N}$. Their main goal is to understand - based on the distribution of those random variables - whether the probability of having an infinite set of individuals knowing the rumor is positive or not.
Junior \textit{et al}~\cite{Junior} manage to write the survival event as a limit of an increasing sequence of events whose probability can be bounded by a nice use of FKG inequality. The use of a non-standard version of Borel-Cantelli lemma helped in the task of finding conditions for the processes to die out. Gallo \textit{et al}~\cite{GGJR} based the proofs of their results on a clever relationship between the rumor processes and a specific discrete time renewal process. With this technique they were able to obtain more precise results for homogeneous versions of the processes.
Consider $\{u_i\}_{i \in {\mathbb N}}$ a set of vertices of ${\mathbb N}$ such that $ 0 < u_1 < u_2 < \cdots $ and a set of independent random variables $\{R_i\}_{i \in {\mathbb N}}$ assuming values in ${\mathbb Z}_+$.
\subsection{Fireworks} \label{S: FP}
At time 0, information travels a distance $R_0$ towards the right side of the origin, in such a way that all vertices $u_i \le R_0$ get informed. In general, at every discrete time $t$ a vertex $u_j$ informed at time $t-1$ passes the information on (whithin $R_j$, its \textit{radius of influence}) and they do this just once, informing the vertices $u_i$ (only those vertices which have not been informed before) $u_j < u_i \le u_j + R_j.$ Observe that, except for the set of vertices $\{u_i\},$ all other vertices are nonactionable, meaning that their \textit{radius of influence} equals 0 almost surely.
\subsubsection{Homogeneous Fireworks} \label{S: hoc}
Consider all the $R_i \sim R$ (having the same distribution) and $u_i=i$ for all $i$.
\begin{teo}[Junior \textit{et al}~\cite{Junior}] \label{T: CriterioJunior} Consider in the \textit{Homogeneous Fireworks Process} \[a_n=\prod_{i=0}^{n}{\mathbb P}(R \leq i).\]
Then \[ \sum_{n=1}^{\infty}a_n = \infty \hbox{ if and only if } {\mathbb P}[V]=0. \] \end{teo}
\begin{teo}[Gallo \textit{et al}~\cite{GGJR}] \label{T: CriterioGallo} For the \textit{Homogeneous Fireworks Process}, \[ {\mathbb P} (V) = \left [ 1 + \sum_{j=1}^{\infty} \prod_{i=0}^{j-1}{\mathbb P}(R \leq i) \right ]^{-1}. \] \end{teo}
Observe that the result presented in Theorem~\ref{T: CriterioJunior} is nicely generalized in Theorem~\ref{T: CriterioGallo}.
\begin{exa} Consider the \textit{Homogeneous Fireworks Process}\ such that \begin{equation*} \label{E: Power Law} {\mathbb P}(R = k) = \frac{2}{(k+2)(k+3)} \textrm { for } k \in {\mathbb N}^*. \end{equation*} Then ${\mathbb P}[V]=\frac{1}{2}.$ \end{exa}
\begin{cor}[Junior \textit{et al}~\cite{Junior}] \label{C: FPHo} For the \textit{Homogeneous Fireworks Process}, consider \[ L = \lim_{n \rightarrow \infty}n{\mathbb P}(R \geq n). \]
We have that \begin{enumerate} \item[\textit{(i)}] If $L > 1$ then ${\mathbb P}[V]>0.$ \item[\textit{(ii)}] If $L < 1$ then ${\mathbb P}[V]=0.$ \item[\textit{(iii)}] If $L = 1$ and there exists $N$ such that for all $n \geq N$ \begin{displaymath} {\mathbb P}(R \geq n) \leq \frac{1}{n-1}, \textit{ then } {\mathbb P}[V]=0. \end{displaymath} \end{enumerate} \end{cor} Let $M$ be the final number of spreaders. \begin{teo}[Gallo \textit{et al}~\cite{GGJR}] \label{T: numinf} If ${\mathbf E}(R) < \infty$ then the random variable $M$ has finite expectation. Besides, $M$ has exponential tail distribution when ${\mathbb P}(R \leq n)$ increases exponentially fast to $1$. \end{teo}
Under more specific assumptions, it is possible to obtain more precise information on the tail distribution. Items \textit{(i)} and \textit{(iii)} of next proposition follows from Proposition B.2 of Gallo \textit{et al.}~\cite{Gallo}, item \textit{(ii)} is due to Remark 5 from Bressaud \textit{et al.}~\cite{Bressaud} and item \textit{(iv)} follows from Theorem 1.1 of Garsia and Lamperti~\cite{Garsia}.
\begin{prop}[Gallo \textit{et al}~\cite{GGJR}] \label{coro1}
We have the following explicit bounds for the tail distributions. \begin{enumerate} \item[\textit{(i)}] If ${\mathbb P}(R > k) \leq C_{r}r^{k},\,k\ge1$, for some $r\in(0,1)$ and a constant $C_{r}\in(0,\log \frac{1}{r})$ then \[ {\mathbb P}(M \ge k)\leq \frac1{C_{r}}(e^{C_{r}}r)^{k}. \] \item[\textit{(ii)}] If ${\mathbb P}(R > k) \sim(\log k)^{\beta}k^{-\alpha}$, $\beta\in {\mathbb R}$, $\alpha>1$, then there exists $C>0$ such that, for large $k$'s, we have ${\mathbb P}(M\ge k)\leq C(\log k)^{\beta}k^{-\alpha}.$ \item[\textit{(iii)}] If ${\mathbb P}(R > k)= \frac{r}{k},\,k\ge1$ where $r\in(0,1)$, there exists $C>0$ such that, for large $k$, we have \[ {\mathbb P}(M\ge k)\leq C\frac{(\ln k)^{3+r}}{(k)^{2-(1+r)^{2}}}. \] \item[\textit{(iv)}] If ${\mathbb P}(R > k) \sim ((k+1)/(k+2))^{\alpha}$, $\alpha\in(1/2,1)$, then there exists $C=C(\alpha)>0$ such that, for large $k$, we have \[ {\mathbb P}(M\ge k)\leq \frac{C}{k^{1-\alpha}}. \] \end{enumerate} \end{prop}
\subsubsection{Heterogeneous Fireworks}
\begin{rem} \label{R: PFHeMorte} Consider the \textit{Heterogeneous Fireworks Process}. One can get a sufficient condition for ${\mathbb P}[V]=0$ (${\mathbb P}[V]>0$) by a coupling argument. Consider ${\mathbb P}(R_i \geq k ) \leq \mathbf{P}(R \geq k)$ (${\mathbb P}(R_i \geq k ) \geq \mathbf{P}(R \geq k)$) for some random variable $R$ whose distribution $\mathbf{P}$ satisfies $\lim_{n \rightarrow \infty}n \mathbf{P}(R \geq n) < 1$ ($\lim_{n \rightarrow \infty}n \mathbf{P}(R \geq n) > 1$). Finally use item ($ii$) (item ($i$)) of Corollary~\ref{C: FPHo}. \end{rem}
\begin{teo}[Junior \textit{et al}~\cite{Junior}] \label{T: PFHeVive} Consider a \textit{Heterogeneous Fireworks Process}\ for which actionable vertices are at integer positions $u_0 = 0 < u_1 < u_2 < \dots $ such that $u_{n+1} - u_n \leq m$, for $m \geq 1.$ Besides, let us assume ${\mathbb P} (R_{n} < m) \in (0,1)$ for all $n.$ Then \begin{enumerate} \item[\textit{(i)}] If $\sum_{n=0}^{\infty}[{\mathbb P}(R_{n} < tm)]^t < \infty$ for some $t \geq 1$ then ${\mathbb P}[V]>0.$ \item[\textit{(ii)}] If for some random variable $R,$ with distribution $\mathbf{P},$ the following conditions hold \begin{itemize} \item $\mathbf{P}(R \geq k) - {\mathbb P}(R_{n} \geq k) \leq b_k$ for all $ k \geq 0$ and all $n \geq 0,$ \item $\lim_{n \rightarrow \infty}n[\mathbf{P}(R \geq n) - b_n] > m,$ \item $\lim_{n \rightarrow \infty}b_n = 0,$ \end{itemize} then ${\mathbb P}[V]>0.$ \item[\textit{(iii)}] ${\mathbb P}(V) \geq \prod_{j=0}^{\infty}\Big[1-\prod_{i=0}^{j}{\mathbb P}(R_{j-i} < (i+1)m)\Big].$ \end{enumerate} \end{teo}
\subsection{Reverse Fireworks} \label{S: RFP}
At time 0, only the origin has the information. At time 1, individuals placed at vertices $u_i$ such that $u_i \le R_i $ get the information from the origin. At time $t \in {\mathbb N}$ the set of vertices $ u_j $ which can find an informed individual at time $t-1$ within a distance $R_j$ to its left, get the information. Let us call this set $A_t$. If for some $t$, $A_t$ is empty the process stops. If the process never stops we say it survives.
Let $S$ be the event ``the reverse process survives". Besides, we denote by $Z$ the final number of spreaders.
\subsubsection{Homogeneous Reverse Fireworks} \label{S: horc}
Consider all the $R_i$ having the same distribution and $u_i=i$ for all $i$.
\begin{teo}[Junior {\it et al}~\cite{Junior}] \label{T: horfpJunior} Consider the \textit{Homogeneous Reverse Fireworks Process}. We have that \begin{enumerate} \item[\textit{(i)}] If $\mathbb{E}(R) = \infty$ then ${\mathbb P}(S) = 1.$ \item[\textit{(ii)}] If $\mathbb{E}(R) < \infty$ then ${\mathbb P}(S) = 0.$ \end{enumerate} \end{teo}
\begin{teo}[Gallo {\it et al}~\cite{GGJR}] \label{T: horfpGallo} Consider the \textit{Homogeneous Reverse Fireworks Process}. If $\mathbb{E}(R) < \infty$ then $Z \sim {\mathcal G}\displaystyle \left (\prod_{k=0}^{\infty}{\mathbb P}(R \leq k) \right )$ in the sense that for $p=\prod_{k=0}^{\infty}{\mathbb P}(R \leq k)$ we have \[{\mathbb P}(Z=k) = p (1-p)^k \hbox{ for all } k.\] \end{teo}
For any $n\ge1$, let $Z(n)$ be the number of spreaders in $\{1,\ldots,n\}$. We will now state limit theorems for the proportion of spreaders within $\{1,\ldots,n\}$, $Z(n)/n$, when $n$ tends to $\infty.$
Let \begin{align*} \label{eq: mu1} \mu&:=1+\sum_{j=1}^{\infty}\prod_{i=0}^{j-1}{\mathbb P}(R \leq i)\ \textrm{ and }\\ \sigma^2&:= \sum_{k=1}^{\infty}k^2{\mathbb P}(R > k-1)\prod_{i=0}^{k-2}{\mathbb P}(R \leq i)-\mu^2. \end{align*}
Notice that $\mu < \infty$ implies that $ \displaystyle \prod_{k=0}^{\infty}{\mathbb P}(R \leq k)= 0$ (this implies ${\mathbf E}{(R)} = \infty$) .
\begin{teo}[Gallo {\it et al}~\cite{GGJR}] \label{theo:Nn}If $\mu<\infty$ then \[ \frac{Z(n)}{n}\stackrel{a.s.}{\longrightarrow}\mu^{-1}, \] and thus, with probability one, $\mu^{-1}$ is the final proportion of spreaders. Moreover, if $\sigma^2 \in(0,\infty)$, then \[ \sqrt{n}\left(\frac{Z(n)}{n}-\mu^{-1}\right)\stackrel{\mathcal{D}}{\rightarrow}\mathcal{N}\left(0,\frac{\sigma^2}{\mu^3}\right). \]
Otherwise, $Z(n)/n \rightarrow 0$. \end{teo}
In particular, observe that if the ${\mathbb P}(R \leq k)$'s satisfy at the same time $\prod_{k=0}^{\infty} {\mathbb P}(R \leq k)=0$ and $\mu=\infty$ (for instance, if they are as in items (iii) and (iv) of Proposition~\ref{coro1}), then the information reaches infinitely many individuals, but the final proportion of informed individuals is zero.
\subsubsection{Heterogeneous Reverse Fireworks} \begin{teo}[Junior {\it et al}~\cite{Junior}] \label{T: PFRHe} Consider the \textit{Heterogeneous Reverse Fireworks Process}. It holds that \begin{enumerate} \item[\textit{(i)}] $\sum_{k=1}^{\infty}{\mathbb P}(R_{n+k} \geq k) = \infty$ for all $n$ if and only if ${\mathbb P}(S) = 1.$ \item[\textit{(ii)}] If $\sum_{n=1}^{\infty}\prod_{k=1}^{\infty}{\mathbb P}(R_{n+k} < k) < \infty$ then ${\mathbb P}(S)> 0.$ \end{enumerate} \end{teo}
\begin{rem} \label{R: PFRHe} By a coupling argument and Theorem~\ref{T: horfpJunior} one can see that if there is a random variable $R$, whose distribution is $\mathbf{P}$, with ${\mathbf E}[R] < \infty$ (${\mathbf E}[R] = \infty$), such that ${\mathbb P}(R_{n} \geq k) \leq \mathbf{P}(R \geq k)$ (${\mathbb P}(R_{n} \geq k) \geq \mathbf{P}(R \geq k)$) for all $k$ then ${\mathbb P}(S) = 0$ (${\mathbb P}(S) = 1$). \end{rem}
\begin{exa} \label{E: um} It is possible to have in the \textit{Heterogeneous Fireworks Process}\ the expectation of the \textit{radius of influence} infinite for all vertices toghether and the process dies out almost surely.
Let $\{b_n\}_{n \in {\mathbb N}}$ be a non-increasing sequence convergent to 0 and such that $b_0<1.$ \begin{itemize} \item[\textit{(i)}] ${\mathbb P}(R_n = 0) = 1 - b_n$ and ${\mathbb P}(R_n = k) = b_{n+k-1} - b_{n+k}$ for $k \geq 1.$ \item[\textit{(ii)}] $\sum_{n=0}^{\infty}b_n = \infty.$ \item[\textit{(iii)}] $\lim_{n \rightarrow \infty} n b_n = 0.$ \end{itemize} Observe that $\mathbb{E}(R_n) = \infty$ for all $n$ from \textit{(ii)}. Besides ${\mathbb P}[V]=0$ from \textit{(iii)}, because For \[V_n = \{ \hbox{The individual at vertex } u_n \hbox{ gets the information}\},\] \begin{equation} \label{E: example} {\mathbb P}(V_n) \leq \sum_{k=0}^{n-1}{\mathbb P}(R_k \geq n-k) = \sum_{k=0}^{n-1}b_{n-1} = (n-1)b_n. \end{equation} and the fact that $V = \lim_{n \to \infty} V_n.$ \end{exa} \begin{exa} It is possible to have in the \textit{Heterogeneous Fireworks Process}\ the expectation of the \textit{radius of influence} finite for all vertices and the process survives with positive probability. Assume that $\sum_{n=0}^{\infty}b_n < \infty,$ while \begin{itemize} \item[\textit{(i)}] ${\mathbb P}(R_n = 0) = b_n$ \item[\textit{(ii)}] ${\mathbb P}(R_n = 1) =1 - b_n$ \end{itemize} Then $\mathbb{E}(R_n) < 1$ for all $n$ and $\mathbb{P}(V) > 0$ by item ($i$) of Theorem~\ref{T: PFHeVive} with $m=t=1$. \end{exa} \begin{exa} Next we present an example where ${\mathbb P}[S]=1$ for a \textit{Heterogeneous Reverse Fireworks Process}\ while ${\mathbb P}[V]=0$ for a \textit{Heterogeneous Fireworks Process}. For this aim consider \begin{enumerate} \item[\textit{(i)}] ${\mathbb P}(R_n = 0) = 1 - b_n$ and ${\mathbb P}(R_n = n) = b_n.$ \item[\textit{(ii)}] $\sum_{n=0}^{\infty}b_n = \infty.$ \item[\textit{(iii)}] $\lim_{n \rightarrow \infty} n b_n = 0.$ \end{enumerate} Observe that even though $\lim_{n \to \infty} {\mathbf E}[R_n] = 0$ and $\lim_{n \to \infty} {\mathbb P}(R_n = 0) = 1,$ from Theorem~\ref{T: PFRHe} and \textit{(ii)} it is true for the \textit{Heterogeneous Reverse Fireworks Process}\ that $\mathbb{P}(S) = 1.$ In the opposite direction, by~(\ref{E: example}) and \textit{(iii)} one have that ${\mathbb P}[V]=0$ for the \textit{Heterogeneous Fireworks Process}. \end{exa}
\section{Cone percolation on ${\mathbb T}_d$} \label{S: MR}
Junior \textit{et al}~\cite{Junior2} consider a process which allows us to associate the dynamic activation on the set of vertices to a discrete rumor process. Individuals become spreaders as soon as they hear the rumor. Next time, they propagate the rumor within their \textit{radius of influence} and immediately become stiflers. Junior \textit{et al}~\cite{Junior2} establish whether the process has positive probability of involving an infinite set of individuals. Besides, they present sharp lower and upper bounds for the probability of that event, depending on the general distribution of the random variables that define the \textit{radius of influence} of each individual. Their proofs are based on comparisons with branching processes.
Pick a $v \in {\mathcal V}({\mathbb T}_d)$ such that $d({\mathcal O},v)=1$ and consider ${\mathbb T}^+_d = {\mathbb T}_d \backslash {\mathbb T}_d^+(v).$ Consider ${\mathbb P}_+$ and ${\mathbb P}$ the probability measures associated to the processes on ${\mathbb T}_d^+$ and ${\mathbb T}_d$ (we do not mention the random variable $R$ unless absolutely necessary). By a coupling argument one can see that for a fixed distribution of $R$ \begin{equation*} {\mathbb P}_+[V] \leq {\mathbb P}[V]. \end{equation*}
Furthermore, by the definition of ${\mathbb T}_d^+$ and its relation with ${\mathbb T}_d$ we have that for a fixed distribution of $R$ \begin{equation*} \label{E: Equal} {\mathbb P}_+[V]=0 \hbox{ if and only if } {\mathbb P}[V]=0. \end{equation*}
Let $p_0 = {\mathbb P}(R=0).$
\begin{teo}[Junior {\it et al}~\cite{Junior2}] \label{T:CSPAH} Consider the \textit{Cone Percolation Model}\ on ${\mathbb T}_d^+$ with \textit{radius of influence} $R.$ \begin{enumerate} \item[\textit{(i)}] If $(1-p_0) d > 1$, then ${\mathbb P}_+[V] > 0,$ \item[\textit{(ii)}] If $(1-p_0) d \le 1$ and ${\mathbf E}(d^R) > 1 + p_0$, then ${\mathbb P}_+[V] > 0,$ \item[\textit{(iii)}] If ${\mathbf E}(d^R) \leq 2 - \frac{1}{d}$, then ${\mathbb P}_+[V] = 0.$ \end{enumerate} \end{teo}
\begin{teo}[Junior {\it et al}~\cite{Junior3}] \label{T: THT1} Consider a \textit{Cone Percolation Model}\ on ${\mathbb T}_d$. Then for $\mathbb{E}(d^R) < 2 - \frac{1}{d}$, we have \begin{displaymath}
\frac{d + \mathbb{E}\displaystyle \left (d^R \right) - p_0 }{d[1-\mathbb{E}\displaystyle \left (d^R \right) + p_0]} \leq \mathbb{E}(|I|) \leq \frac{\mathbb{E}\displaystyle \left (d^R \right)+d-2}{2d - 1 - d\mathbb{E}\displaystyle \left (d^R \right)}. \end{displaymath} \end{teo}
\begin{exa}[Junior {\it et al}~\cite{Junior3}] Consider $R \sim {\mathcal G}(1-p)$, a \textit{radius of influence} satisfying \[ {\mathbb P}(R = k) = (1-p)p^{k}, k=0,1,2,\dots \] and assume also $pd < \frac{1}{2}$. So we have \begin{displaymath}
\frac{1-dp +p -p^2}{1-2dp+dp^2} \leq \mathbb{E}(|I|) \leq \frac{1-dp-p}{1-2dp}. \end{displaymath} That gives us a fairly sharp bound even when we pick $p$ and $d$ such that $pd$ is very close to $\frac{1}{2}$ as, for example, $p = 10^{-6}$ and $d = 499,000$.
For these parameters we get $250.438 \leq \mathbb{E}(|I|) \leq 250.501$. \end{exa}
Let $\rho$ and $\psi$ be, respectively, the smallest non-negative roots of the equations \begin{align*} & {\mathbf E}(\rho^{d^R}) + (1 - \rho)p_0 = \rho, \\ & {\mathbf E}(\psi^{\frac{d}{d-1}(d^{R}-1)}) = \psi. \end{align*}
\begin{teo}[Junior {\it et al}~\cite{Junior2}] \label{T: SobrevivenciaTd+} Consider the \textit{Cone Percolation Model}\ on ${\mathbb T}_d^+.$ Then \begin{displaymath} 1 - \rho \leq {\mathbb P}_+(V) \leq 1 - \psi. \end{displaymath} \end{teo}
\begin{teo}[Junior {\it et al}~\cite{Junior2}] \label{T: ViveTd} For the \textit{Cone Percolation Model}\ on ${\mathbb T}_d$ with \textit{radius of influence} $R$, it holds that \begin{equation*} \label{E: ViveTd} 1 - \displaystyle \left(1 - \rho^{\frac{d+1}{d}}\right)p_0 - {\mathbf E}\displaystyle \left(\rho^{\frac{(d+1)}{d}d^{R}}\right) \leq {\mathbb P}[V] \leq 1 - {\mathbf E}\displaystyle \left(\psi^{\frac{(d+1)}{d-1}(d^{R}-1)}\right). \end{equation*} \end{teo}
Consider $d=2$ and $R$ following a Binomial distribution with parameters $4$ and $\frac{1}{2}$ ($R \sim \mathcal{B}(4,\frac{1}{2})$). Therefore $\rho$ and $\psi$ are, respectively, solutions of \begin{align*} & x^{16} + 4x^8 + 6x^4 + 4x^2 - 16x + 1 = 0, \\& x^{30} + 4x^{14} + 6x^6 + 4x^2 - 16x + 1 = 0. \end{align*} So $\rho$ = 0.0635146 and $\psi$ = 0.06350850, which implies that \[ 0.937435919 \leq {\mathbb P}[V] \leq 0.937435962. \]
\begin{teo}[Junior {\it et al}~\cite{Junior2}] \label{aest} The \textit{Heterogeneous Cone Percolation Process}\ in ${\mathbb T}^+_d$ has a giant component with positive pro\-ba\-bi\-li\-ty if for some fixed $n$, \begin{equation} \label{eq1} \liminf_{j \rightarrow \infty} d^n \prod_{k=0}^{n-1}[1-\prod_{i=0}^{k}{\mathbb P}_+[R_{jn+i} < k+1-i]] > 1. \end{equation} \end{teo}
A consequence of Theorem 5.4 from Bertacchi and Zucca~\cite{BZ} is the following result
\begin{cor} \label{C: novo} Consider a \textit{Homogeneous Reverse Fireworks Process}\ on ${\mathbb T}_d$. Then \[ {\mathbb P}(S) = 1 \textrm { if and only if } \sum_{n=1}^{\infty} d^n {\mathbb P}(R \geq n) = \infty. \] \[ {\mathbb P}(S) = 0 \textrm { if and only if } \sum_{n=1}^{\infty} d^n {\mathbb P}(R \geq n)\prod_{j=1}^{n-1}[1 - {\mathbb P}(R \geq j)] \leq 1 \] \end{cor}
\begin{teo}[Junior {\it et al}~\cite{Junior3}] \label{T: esferic} For a \textit{Cone Percolation Model}\ in ${\mathbb T}_S$ and $R$, the \textit{radius of influence}, ${\mathbb P}(V) > 0$ if \begin{equation*} \label{E: esferic} \lim_{n \rightarrow \infty}\sqrt[n]{\rho_n} > e^{-\textrm{dim\ inf\ } \partial {\mathbb T}_S} \end{equation*} where \begin{equation*} \rho_n := \prod_{k=0}^{n-1} [1-\prod_{i=0}^{k}{\mathbb P}(R < i+1)]. \end{equation*} \end{teo}
\begin{cor}[Junior {\it et al}~\cite{Junior3}] \label{C: esferic1} For a \textit{Cone Percolation Model}\ in ${\mathbb T}_S$ and $R$, a \textit{radius of influence} satisfying ${\mathbb P}(R \leq k) = 1$ for some $k \in {\mathbb N}$, ${\mathbb P}(V) > 0$ if
\begin{equation*} \label{E: esferic1} \textrm{dim\ inf\ } \partial {\mathbb T}_S > \ln \displaystyle \left[\frac{1}{1 - \prod_{j=1}^{k}{\mathbb P}(R < j)} \right]. \end{equation*} \end{cor}
\begin{defn} A $k$\textit{-periodic tree} with degree $\tilde{d} = (d_1, \cdots, d_k)$, $d_i \geq 2$ for all $i=1,2,\cdots, k$, is as tree such that for any vertex whose distance to the origin is $nk+i-1$ for some $ n \in{\mathbb N} $ has degree $d_i + 1$. We refer to this tree as ${\mathbb T}_{\tilde{d}}$. \end{defn}
\begin{exa}[Junior {\it et al}~\cite{Junior3}] \label{E: B(p)} Consider a \textit{Cone Percolation Model}\ in ${\mathbb T}_S$ with $R \sim {\mathcal B}(p),$ a \textit{radius of influence} sa\-tis\-fying \[ {\mathbb P}(R = 1) = p = 1 - {\mathbb P}(R = 0). \] \begin{itemize} \item[\textit{(i)}] If dim\ inf\ $\partial {\mathbb T}_S > -\ln p$ then ${\mathbb P}(v)>0,$
\item[\textit{(ii)}] If ${\mathbb T}_S = {\mathbb T}_{\tilde{d}}$ and $ \sqrt[k]{\prod_{j=1}^{k}d_{j}} > \frac{1}{p}$ then ${\mathbb P}(V)>0.$ \end{itemize} \end{exa}
\section{Random environments} \label{S: RPRE}
In this section we review the Fireworks and the Reverse Fireworks processes, with a random number of stations at each vertex. Bertacchi and Zucca~\cite{BZ} consider an extra source of randomness: the number of individuals sitting on each vertex. They consider two families of random variables $\{N_x\}_{x \in {\mathcal G}}$ and $\{R_{x,i}\}^{i \in {\mathbb N}}_{x \in {\mathcal G}}$ such that $\{N_x, R_{x,i}\}$ are independent and $\{R_{x,i}\}_{i \in {{\mathbb N}}}$ are identically distributed for all $x \in {\mathcal G}$ that is $R_{x,i} \sim R_x.$ In their paper $N_x$ represents the random number of individuals at vertex $x$ (in particular $N_{\mathcal O}$ is the number of individuals at the origin) while $\{R_{x,i}\}_{i=1}^{N_x}$ are their \textit{radius of influence}. The main question about this model is to understand under which conditions, the signal, starting from one vertex of a graph (${\mathbb N}$ or a Galton-Watson tree), will spread indefinitely with positive probability or die out almost surely in a finite number of steps.
Bertacchi and Zucca~\cite{BZ} rely in their analisys on associating the processes with random numbers of stations (fireworks or reverse fireworks), with processes with one station per vertex as in Junior \textit{et al}~\cite{Junior} . Indeed, they consider processes with one station on each vertex $x$ and \textit{radius of influence} ${\tilde{R}}_x = {\mathbf 1}_{\{N_x \ge 1\}} \max\{R_{x,j}: j = 1, \dots, N_x\}.$ They call this process, the \textit{deterministic counterpart} or \textit{annealed counterpart} of the original process. They observe that the \textit{annealed counterpart} does not retain any information about the environment, nevertheless the probability of survival for the original process and for its \textit{annealed counterpart} are the same.
\subsection{Fireworks}
For $x \in {\mathcal G},$ let us define \begin{displaymath} \varphi_{N_x}(t) := {\mathbf E}(t^{N_x}) = \sum_{j=0}^{\infty} {\mathbb P}(N_x=j) t^j \end{displaymath}
\subsubsection{Homogeneus Fireworks} Consider $R_i \sim R$ and $N_x \sim N$ for all $x \in {\mathcal G}.$ Let us define \begin{displaymath} f_{R,N}(n) := n \{1 - \varphi_{N}(\mathbb{P}(R < n))\}. \end{displaymath}
\begin{teo}[Bertacchi and Zucca~\cite{BZ}] \label{T: NAT}
\begin{itemize} \item[\textit{(i)}] If $ \displaystyle \limsup_{n \rightarrow \infty}f_{R, N}(n) < 1$
then $\mathbb{P}(V) = 0$. \item[\textit{(ii)}] If $ \displaystyle \liminf_{n \rightarrow \infty}f_{R, N}(n) > 1$
then $\mathbb{P}(V) > 0$. \item[\textit{(iii)}] If $\mathbb{E}(N) < \infty$ and $\displaystyle \limsup_{n \to \infty} n \mathbb{P}( R \geq n) < \frac{1}{\mathbb{E}(N)}$ then $\mathbb{P}(V) = 0$. \item[\textit{(iv)}] If $\mathbb{E} (N) < \infty$ and $\mathbb{E} (R) < \infty$ then $\mathbb{P}(V) = 0$. \item[\textit{(v)}] If $ \displaystyle \liminf_{n \rightarrow \infty}n \mathbb{P}( R \geq n)\varphi^{\prime}_{N}(\mathbb{P}(R < n)) > 1$
then $\mathbb{P}(V) > 0$. \end{itemize} \end{teo}
A consequence of Theorem 1 from Gallo \textit{et al}~\cite{GGJR} is the following result
\begin{cor} \begin{displaymath} \mathbb{P}(V) = \left [ 1 + \sum_{j=1}^{\infty} \prod_{i=0}^{j-1}\displaystyle \varphi_{N} \left ( \displaystyle \mathbb{P}(R \leq i) \right )\right ]^{-1} \end{displaymath} \end{cor}
\begin{rem} It is possible to have $\mathbb{E}(N) = \infty$, $\mathbb{E}(R) = \infty$ and $\mathbb{P}(V) = 0$. Take $\mathbb{P}( N \geq n) \sim \frac{1}{n}$ when $n \to \infty$ and $\mathbb{P}( R \geq n) = \frac{1}{n \ln n \ln (ln n)}$ \end{rem}
\subsubsection{Heterogeneous Fireworks}
\begin{teo}[Bertacchi and Zucca~\cite{BZ}] In the heterogeneous case, if \[ \sum_{n=0}^{\infty} \prod_{i=0}^{n} \varphi_{N_i} \left ( \displaystyle \mathbb{P}(R_i < n-i+1) \right ) \] then $\mathbb{P}(V) > 0$. \end{teo}
Adapting the proof of Theorem 2.3 from Junior {\it et al}~\cite{Junior} we have
\begin{teo} In the heterogeneous case, if \begin{itemize} \item[\textit{(i)}] $ \varphi_{N_i}(\mathbb{P}(R_i < 1)) \in (0,1).$ \item[\textit{(ii)}] $ \lim_{n \to \infty} \prod_{i=0}^{n-1} \varphi_{N_i} \left ( \displaystyle \mathbb{P}(R_i < 2n-1) \right ) = 1.$ \item[\textit{(iii)}] $ \lim_{n \to \infty} \prod_{i=n}^{2n-1} \varphi_{N_i} \left ( \displaystyle \mathbb{P}(R_i < 2n-1) \right ) > 0.$ \end{itemize} then $\mathbb{P}(V) = 0$. \end{teo}
\subsection{Reverse Fireworks}
\subsubsection{Homogeneous Reverse Fireworks}
Let us define
\[ W = \sum_{n=0}^{\infty} \left [ 1 - \varphi_{N} \left ( \displaystyle \mathbb{P}(R < n) \right ) \right ] \]
\begin{teo}[Bertacchi and Zucca~\cite{BZ}] \label{T: NATR}
\begin{itemize} \item[\textit{(i)}] If $W = \infty$ then $\mathbb{P}(S) = 1.$ \item[\textit{(ii)}] If $W < \infty$ then $\mathbb{P}(S) = 0.$ \end{itemize} \end{teo}
Theorem~\ref{T: NATR} can also be obtained as a consequence of Theorem 3.2 from Junior {\it et al}~\cite{Junior2} or as a consequence of Theorem 2 from Gallo {\it et al}~\cite{GGJR}.
\begin{rem}[Bertacchi and Zucca~\cite{BZ}] \label{R: BZ-RandN} Theorems~\ref{T: NAT} and~\ref{T: NATR} admit a similar corolary \begin{itemize} \item[\textit{(i)}] For every unbounded random variable $R$ there exists a random variable $N$ such that $\mathbb{P}(V) > 0$ ($\mathbb{P}(S) = 1$). For $\epsilon > 0$ and $\delta \in (0,1)$ consider $N$ satisfying \[ \mathbb{P} \left (N \geq \frac{\ln(1-\delta)}{\ln (\mathbb{P} (R < n))} \right) \geq \frac{1 + \epsilon}{n \delta}. \] \item[\textit{(ii)}] For every random variable $N$ such that $\mathbb{P}(N = 0) < 1$ there exists a random variable $R$ such that $\mathbb{P}(V) > 0$ ($\mathbb{P}(S) = 1$). Take $R$ satisfying $\mathbb{P} (R \geq n) = p_n$, where $p_n = \inf \{ t \geq 0; \varphi_{N} \left ( 1-t\right ) \leq 1 - \frac{2}{n} $\}. \end{itemize}
\end{rem}
\subsubsection{Heterogeneous Reverse Fireworks}
\begin{teo}[Bertacchi and Zucca~\cite{BZ}] In the heterogeneous case, \[ \sum_{k=0}^{\infty} \left [ 1 - \varphi_{N_{n+k}} \left ( \displaystyle \mathbb{P}(R_{n+k} < k) \right ) \right ] = \infty, \textrm { if and only if } \mathbb{P}(S) = 1. \] By other hand, if \[ \sum_{n=0}^{\infty} \prod_{k=1}^{\infty} \varphi_{N_{n+k}} \left ( \displaystyle \mathbb{P}(R_{n+k} < k) \right ) < \infty, \mathbb{P}(S) > 0. \]
\end{teo}
\subsection{Galton Watson} Let us define the space of unlabelled GW-trees (the usual GW-trees). Consider a GW-process, with offspring distribution $ {\mathbb P}(D = d),\ 0 \le d < \infty $. We assume that ${\mathbb P}(D = 1) < 1$ (otherwise the resulting random tree is ${\mathbb N}$) and we suppose that $ \mu_D := \displaystyle \sum_{d=0}^{\infty} d {\mathbb P}(D = d) > 1$ (the supercritical case). The underlying random graph will be a GW-tree generated by this process. Let $ \displaystyle g(s): = \sum_{d=0}^{\infty} s^d {\mathbb P}(D = d) $ be the generating function of $D$ and let $\pi \in [0, 1]$ be the smallest nonnegative fixed point of $g$. If ${\mathbb P}(D > k) = 0$ for some $k$ we say that the GW-tree has maximum degree $k$ or that it is $k$-bounded.
\subsubsection{Homogeneous Fireworks} In this case, the random number of stations are independent and identically distributed ${\mathbb N}$-valued random variables with common law $N$. Analogously, The radii of the stations are independent and identically distributed with distribution $R$ (either discrete or continuous random variable).
\begin{defn} We define \[ \Phi(t) : = \varphi_N({\mathbb P}(R < 1)) + \sum_{n=1}^{\infty}[\varphi_N({\mathbb P}(R < n+1))-\varphi_N({\mathbb P}(R < n))]t^n.\] \end{defn}
In particular observe that \[ \Phi(0) = \varphi_N({\mathbb P}(R < 1)) \textrm { and the case } N =1 \textrm { a.s.},\] \[ \Phi(t) = \sum_{n=0}^{\infty}[{\mathbb P}( n \leq R < n+1) ]t^n. \]
\begin{teo}[Bertacchi and Zucca~\cite{BZ}] Consider a \textit{Homogeneous Fireworks Process}. We have that \begin{itemize} \item[\textit{(i)}] If $\Phi(\mu_D)- 1 > \Phi(0) = \varphi_N(P(R < 1))$ and ${\mathbb P}(N = 0) = 0 $ then for the Fireworks process there is survival with positive probability for almost every realization of the environment such that the underlying tree is infinite and there is at least one station at the root. \item[\textit{(ii)}] If $\Phi(\mu_D) - 1 > \Phi(0) = \varphi_N(P(R < 1))$ and ${\mathbb P}(N = 0) > 0$ then for the Fireworks process
${\mathbb P}(V|\tau = T,N_{\mathcal{O}} = n) > 0$ for almost every $(T, n)$ such that $T$ is an infinite (unlabelled) tree and $n \geq 1$. \item[\textit{(iii)}] If the GW-tree is $k$-bounded and $\Phi(k) \leq 2- \frac{1}{k}$ then the Fireworks process becomes extinct a.s. for almost every realization of the environment. \end{itemize} \end{teo}
\subsubsection{Homogeneous Reverse Fireworks} In this case, the random number of stations are independent and identically distributed ${\mathbb N}$-valued random variables with common law $N$, except by numbers of station the root $\mathcal{O}$. For the root, we take $N_{\mathcal{O}} = \min\{ n > 0: {\mathbb P}(N = n) > 0 \} $ Besides, the radii of the stations are independent and identically distributed with distribution $R$ (either discrete or continuous random variable).
\begin{defn} We define \[ \phi_1(t) : = \sum_{n=1}^{\infty}[1 - \varphi_N({\mathbb P}(R < n)) ]{\mu_D}^n \] \[ \phi_2(t) : = \sum_{n=1}^{\infty}[1 - \varphi_N({\mathbb P}(R < n)) ]{\mu_D}^n \prod_{j=1}^{n-1}\varphi_N({\mathbb P}(R < j)) \] \end{defn}
\begin{teo}[Bertacchi and Zucca~\cite{BZ}] Consider a \textit{Homogeneous Reverse Fireworks Process}. The following hold \begin{itemize} \item[\textit{(i)}] If $\phi_1(\mu_D) = \infty$ then there is survival with probability 1 for the Reverse Fireworks process for almost all realizations of the environment such that the underlying tree is infinite. \item[\textit{(ii)}] If ${\mathbb P}(N = 0) = 0$, $\phi_1(\mu_D) < \infty$ and $\phi_2(\mu_D) > 1$ then there is survival with positive probability (strictly smaller than 1) for the Reverse Fireworks process for almost all realizations of the environment such that the underlying tree is infinite.
\item[\textit{(iii)}] If ${\mathbb P}(N = 0) > 0$, $\phi_1(\mu_D) < \infty$ and $\phi_2(\mu_D) > 1$ then ${\mathbb P}(S|\tau = T) \in (0, 1)$ for almost every infinite (unlabelled) tree $T$. \item[\textit{(iv)}] If $\phi_1(\mu_D) < \infty$ and $\phi_2(\mu_D) \leq 1$ then there is a.s. extinction for the Reverse Fireworks process for almost all realizations of the environment. \end{itemize} \end{teo}
\begin{defn} We define \[ \mathcal{M}_c := \left [ \limsup_{n \to \infty} \sqrt[n]{1 - \varphi_N({\mathbb P}(R < n))} \right ]^{-1}.\] \end{defn}
\begin{cor}[Bertacchi and Zucca~\cite{BZ}] There exists a critical value $\mu_c \in [1,\infty), \mu_c \leq \mathcal{M}_c$ such that \begin{itemize} \item[\textit{(i)}] $\mu_D < \mu_c$ implies a.s. extinction for almost all realizations of the environment. \item[\textit{(ii)}] $\mu_c < \mu_D < \mathcal{M}_c$ and ${\mathbb P}(N = 0) = 0$ implies survival with positive probability for almost all realizations of the environment such that the underlying tree is infinite. \item[\textit{(iii)}] $\mu_c < \mu_D < \mathcal{M}_c$ and ${\mathbb P}(N = 0) > 0$ implies survival with positive probability for almost every infinite (unlabelled) tree. \item[\textit{(iv)}] $\mathcal{M}_c < \mu_D$ implies survival with probability 1 for almost all realizations of the environment such that the underlying tree is infinite. \item[\textit{(v)}] If $\mu_D = \mu_c < \mathcal{M}_c$ then there is a.s. extinction for almost all realizations of the environment. \end{itemize} \end{cor}
\section{Open problems} \label{S: OQ}
Some natural extensions for these models are those considering
\begin{itemize} \item[\textit{(i)}] Fireworks processes (direct and reverse) on ${{\mathbb Z}^d}^+$. An especially interesting case is when $d=2$ and the \textit{boxes of influence} are distributed as $[0, R_x) \times [0, R_y)$ with $R_x$ independent of $R_y$ and the rumor starting from $(0,0)$ or from every $(x,y)$ such that $x=0$ or $y=0$; \item[\textit{(ii)}] Fireworks processes on ${\mathbb Z}$. Heterogenous versions with \textit{radius of influence} non i.i.d. and with in\-di\-vi\-duals being initially placed following a renewal process or a Markovian process. \item[\textit{(iii)}] Reverse fireworks processes on ${\mathbb Z}$. Individuals throw their \textit{radius of influence} to every direction as in~(\ref{E: defBu}) (See Gallo \textit{et all}~\cite{GGJR}). They believe that conditions for survival will be the same but the final proportion of informed individual will be strictly larger. \item[\textit{(iv)}] Cone Percolation on Spherically Symmetric and on Galton Watson trees. Lower and upper bounds for the survival probability and for the extinction time. \end{itemize}
\noindent Acknowledgments: F.P.M. wishes to thank NYU-Shanghai China and V.V.J. and K.R. wish to thank Instituto de Matem\'atica e Estat\'{\i}stica-USP Brazil for kind hospitality.
\end{document} | arXiv |
Wanxiong Shi
Wanxiong Shi (Chinese: 施皖雄; pinyin: Shī Wǎnxióng; Pe̍h-ōe-jī: Si Oán-hiông; 6 October 1963 - 30 September 2021) was a Chinese mathematician. He was known for his fundamental work in the theory of Ricci flow.
Wanxiong Shi
施皖雄
Born(1963-10-06)October 6, 1963
China
DiedSeptember 30, 2021(2021-09-30) (aged 57)
Washington D.C., USA
Alma materUniversity of Science and Technology of China
Chinese Academy of Sciences
Harvard University (Ph.D.)
Scientific career
FieldsMathematics
InstitutionsPurdue University
ThesisRicci Deformation Of The Metric On Complete Noncompact Kahler Manifolds
Doctoral advisorShing-Tung Yau
Education
Shi was a native of Quanzhou, Fujian. In 1978, Shi graduated from Quanzhou No. 5 Middle School, and entered the University of Science and Technology of China. Shi earned his bachelor's degree in mathematics in 1982, then he went to the Institute of Mathematics of Chinese Academy of Sciences and obtained his master's degree in mathematics in 1985 under the guidance of Lu Qikeng (Chinese: 陆启铿) and Zhong Jiaqing (Chinese: 钟家庆). Then Shi was recruited by Shing-Tung Yau to study under him at the University of California, San Diego.[1] In 1987, Shi followed Yau to Harvard University and obtained his Ph.D. there in 1990.[2]
Since Shi was stronger in geometric analysis than other Chinese students, having an impressive ability to carry out highly technical arguments, he was assigned by Yau to investigate Ricci flow in the challenging case of noncompact manifolds.[3] Shi made significant breakthroughs and was highly regarded by researchers in the field. Richard Hamilton, the founder of Ricci flow theory, liked his work very much.[4]
Academic career and later life
Upon his graduation, several prominent universities were interested in offering him a faculty position. Hung-Hsi Wu (Chinese: 伍鸿熙) from the University of California, Berkeley asked Yau if Shi could come to Berkeley. Without seeking opinion from Yau, Shi applied to and got tenure track assistant professorship offers from the University of California, San Diego, where Richard Hamilton was working at, and Purdue University.
Shi decided to join Purdue University. He published several important papers there, and was awarded three grants from the NSF in 1991, 1994 and 1997.[5][6][7] However, Shi did not pass the tenure review in 1997, so he had to leave the university. (The principal investigator of the NSF grant of 1997 was changed because of this.) Yau believes that the failure was due to the faculty members not realising the importance of Ricci flow theory. Hamilton sent a belated reference letter to Purdue University in which he rebuked the decision, but to no avail.[4]
Shi then left academia and moved to Washington D.C., where he lived a frugal and secluded life in solitude, and had less and less contact with his friends. He turned down some offers from other universities.[4] Yau and former classmates of Shi tried to persuade Shi and help him return to academia, but he rejected.[8] Yau felt sorry for Shi's leaving academia, since among the four students of Yau who worked on Ricci flow, Shi had done the best work. Shi died from a sudden heart attack in the evening of September 30, 2021.[9][10]
Work
Shi initiated the study of Ricci flow theory on noncompact complete manifolds. He proved local derivative estimates for the Ricci flow, which are fundamental to many arguments of the theory, including Perelman's proof of the Poincaré conjecture using Ricci flow in 2002.[11]
Publications
• Shi, Wan-Xiong (1989). "Complete noncompact three-manifolds with non-negative Ricci curvature". J. Differ. Geom. 29 (2): 353–360. doi:10.4310/jdg/1214442879.
• Shi, Wan-Xiong (1989). "Deforming the metric on complete Riemannian manifolds". J. Differ. Geom. 30 (1): 223–301. doi:10.4310/jdg/1214443292.
• Shi, Wan-Xiong (1989). "Ricci deformation of the metric on complete non-compact Riemannian manifolds". J. Differ. Geom. 30 (2): 303–394. doi:10.4310/jdg/1214443595.
• Shi, Wan-Xiong (1990). "Complete noncompact Kähler manifolds with positive holomorphic bisectional curvature". Bull. Am. Math. Soc. New Ser. 23 (2): 437–440. doi:10.1090/S0273-0979-1990-15954-3.
• Shi, Wanxiong and Yau, S. T. (1994). "Harmonic maps on complete noncompact Riemannian manifolds". A tribute to Ilya Bakelman. Proceedings of a conference, College Station, TX, USA, October 1993. College Station, TX: Texas A & M University. pp. 79–120. ISBN 0-9630728-2-X.{{cite conference}}: CS1 maint: multiple names: authors list (link)
• Shi, Wan-Xiong and Yau, S.-T. (1996). "A note on the total curvature of a Kähler manifold". Math. Res. Lett. 3 (1): 123–132. doi:10.4310/MRL.1996.v3.n1.a12.{{cite journal}}: CS1 maint: multiple names: authors list (link)
• Shi, Wan-Xiong (1997). "Ricci flow and the uniformization on complete noncompact Kähler manifolds". J. Differ. Geom. 45 (1): 94–220. doi:10.4310/jdg/1214459756.
• Shi, Wan-Xiong (1998). "A uniformization theorem for complete Kähler manifolds with positive holomorphic bisectional curvature". J. Geom. Anal. 8 (1): 117–142. doi:10.1007/BF02922111. S2CID 121610392.
References
1. Shing-Tung Yau; Steve Nadis (2019). The Shape of a Life: One Mathematician's Search for the Universe's Hidden Geometry. Yale University Press. p. 171. ISBN 9780300245523.
2. "才学一流 破百年猜想 贡献卓著 人生几何 解千古疑惑 淡泊超然". School of Mathematical Sciences USTC. 9 October 2021. Retrieved 7 May 2022.
3. 郑方阳 Zheng Fangyang (6 October 2021). "忆皖雄同学". Wechat official accounts platform. Retrieved 7 May 2022.
4. Shing-Tung Yau (7 October 2021). "丘成桐：悼念我的学生施皖雄". sohu.com. Retrieved 7 May 2022.
5. "Award Abstract # 9103140 Mathematical Sciences: "Studying the Topology of Riemannian Manifolds through Ricci Deformation of the Metric"". National Science Foundation. 10 May 1991. Retrieved 9 May 2022.
6. "Award Abstract # 9403405 Mathematical Sciences: Heat Flow on Riemannian Manifolds". National Science Foundation. 13 July 1994. Retrieved 9 May 2022.
7. "Award Abstract # 9703656 Global Analysis on Riemannian Manifolds". National Science Foundation. 14 February 1997. Retrieved 9 May 2022.
8. 王晓林 Wang Xiaolin (12 October 2021). "【情系科大】特立独行，不离不弃——怀念皖雄同学". Wechat official accounts platform. Retrieved 7 May 2022.
9. "58岁数学家施皖雄去世，曾为解决庞加莱猜想做出基础性贡献". thepaper.cn. 7 October 2021. Retrieved 7 May 2022.
10. "师友追忆旅美数学家施皖雄——"我们一起生活，对于将来是荣幸的"". 上海科技报 Shanghai Keji Bao. 21 October 2021. Retrieved 7 May 2022.
11. Richard S. Hamilton. "Prof. Hamilton's speech about Poincare conjecture in Beijing (8 June, 2005)". The Institute of Mathematical Sciences, CUHK. Retrieved 7 May 2022.
External links
• Wan-Xiong Shi at the Mathematics Genealogy Project
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
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>Bulletin of the Australian Mathematical Society
>ON A CERTAIN GENERALISATION OF THE ITERATED FUNCTION...
Bulletin of the Australian Mathematical Society
ON A CERTAIN GENERALISATION OF THE ITERATED FUNCTION SYSTEM
Part of: Classical measure theory
Published online by Cambridge University Press: 14 August 2012
FILIP STROBIN and
JAROSŁAW SWACZYNA
FILIP STROBIN*
Institute of Mathematics, Łódź University of Technology, Wólczańska 215, 93-005 Łódź, Poland (email: [email protected])
Institute of Mathematics, Łódź University of Technology, Wólczańska 215, 93-005 Łódź, Poland (email: [email protected])
✉For correspondence; e-mail: [email protected]
We follow the idea of generalising the notion of classical iterated function systems, as presented by Mihail and Miculescu. We give their deliberations a more general setting and, using this general approach, study the generic aspect of the problem of existence of an attractor of a function system.
fractalsiterated function systemsfixed pointsBaire categoryporosity
MSC classification
Secondary: 28A80: Fractals
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 1 , February 2013 , pp. 37 - 54
Copyright © Australian Mathematical Publishing Association Inc. 2012
[B]Barnsley, M. F., Fractals Everywhere (Academic Press, Boston, 1993).Google Scholar
[Br]
[Br]Browder, F., 'On the convergence of successive approximations for nonlinear functional equations', Indag. Math. 30 (1968), 27–35.CrossRefGoogle Scholar
[DM]
[DM]De Blasi, F. and Myjak, J., 'Sur la porosité de l'ensemble des contractions sans point fixe', C. R. Acad. Sci. Paris 308 (1989), 51–54.Google Scholar
[GD]
[GD]Granas, A. and Dugundji, J., Fixed Point Theory, Springer Monographs in Mathematics (Springer, New York, 2003).CrossRefGoogle Scholar
[H]
[H]Hutchinson, J., 'Fractals and self-similarity', Indiana Univ. Math. J. 30(5) (1981), 713–747.CrossRefGoogle Scholar
[J]
[J]Jachymski, J. and Jóźwik, I., 'Nonlinear contractive conditions: a comparison and related problems', Banach Center Publ. 77 (2007), 123–146.CrossRefGoogle Scholar
[M]Mihail, A., 'Recurrent iterated function systems', Rev. Roumaine Math. Pures Appl. 53(1) (2008), 43–53.Google Scholar
[Ma]
[Ma]Máté, L., 'The Hutchinson–Barnsley theorey for certain noncontraction mappings', Period. Math. Hungar. 27(1) (1993), 21–33.CrossRefGoogle Scholar
[MM1]
[MM1]Mihalil, A. and Miculescu, R., 'Applications of fixed point theorems in the theory of generalized IFS', Fixed Point Theory and Applications 2008 (2008), article ID 312876, 11 pp;doi:10.1155/2008/312876.Google Scholar
[MM2]Michalil, A. and Miculescu, R., 'Generalized IFSs on noncompact spaces', Fixed Point Theory and Applications 2010 (2010), article ID 584215, 11 pp; doi:10.1155/2010/584215.Google Scholar
[S1]
[S1]Strobin, F., Genericity and Porosity of Some Subsets of Function Spaces, Doctorial Dissertation, Polish Academy of Sciences, 2011.Google Scholar
[S2]Strobin, F., 'Some porous and meager sets of continuous mappings', J. Nonlinear Convex Anal. 13(2) (2012), 351–361.Google Scholar
[S3]Strobin, F., 'σ-porous sets of generalized nonexpansive mappings', Fixed Point Theory, to appear.Google Scholar
[Se]
[Se]Serban, M., 'Fixed point theorems for operators on Cartesian product spaces and applications', Semin. Fixed Point Theory Cluj-Napoca 3 (2002), 163–172.Google Scholar
[RZ1]
[RZ1]Reich, S. and Zaslavski, A., 'Almost all nonexpansive mappings are contractive', C. R. Math. Acad. Sci. Soc. R. Can. 22(3) (2000), 118–124.Google Scholar
[RZ2]Reich, S. and Zaslavski, A., 'The set of noncontractive mappings is σ-porous in the space of all nonexpansive mappings', C. R. Acad. Sci. Paris Sér. I Math. 333(6) (2001), 539–544.CrossRefGoogle Scholar
[Z1]
[Z1]Zajíček, L., 'Porosity and σ-porosity', Real Anal. Exchange 13(2) (1987/88), 314–350.Google Scholar
[Z2]Zajíček, L., 'On σ-porous sets in abstract spaces', Abstr. Appl. Anal. 5 (2005), 509–534.CrossRefGoogle Scholar
Miculescu, Radu and Mihail, Alexandru 2013. Alternative characterization of hyperbolic affine infinite iterated function systems. Journal of Mathematical Analysis and Applications, Vol. 407, Issue. 1, p. 56.
Secelean, Nicolae-Adrian 2013. Iterated function systems consisting of F-contractions. Fixed Point Theory and Applications, Vol. 2013, Issue. 1,
Secelean, Nicolae-Adrian 2014. Generalized iterated function systems on the spacel∞(X). Journal of Mathematical Analysis and Applications, Vol. 410, Issue. 2, p. 847.
Chiţescu, Ion Ioana, Loredana and Miculescu, Radu 2014. Type $${\mathcal{A}}$$ A Sets and the Attractors of Infinite Iterated Function Systems. Results in Mathematics, Vol. 66, Issue. 3-4, p. 511.
Miculescu, Radu 2014. Generalized Iterated Function Systems with Place Dependent Probabilities. Acta Applicandae Mathematicae, Vol. 130, Issue. 1, p. 135.
Strobin, Filip 2015. Attractors of generalized IFSs that are not attractors of IFSs. Journal of Mathematical Analysis and Applications, Vol. 422, Issue. 1, p. 99.
Dumitru, Dan Ioana, Loredana Sfetcu, Răzvan-Cornel and Strobin, Filip 2015. Topological version of generalized (infinite) iterated function systems. Chaos, Solitons & Fractals, Vol. 71, Issue. , p. 78.
Secelean, Nicolae-Adrian 2015. Generalized F-iterated function systems on product of metric spaces. Journal of Fixed Point Theory and Applications, Vol. 17, Issue. 3, p. 575.
Miculescu, Radu and Mihail, Alexandru 2016. Reich-type iterated function systems. Journal of Fixed Point Theory and Applications, Vol. 18, Issue. 2, p. 285.
Ri, Song-il 2016. A new fixed point theorem in the fractal space. Indagationes Mathematicae, Vol. 27, Issue. 1, p. 85.
Jaros, Patrycja Maślanka, Łukasz and Strobin, Filip 2016. Algorithms generating images of attractors of generalized iterated function systems. Numerical Algorithms, Vol. 73, Issue. 2, p. 477.
Balu, Rinju Mathew, Sunil and Secelean, Nicolae Adrian 2017. Separation properties of (n, m)-IFS attractors. Communications in Nonlinear Science and Numerical Simulation, Vol. 51, Issue. , p. 160.
Georgescu, Flavian 2017. IFSs consisting of generalized convex contractions. Analele Universitatii "Ovidius" Constanta - Seria Matematica, Vol. 25, Issue. 1, p. 77.
Georgescu, Flavian 2017. Iterated Function Systems Consisting of Generalized Convex Contractions in the Framework of Complete Strong b-metric Spaces. Annals of West University of Timisoara - Mathematics and Computer Science, Vol. 55, Issue. 2, p. 119.
Ri, SongIl 2018. A new idea to construct the fractal interpolation function. Indagationes Mathematicae, Vol. 29, Issue. 3, p. 962.
Maślanka, Łukasz and Strobin, Filip 2018. On generalized iterated function systems defined on ℓ∞-sum of a metric space. Journal of Mathematical Analysis and Applications, Vol. 461, Issue. 2, p. 1795.
Miculescu, Radu and Urziceanu, Silviu-Aurelian 2018. The canonical projection associated with certain possibly infinite generalized iterated function systems as a fixed point. Journal of Fixed Point Theory and Applications, Vol. 20, Issue. 4,
Miculescu, Radu and Mihail, Alexandru 2018. A Generalization for a Finite Family of Functions of the Converse of Browder's Fixed Point Theorem. Bulletin of the Brazilian Mathematical Society, New Series, Vol. 49, Issue. 4, p. 673.
Oliveira, Elismar R. and Strobin, Filip 2018. Fuzzy attractors appearing from GIFZS. Fuzzy Sets and Systems, Vol. 331, Issue. , p. 131.
RI, SONG-IL 2018. A NEW NONLINEAR BIVARIATE FRACTAL INTERPOLATION FUNCTION. Fractals, Vol. 26, Issue. 04, p. 1850054.
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\begin{document}
\title[Properness for scaled gauged maps]{Properness for scaled gauged
maps}
\maketitle
\begin{abstract}
We prove properness of moduli stacks of gauged maps satisfying a
stability condition introduced by Mundet \cite{mund:corr}, Schmitt
\cite{schmitt:univ} and Ziltener \cite{zilt:qk}. The proof combines
a git construction of Schmitt \cite{schmitt:univ}, properness for
twisted stable maps by Abramovich-Vistoli
\cite{abramovich:compactifying}, a variation of a boundedness
argument due to Ciocan-Fontanine-Kim-Maulik \cite{cf:st}, and a
removal of singularities for bundles on surfaces in
Colliot-Th\'el\`ene-Sansuc \cite{ciollot}. \end{abstract}
\tableofcontents
\section{Introduction}
The moduli stack of maps from a curve to the stack quotient of a smooth projective variety by the action of a complex reductive group has a natural stability condition introduced by Mundet in \cite{mund:corr} and investigated further in Schmitt \cite{schmitt:univ,schmitt:git}; the condition generalizes stability for bundles over a curve introduced by Mumford, Narasimhan-Seshadri and Ramanathan \cite{ra:th}. In an earlier paper \cite{cross} the first and third authors used the moduli of Mundet-stable maps to give a formula that relates the genus zero gauged Gromov-Witten invariants and Gromov-Witten invariants of the git quotient of a smooth projective variety with reductive group action, termed a quantum analog of Witten's localization theorem. The proof of the formula depended on the properness of the stack. This properness was proved via symplectic geometry and results of Ziltener \cite{zilt:qk} and Ott \cite{ott:remov}. In this paper we give a purely algebraic proof of properness via the valuative criterion for stacks \cite[Chapter
7]{la:ch}.
The stability condition for maps to quotient stacks combines several stability conditions already present in the literature, and leads to a notion of gauged Gromov-Witten invariant. Let $X$ be a smooth projective $G$-variety such that the semi-stable locus is equal to the stable locus, and $X/G$ the quotient stack. By definition a map from a curve $C$ to $X/G$ is a pair that consists of a bundle $P \to C$ and a section $u$ of the associated bundle $P \times_G X \to C$. We denote by $\pi: X/G \to \operatorname{pt}/G =: BG$ the projection to the classifying space. In case $X$ is a point, a stability condition for $ \on{Hom}(C,X/G)$, bundles on $C$, was introduced by Ramanathan \cite{ra:th}. A stability condition that combines bundle and target stability was introduced by Mundet \cite{mund:corr}. There is a compactified moduli stack $\overline{\M}^G_n(C,X,d)$ whose open locus consists of Mundet semistable maps of class $d \in H_2^G(X,\mathbb{Z})$ with markings:
$$ C \to S, \quad v: C \to X/G, \quad (z_1,\ldots, z_n): S \to C^n \ \text{distinct} .$$
The compactification uses the notion of Kontsevich stability for maps \cite{qk1}, \cite{qk2}, \cite{qk3}. The stack admits evaluation maps to the quotient stack
$$ \ev: \overline{\M}^G_n(C,X,d) \to (X/G)^n,\quad (\hat{C},P,u,\ul{z})\mapsto (z_i^* P, u \circ z_i) .$$
In addition, assuming stable=semistable there is a virtual fundamental class constructed via the machinery of Behrend-Fantechi \cite{bf:in}.
Let $\widehat{QH}_G(X)$ denote the formal completion of $QH_G(X)$ at $0$. The {\em gauged Gromov-Witten trace} is the map
\begin{equation} \label{gtrace} \tau_X^G : \widehat{QH}_G(X) \to
\Lambda_X^G, \quad \alpha \mapsto \sum_{n,d} \frac{q^d}{n!}
\int_{\overline{\M}_n^G(C,X,d)} \ev^* (\alpha,\ldots,\alpha)
.\end{equation}
The derivatives of the potential will be called gauged Gromov-Witten invariants. For toric varieties, the potential $\tau_X^G$ already appears in Givental \cite{gi:eq} and Lian-Liu-Yau \cite{lly:mp1} under the name of {\em quasimap potential}.\footnote{We are simplifying
things a bit for the sake of exposition; actually the quasimap
potentials in those papers involve an additional determinant line
bundle in the integrals.} In those papers (following earlier work of Morrison-Plesser \cite{mp:si}) the gauged potential is explicitly computed in the toric case, and questions about Gromov-Witten invariants of toric varieties or complete intersections therein reduced to a computation of quasimap invariants. We wish re-prove and extend the results of those papers in a uniform and geometric way that extends to quantum K-theory and non-abelian quotients and does not use any assumption such as the existence of a torus action with isolated fixed points. The splitting axiom for the gauged invariants is somewhat different than the usual splitting axiom in Gromov-Witten theory: the potential $\tau_X^G$ is a non-linear version of a {\em trace} on the Frobenius manifold $QH_G(X)$. Note that there are several other notions of gauged Gromov-Witten invariants, for example, Ciocan-Fontanine-Kim-Maulik \cite{cf:st}, Frenkel-Teleman-Tolland \cite{toll:gw1}, as well as a growing body of work on gauged Gromov-Witten theory with potential \cite{tianxu}, \cite{glsm}.
The gauged Gromov-Witten invariants so defined are closely related to, but different from in general, the Gromov-Witten invariants of the stack-theoretic geometric invariant theory quotient. The stack of marked maps to the git quotient
$$ v: C \to X \qu G, \quad (z_1,\ldots, z_n) \in C^n \ \text{distinct} $$
is compactified by the {\em graph space} \label{graphs}
$$ \overline{\M}_n(C, X \qu G, d) := \overline{\M}_{g,n}(C \times X \qu G, (1,d))$$
the moduli stack of stable maps to $C \times X \qu G$ of class $(1,d)$; in case $X \qu G$ is an orbifold the domain is allowed to have orbifold structures at the nodes and markings as in \cite{cr:orb}, \cite{agv:gw}. The stack admits evaluation maps
$$ \ev: \overline{\M}_n(C,X \qu G,d) \to (\overline{\cI}_{X \qu G})^n $$
where $\overline{\cI}_{X\qu G}$ is the {\em rigidified inertia stack} of $X\qu G$. The {\em graph trace} is the map
$$ \tau_{X \qu G}: \widehat{QH}_{\mathbb{C}^\times}(X \qu G) \to \Lambda_X^G, \quad \alpha \mapsto \sum_{n,d} \frac{q^d}{n!} \int_{\overline{\M}_n(C,X \qu G,d)} \ev^* (\alpha,\ldots,\alpha) $$
and where the equivariant parameter for the $\mathbb{C}^\times$-action is interpreted as a $\psi$-class at the corresponding marking. The relationship between the graph Gromov-Witten invariants of $X \qu G$ and Gromov-Witten invariants arising from stable maps to $X \qu G$ in the toric case is studied in \cite{gi:eq}, \cite{lly:mp1}, and other papers.
The goal of this paper is to construct, using only algebraic geometry, a proper algebraic cobordism between the moduli stack of Mundet semistable maps and the moduli stack of stable maps to the git quotient with corrections coming from ``affine gauged maps''. Affine gauged maps are maps
$$ v: \mathbb{P}^1 \to X /G, \quad u(\infty) \in X^{\on{ss}}/G, \quad z_1,\ldots,z_n \in \mathbb{P}^1 - \{ \infty \} \ \text{distinct}$$
where $\infty = [0,1] \in \mathbb{P}^1$ is the point ``at infinity'', modulo {\em affine} automorphisms, that is, automorphisms of $\mathbb{P}^1$ which preserve the standard affine structure on $\mathbb{P}^1 - \{ 0 \}$. Denote by $\overline{\M}^G_{n,1}(\mathbb{A},X)$ the compactified moduli stack of such affine gauged maps to $X$; we use the notation $\mathbb{A}$ to emphasize that the equivalence only uses affine automorphisms of the domains. A table with the different kinds of stable maps to quotients stacks is presented in Section \ref{table}.
\label{scaledaffine} Evaluation at the markings defines a morphism
$$ \ev \times \ev_\infty: \overline{\M}^G_{n,1}(\mathbb{A},X,d) \to (X/G)^n \times \overline{\cI}_{X \qu G} .$$
In the case $d = 0$, the moduli stack $\overline{\M}^G_{0,1}(\mathbb{A},X,d)$ is isomorphic to $\overline{\cI}_{X \qu G}$ via evaluation at infinity. The {\em quantum Kirwan map} is the map
$$ \kappa_X^G: \widehat{QH}_G(X) \to QH_{\mathbb{C}^\times}(X \qu G) $$
defined as follows. Let $\ev_{\infty,d}: \overline{\M}_n^G(\mathbb{A},X,d) \to \overline{\cI}_{X \qu G}$ be evaluation at infinity restricted to affine gauged maps, and
$$ \ev_{\infty,d,*}: H( \overline{\M}_n^G(\mathbb{A},X,d)) \otimes_\mathbb{Q} \Lambda_X^G \to H_G(\overline{\cI}_{X \qu G}) \otimes_\mathbb{Q} \Lambda_X^G $$
push-forward using the virtual fundamental class. The quantum Kirwan map is
$$ \kappa_X^G: \widehat{QH}_G(X) \to QH_{\mathbb{C}^\times}(X \qu G), \quad \alpha \mapsto \sum_{n,d} \frac{q^d}{n!} \ev_{\infty,d,*} \ev^* (\alpha,\ldots, \alpha) .$$
As a formal map each term in the Taylor series of $\kappa_X^G$ is well-defined on $QH_G(X)$, but in general the sum of terms may have convergence issues. The $q =0 $ specialization of $\kappa_X^G$ is the Kirwan map to the cohomology of a git quotient studied in \cite{ki:coh}.
The cobordism relating stable maps to the quotient with Mundet semistable maps is itself a moduli stack of gauged maps with scaling defined by allowing the linearization to tend towards infinity. In order to determine which stability condition to use, the source curves must be equipped with additional data of a {\em scaling}: a section
$$ \delta: \hat{C} \to \mathbb{P} \left(\omega_{\hat{C}/(C \times S)} \oplus
\mO_{\hat{C}} \right) $$
of the projectivized relative dualizing sheaf. If the section is finite, one uses the Mundet semistability condition, while if infinite one uses the stability condition on the target. The possibility of constructing a cobordism in this way was suggested by a symplectic argument of Gaio-Salamon \cite{ga:gw}. A {\em scaled gauged map} is a map to the quotient stack whose domain is a curve equipped with a section of the projectivized dualizing sheaf and a collection of distinct markings: A datum
$$ \hat{C} \to S, \quad v:\hat{C} \to C \times X/G, \quad \delta: \hat{C} \to \mathbb{P} \left(\omega_{\hat{C}/(C \times S)} \oplus
\mO_{\hat{C}} \right), \quad z_1,\ldots, z_n \in \hat{C} $$
where $\hat{C} \to S$ is a nodal curve of genus $g=\operatorname{genus}{C}$, $v = (P,u)$ is a morphism to the quotient stack $X/G$ that consists of a principal $G$-bundle $P \to \hat{C}$ and a map $u: \hat{C} \to P \times_G X$ of whose class projects to $[C] \in H_2(C)$, and $\delta$ is a section of the projectivization of the relative dualizing sheaf $\omega_{\hat{C}/(C \times S)}$ satisfying certain properties. In the case that $X \qu G$ is an orbifold, the domain $\hat{C}$ is allowed to have orbifold singularities at the nodes and markings and the morphism is required to be representable. The moduli stack of stable scaled gauged maps $\overline{\M}^G_{n,1}(C,X,d)$ \label{scaledproj} with $n$ markings and class $d \in H_2^G(X,\mathbb{Q})$ is equipped with a forgetful map
$$ \rho: \overline{\M}^G_{n,1}(C,X,d) \to \overline{\M}_{0,1} \cong \mathbb{P}^1, \quad [\hat{C}, u, \delta, \ul{z}] \mapsto \delta .$$
The fibers of $\rho$ over zero $0,\infty \in \mathbb{P}^1$ consist of either Mundet semistable gauged maps, in the case $\delta = 0$, or stable maps to the git quotient together with affine gauged maps, in the case $\delta = \infty$: In notation,
\begin{multline} \label{fibers} \rho^{-1}(0) = \overline{\M}^G_n(C,X,d),
\quad \rho^{-1}(\infty) = \bigcup_{d_0 + \ldots + d_r = d}
\bigcup_{I_1 \cup \ldots \cup I_r =\{1,\ldots,n\} } \\ \left(
\overline{\M}^{{\on{fr}}}_{g,r}(C \times X \qu G,(1,d_0)) \times_{(\overline{\cI}_{X
\qu G})^r} \prod_{j=1}^r \overline{\M}^G_{|I_j|,1} (\mathbb{A},X,d_j)
\right) / (\mathbb{C}^\times)^r \end{multline}
where the superscipt ${\on{fr}}$ indicates the inclusion of framings at the tangent spaces to the markings, $(\mathbb{C}^\times)^r$ acts diagonally on the framings and on the scalings, and we identify $H_2(X \qu G)$ as a subspace of $H_2^G(X)$ via the inclusion $X \qu G \subset X / G$. The properness of these moduli stacks was argued via symplectic geometry in \cite{qk2}. The advantage of the symplectic proof is that the compactness is somewhat more natural; it follows by a combination of Gromov and Uhlenbeck compactness theorems as in Ott \cite{ott:remov} and also applies in the presence of Lagrangian boundary conditions as in Xu \cite{xu:compact}, for an arbitrary symplectic manifold. However, in the setting of virtual fundamental classes constructed algebraically, one prefers to stay in the framework of algebraic geometry. This is especially true in a subsequent paper of the first and third authors in which we extend the results to quantum K-theory, and in particular give presentations of the quantum K-theory ring of toric stacks; the definition of quantum K-theory is at the moment heavily algebraic, and there is no known definition purely in terms of symplectic geometry. Also, it is good to have several proofs. In this paper we give an algebraic proof of the following:
\begin{theorem} \label{main} For any real $E > 0$, the union of components $\overline{\M}_{n,1}^G(C,X,d)$, $\overline{\M}_n^G(C,X,d)$, and
$\overline{\M}_{n,1}^G(\mathbb{A},X,d)$ with $(d, c_1^G(\tilde{X})) < E$ is proper. \end{theorem}
\noindent The proof is a combination of boundedness arguments and valuative criteria. By integration over the moduli stack of stable scaled gauged maps one obtains the following identity: Let $\tau_X^{G,k}$ denote the gauged potential of \eqref{gtrace} defined using the polarization $\tilde{X}^k$ for $k$ a positive integer.
\begin{equation} \label{qwit} \lim_{k \to \infty} \tau_X^{G,k} = \tau_{X
\qu G} \circ \kappa_X^G .\end{equation}
This is called in \cite{ga:gw} and \cite{qk1} the {\em adiabatic limit
theorem}.
\section{Scaled curves}
Scaled curves are curves with a section of the projectivized dualizing sheaf incorporated, intended to give complex analogs of spaces introduced by Stasheff \cite{st:hs} such as the multiplihedron, cyclohedron etc. Recall from Deligne-Mumford \cite{dm:irr} and Behrend-Manin \cite[Definition 2.1]{bm:gw} the definition of stable and prestable curves. A {\em prestable curve} over the scheme $S$ is a flat proper morphism $\pi: C \to S$ of schemes such that the geometric fibers of $\pi$ are reduced, connected, one-dimensional and have at most ordinary double points (nodes) as singularities. A {\em
marked prestable curve} over $S$ is a prestable curve $\pi: C \to S$ equipped with a tuple $\ul{z} = (z_1,\ldots,z_n): S \to C^n$ of distinct non-singular sections. A {\em morphism} $p : C \to D$ of prestable curves over $S$ is an $S$-morphism of schemes, such that for every geometric point $s$ of $S$ we have (a) if $\eta$ is the generic point of an irreducible component of $D_s$, then the fiber of $p_s$ over $\eta$ is a finite $\eta$-scheme of degree at most one, (b) if $C'$ is the normalization of an irreducible component of $C_s$, then $p_s(C')$ is a single point only if $C'$ is rational. A prestable curve is {\em stable} if it has finitely many automorphisms. Denote by $\overline{\M}_{g,n}$ the proper Deligne-Mumford stack of stable curves of genus $g$ with $n$ markings \cite{dm:irr}. The stack $\overline{\MM}_{g,n}$ of prestable curves of genus $g$ with $n$ markings is an Artin stack locally of finite type \cite[Proposition 2]{be:gw}.
\begin{figure}
\caption{Associativity divisor relation}
\label{assoc}
\end{figure}
The following constructions give complex analogs of the spaces constructed in Stasheff \cite{st:hs}. For any family of possibly nodal curves $C \to S$ we denote by $\omega_C$ the relative dualizing sheaf defined for example in Arbarello-Cornalba-Griffiths \cite[p. 97]{ar:alg2}. Similarly for any morphism $\hat{C} \to C$ we denote by $ \omega_{\hat{C}/C}$ the relative dualizing sheaf and $\mathbb{P}(\omega_{\hat{C}/C} \oplus \mO_{\hat{C}}) \to \hat{C} $ the projectivization. A {\em scaling} is a section
$$ \delta: \hat{C} \to \mathbb{P}(\omega_{\hat{C}/C} \oplus \mO_{\hat{C}}), \quad \mathbb{P}(\omega_{\hat{C}/C} \oplus \mO_{\hat{C}}) = (\omega_{\hat{C}/C} \oplus \mO_{\hat{C}})^\times / \mathbb{C}^\times .$$
If $\hat{C} \to C$ is an isomorphism then $\omega_{\hat{C}/C}$ is trivial:
$$ (\hat{C} \cong C) \implies (\mathbb{P}(\omega_{\hat{C}/C} \oplus \mO_{\hat{C}}) \cong C \times \mathbb{P}^1) .$$
In this case a scaling $\delta$ is a section $C \to \mathbb{P}^1$, and $\delta$ is required to be constant. Thus the space of scalings on an unmarked, irreducible curve is $\mathbb{P}^1$.
Scalings on nodal curves with markings are required to satisfy the following properties. First, $\delta$ should satisfy the {\em
affinization} property that on any component $\hat{C}_i$ of $\hat{C}$ on which $\delta$ is finite and non-zero, $\delta$ has no zeroes and a single double pole. In particular, this implies that in the case $\hat{C} \cong C$, then $\delta$ is a constant section as in the last paragraph, while on any component $\hat{C}_i$ of $\hat{C}$ with finite non-zero scaling which maps to a point in $C$, $\delta$ defines an affine structure on the complement of the pole. To define the second property, note that any morphism $\hat{C} \to C$ of class $[C]$ defines a {\em rooted tree} whose vertices are components $\hat{C}_i$ of $\hat{C}$, whose edges are nodes $w_j \in \hat{C}$, and whose root vertex is the vertex corresponding to the component $\hat{C}_0$ that maps isomorphically to $C$. Let ${\mathcal{T}}$ denote the set of indices of {\em terminal} components $\hat{C}_i$ that meet only one other component of $\hat{C}$:
$$ {\mathcal{T}} = \{ i \ | \ \# \{ j \neq i | \hat{C}_j \cap \hat{C}_i \neq \emptyset \} = 1 \} $$
as in Figure \ref{leaves}. The {\em bubble components} are the components of $\hat{C}$ mapping to a point in $C$.
\begin{figure}
\caption{A scaled marked curve}
\label{leaves}
\end{figure}
For each terminal component $\hat{C}_i, i \in {\mathcal{T}}$ there is a canonical non-self-crossing path of components $\hat{C}_{i,0} = \hat{C}_0,\ldots, \hat{C}_{i,k(i)} = \hat{C}_{i}$. Define a partial order on components by $\hat{C}_{i,j} \preceq \hat{C}_{i,k}$ for $j \leq k$. The {\em monotonicity property} requires that $\delta$ is finite and non-zero on at most one of these (gray shaded) components, say $\hat{C}_{i, f(i)}$, and
\begin{equation} \label{monotone} \delta | \hat{C}_{i,j} = \begin{cases} \infty & j < f(i) \\
0 & j > f(i) \end{cases}
. \end{equation}
We call $\hat{C}_{i,f(i)}$ a {\em transition component}. That is, the scaling $\delta$ is infinite on the components before the transition components and zero on the components after the transition components, in the ordering $\preceq$. See Figure \ref{leaves}. In addition the {\em marking condition } requires that the scaling is finite at the markings:
$$\delta(z_i) < \infty, \quad \forall i =1,\ldots, n .$$
\begin{definition} A {\em prestable scaled curve} with target a smooth
projective curve $C$ is a morphism from a prestable map $\hat{C}$ to
$C$ of class $[C]$ equipped with section $\delta$ and $n$ markings
$\ul{z} = (z_1,\ldots, z_n)$ satisfying the affinization,
monotonicity and marking properties. Isomorphisms of prestable scaled
curves are diagrams
$$ \begin{diagram} \node{ \hat{C}_1} \arrow{s} \arrow{e,t}{\varphi}
\node{ \hat{C}_2 } \arrow{s} \\ \node{S_1} \arrow{e}
\node{S_2} \end{diagram}, \quad (D\varphi^*) \varphi^*( \delta_2) = \delta_1, \quad \varphi(z_{i,1}) = z_{i,2}, \ \ \forall i = 1,\ldots, n $$
where the top arrow is an isomorphism of prestable curves and
$$ D\varphi^* : \varphi^* \mathbb{P}(\omega_{\hat{C}_2/C} \oplus \mO_{\hat{C}_2}) \to \mathbb{P}(\omega_{\hat{C}_1/C} \oplus \mO_{\hat{C}_1})$$
is the associated morphism of projectivized relative dualizing sheaves. A scaled curve is {\em stable} if on each bubble component $\hat{C}_i \subset \hat{C}$ (that is, component mapping to a point in $C$) there are at least three special points (markings or nodes),
$$ (\delta| \hat{C}_i \in \{0, \infty\} ) \ \implies \ \# (( \{ z_i \} \cup \{ w_j \} ) \cap \hat{C}_i) \ge 3 $$
or the scaling is finite and non-zero and there are least two special points
$$ (\delta| \hat{C}_i \notin \{0, \infty\} ) \ \implies \ \# (( \{ z_i \} \cup \{ w_j \} ) \cap \hat{C}_i) \ge 2 .$$
\end{definition}
Introduce the following notation for moduli spaces. Let $\overline{\MM}_{n,1}(C)$ denote the category of prestable $n$-marked scaled curves and $\overline{\M}_{n,1}(C)$ the subcategory of stable $n$-marked scaled curves.
The {\em combinatorial type} of a prestable marked scaled curve is defined as follows. Given such $(\hat{C},u: \hat{C} \to C, \ul{z},\delta)$ Let $\Gamma$ be the graph whose vertex set $\on{Vert}(\Gamma)$ is the set of irreducible components of $C$, finite edges $\on{Edge}_{< \infty}(\Gamma)$ correspond to nodes, semi-infinite edges $\on{Edge}_\infty(\Gamma)$ correspond to markings, and equipped with the labelling of semi-infinite edges by $\{ 1,\ldots , n\}$ a distinguished {\em root vertex} $v_0 \in \on{Vert}(\Gamma)$ corresponding to the root component and a set of {\em transition vertices} $\on{Vert}^t(\Gamma) \subset \on{Vert}(\Gamma)$ corresponding to the transition components. Graphically we represent a combinatorial type as a graph with transition vertices shaded by grey, and the vertices lying on three levels depending on whether they occur before or after the transition vertices. See Figure \ref{spider}. Note that the combinatorial type is functorial; in particular any automorphism of prestable marked scaled curves induces an automorphism of the corresponding type, that is, an automorphism of the graph preserving the additional data.
We note that the graphical representation of the combinatorial type of a curve can be viewed as the graph of a Morse/height function on the curve. In general this gives a spider like figure with the root component being the body of the spider. From this perspective the paths used in the monotonicity property of scalings are the legs of the spider.
\begin{figure}
\caption{Combinatorial type of a scaled marked curve}
\label{spider}
\end{figure}
\begin{example} \begin{enumerate} \item For $n = 0$, no bubbling is possible and $\overline{\M}_{0,1}(C)$ is
the projective line, $ \overline{\M}_{0,1}(C) \cong \mathbb{P}^1 .$
\item For $n = 1$, $\overline{\M}_{1,1}(C)$ consists of configurations
$\M_{1,1}(C) \cong C \times \mathbb{C}$ with irreducible domain and finite
scaling; a configurations $\overline{\M}_{1,1} - \M_{1,1}$ with one
component $\hat{C}_0 \cong C$ with infinite scaling $\delta |
\hat{C}_0$, and another component $\hat{C}_1$ mapping trivially to
$C$, equipped with a one-form $\delta | \hat{C}_1$ with a double
pole at the node and a marking $z_1 \in \hat{C}_1$. Thus $
\overline{\M}_{1,1}(C) \cong C \times \mathbb{P}^1 .$
\item For $n= 2$, $\overline{\M}_{2,1}(C)$ consists of configurations
$\M_{2,1}(C)$ with two distinct points $z_1, z_2 \in C$ and a
scaling $\delta \in \mathbb{P}^1$; configurations $\M_{2,1,\Gamma_1}$ where
the two points $z_1,z_2$ have come together and bubbled off onto a
curve $z_1,z_2 \in \hat{C}_1$ with zero scaling $\delta |
\hat{C}_1$, so that $\M_{2,1,\Gamma_1} \cong C \times \mathbb{P}^1$;
configurations $\M_{2,1,\Gamma_2}$ with a root component $\hat{C}_0$
with infinite scaling $\delta | \hat{C}_0$, and two components
$\hat{C}_1,\hat{C}_2$ with non-trivial scalings $\delta | \hat{C}_1,
\delta | \hat{C}_2$ containing markings $z_1 \in \hat{C}_1, z_2 \in
\hat{C}_2$; a stratum $\M_{2,1,\Gamma_2}$ of configurations with a
component $\hat{C}_1$ containing two markings $z_1,z_2 \in
\hat{C}_1$ and $\delta | \hat{C}_1$ non-zero; a stratum
$\M_{2,1,\Gamma_3}$ containing with three components, one
$\hat{C}_0$ mapping isomorphically to $C$; one $\hat{C}_1$ with two
nodes and a one form $\delta | \hat{C}_1$ with a double pole at the
node attaching to $\hat{C}_0$; and a component $\hat{C}_2$ with two
markings $z_1,z_2 \in \hat{C}_2$, a node, and vanishing scaling
$\delta | \hat{C}_2$; and a stratum a stratum $\M_{2,1,\Gamma_4}$
containing the root component $\hat{C}_0$, a component $\hat{C}_1$
with infinite scaling with three nodes, and two components
$\hat{C}_2, \hat{C}_3$ with finite, non-zero scaling, each
containing a node and a marking. The two evaluation maps at the
markings, together with the forgetful map to $\overline{\M}_{0,1}(C)$,
define an isomorphism $ \overline{\M}_{2,1}(C) \to C \times C \times \mathbb{P}^1
.$
\end{enumerate} \end{example}
\begin{remark} The extension of the one-form in a family of scaled
curves may be explicitly described as follows. On each component of
the limit, the one-form is determined by the limiting behavior of
the product of deformation parameters for the nodes connecting that
component to the root component of the limit:
Let
$$\hat{C} \to S,\delta : \hat{C} \to \mathbb{P}(\omega_{\hat{C}/C \times S} \oplus
\mO_{\hat{C}}),\ul{z}: S \to \hat{C}^n$$
be a family of scaled curves over a
punctured curve $S = \overline{S} - \{ \infty \} $ and
$\hat{C}_\infty$ a curve over $\infty$ extending the family $\hat{C}$. Let $\Def(\hat{C}_\infty)/\Def_\Gamma(\hat{C}_\infty)$ denote the deformation space of the curve $\hat{C}_\infty$ normal to the stratum of curves of the same combinatorial type $\Gamma$ as $\hat{C}_\infty$. This normal deformation space is canonically identified with the sum of products of cotangent lines at the nodes
$$ \Def(\hat{C}_\infty)/\Def_\Gamma(\hat{C}_\infty) = \sum_{w} T^\vee_w \hat{C}_{i_-(w)} \otimes T^\vee_w \hat{C}_{i_+(w)} $$
where $\hat{C}_{i_\pm(w)}$ are components of $\hat{C}_\infty $ adjacent to $w$, see \cite[p. 176]{ar:alg2}. Over the deformation space $\Def(\hat{C}_\infty)$ lives a semiversal family, universal if the curve is stable. Given family of curves $\hat{C} \to S$ as above the curve $\hat{C}$ is obtained by pull-back of the semiversal family by a map
$$ S \to \sum_{w} T^\vee_w \hat{C}_{i_-(w)} \otimes T^\vee_w \hat{C}_{i_+(w)}, \quad z \mapsto (\delta_w(z)) $$
describing the curves as local deformations (non-uniquely, since the curves themselves may be only prestable.) Let
$$ \hat{C}_0 = \hat{C}_{i,0}, \ldots, \hat{C}_{i,l(i)} := \hat{C}_i $$
denote the path of components from the root component, and
$$ w_{i,0},\ldots, w_{i,l(i)-1} \in \hat{C}_\infty $$
the corresponding sequence of nodes. The nodes $w_{i,j}, w_{i,j+1}$
lie
in the same component $C_{i,j+1}$ and we have a canonical isomorphism
$$ T_{w_{i,j}}^\vee C_{i,j+1} \cong T_{w_{i,j+1}} C_{i,j+1} $$
corresponding to the relation of local coordinates $z_+ = 1/z_-$ near $w_{i,j}$. Deformation parameters for this chain lie in the space
\begin{multline}
\on{Hom}(T_{w_{i,0}}^{\vee} \hat{C}_{i,0}, T_{w_{i,1}}^{\vee} \hat{C}_{i,1})
\oplus \on{Hom}(T_{w_{i,1}}^{\vee} \hat{C}_{i,1}, T_{w_{i,2}}^{\vee}
\hat{C}_{i,2}) \ldots \\ \oplus \on{Hom}(T_{w_{i,l(i) - 2}}^{\vee}
\hat{C}_{i,l(i)-2}, T_{w_{i,l(i)-1}}^{\vee} \hat{C}_{i,l(i)-1})
.\end{multline}
In particular, the product of deformation parameters
\begin{equation} \label{product} \gamma_{w_{i,0}}(z) \cdots \ldots \cdot \gamma_{w_{i,l(i) -1}}(z) \in \on{Hom}(T_{w_{i,0}}^{\vee} \hat{C}_{i,0}, T_{w_{i,l(i)-1}}^{\vee} \hat{C}_{i,l(i)-1}) \end{equation}
is well-defined. The product represents the {\em scale} at which the bubble component $\hat{C}_i$ forms in comparison with $\hat{C}_0 = \hat{C}_{i,0}$, that is, the ratio between the derivatives of local coordinates on $\hat{C}_i$ and $\hat{C}_0$. If $z$ is a point in $\hat{C}_i$ then we also have a canonical isomorphism
$ T_z^\vee \hat{C}_i \to T_{w_{i,0}} \hat{C}_0 .$
The product \eqref{product} gives an isomorphism
$ T_z^\vee \hat{C}_i \to T_{w_0}^{\vee} \hat{C}_{0} .$
\begin{equation} \label{oneform} \delta | \hat{C}_i = \lim_{z \to 0}
\delta(z) (\gamma_{w_{i,0}}(z) \cdots \ldots \cdot \gamma_{w_{i,l(i)
-1}}(z)) \end{equation}
the ratio of the scale of the bubble component with the parameter $\delta(z)^{-1}$. This ends the Remark. \end{remark}
One may view a scaled curve with infinite scaling on the root component as a nodal curve formed from the root component and a collection of bubble trees as follows.
\begin{definition} \label{affine} An {\em affine prestable
scaled curve} consists of a tuple $(C,\delta,\ul{z})$ where $C$ is a
connected projective nodal curve, $\delta: C \to \mathbb{P}( \omega_C \oplus
\mO_C)$ a section of the projectivized dualizing sheaf, and $\ul{z}
= (z_0,\ldots,z_n)$ non-singular, distinct points, such that
\begin{enumerate} \item $\delta$ is monotone in the following sense: For each terminal
component $\hat{C}_{i}, i \in {\mathcal{T}}$ there is a canonical
non-self-crossing path of components
$$\hat{C}_{l(i),0} = \hat{C}_0,\ldots,
\hat{C}_{i,k(i)} = \hat{C}_i .$$
The monotonicity condition is for any such non-self-crossing path of components starting with a root component, that $\delta$ is finite and non-zero on at most one of these {\em transition components}, say $\hat{C}_{i, f(i)}$, and the scaling is infinite for all components before the transition component and zero for components after the transition component:
$$ \delta | \hat{C}_{i,j} = \begin{cases} \infty & j < f(i) \\ 0 & j
> f(i) \end{cases} . $$
\item $\delta$ is infinite at $z_0$, and finite at $z_1,\ldots, z_n$.
\end{enumerate}
A prestable affine scaled curve is {\em stable} if it has finitely many automorphisms, or equivalently, if each component ${C}_i \subset {C}$ has at least three special points (markings or nodes),
$$ (\delta| {C}_i \in \{0, \infty\} ) \ \implies \ \# (( \{ z_i \} \cup \{ w_j \} ) \cap {C}_i) \ge 3 $$
or the scaling is finite and non-zero and there are least two special points
$$ (\delta| {C}_i \notin \{0, \infty\} ) \ \implies \ \# (( \{ z_i \} \cup \{ w_j \} ) \cap {C}_i) \ge 2 .$$
\end{definition}
We will see below in Theorem \ref{scaledproper} that scaled marked curves have no automorphisms. Examples of stable affine scaled curves are shown in Figure \ref{affine}. Denote the moduli stack of prestable affine scaled curves resp. stable affine $n$-marked scaled curves by $\overline{\MM}_{n,1}(\mathbb{A})$ resp. $\overline{\M}_{n,1}(\mathbb{A})$.
\begin{figure}
\caption{Examples of stable affine scaled curves}
\label{affine}
\end{figure}
\begin{theorem} \label{scaledproper} For each $n \ge 0$ and smooth
projective curve $C$ the moduli stack $\overline{\M}_{n,1}(C)$
resp. $\overline{\M}_{n,1}(\mathbb{A})$ of stable scaled affine curves is a
proper scheme locally isomorphic to a product of a number of copies
of $C$ with a toric variety. The stack $\overline{\MM}_{n,1}(C)$
resp. $\overline{\MM}_{n,1}(\mathbb{A})$ of prestable scaled curves is an Artin
stack of locally finite type. \end{theorem}
\begin{proof}
Standard arguments on imply that $\overline{\M}_{n,1}(C)$ and
$\overline{\MM}_{n,1}(C)$ are stacks, that is, categories fibered in
groupoids satisfying effective descent for objects and for which
morphisms form a sheaf. An object $(\hat{C},\ul{z},\delta)$ of
$\overline{\M}_{n,1}(C)$ over a scheme $S$ is a family of curves with
sections. Families of curves with markings and sections satisfy the
gluing axioms for objects; similarly morphisms are determined
uniquely by their pull-back under a covering. Standard results on
hom-schemes imply that the diagonal for $\overline{\MM}_{n,1}(C)$, hence
also $\overline{\M}_{n,1}(C)$, is representable, see for example
\cite[1.11]{dm:irr} for similar arguments, hence the stacks
$\overline{\MM}_{n,1}(C)$ and $\overline{\M}_{n,1}(C)$ are algebraic.
In preparation for showing that $\overline{\M}_{n,1}(C)$ is a variety we
claim that for any object $(\hat{C},\ul{z},\delta)$ of the moduli
stack $\overline{\M}_{n,1}(C)$ the automorphism group is trivial. Let
$\Gamma$ be the combinatorial type. The association of $\Gamma$ to
$(\hat{C},\ul{z},\delta)$ is functorial and any automorphism of
$(\hat{C},\ul{z},\delta)$ induces an automorphism of $\Gamma$. The
graph $\Gamma$ is a tree with labelled semi-infinite edges, each
vertex is determined uniquely by the partition of semi-infinite edges
given by removing the vertex; hence the automorphism acts trivially
on the vertices of $\Gamma$. Each component has at least three
special points, or two special points and a non-trivial scaling and
so has trivial automorphism group fixing the special points. Thus
the automorphism is trivial on each component of $\hat{C}$. The
claim follows.
The moduli space of stable scaled curves has a canonical covering by
varieties corresponding to the versal deformations of prestable
curves constructed by gluing. Suppose that $(u: \hat{C} \to C,
\ul{z}, \delta)$ is an object of $\overline{\M}_{n,1}(C)$ of combinatorial
type $\Gamma$. Let $\rho: \overline{\M}_{n,1}(C) \to \overline{\M}_{0,1}(C)
\cong \mathbb{P}^1$ denote the forgetful morphism. The locus $\rho^{-1}(\mathbb{C})
\subset \overline{\M}_{n,1}(C)$ of curves with finite scaling is isomorphic
to $\overline{\M}_n(C) \times \mathbb{C}$, where the last factor denotes the
scaling. In the case that the root component has infinite scaling,
let $\Gamma_1,\ldots,\Gamma_k$ denote the (possibly empty)
combinatorial types of the bubble trees attached at the special
points. The stratum ${\M}_{n,1,\Gamma}(C)$ is the product of $C^k$
with moduli stacks of scaled affine curves
${\M}_{n_i,1,\Gamma_i}(\mathbb{A})$ for $i =1,\ldots, k$, each isomorphic to
an affine space given by the number of markings and scalings minus
the dimension of the automorphism group $(n_i + 1) + 1 -
\dim( \on{Aut} (\mathbb{P}^1)) = n_i - 1$ \cite{mau:mult}. Let
$$\gamma_e \in T^\vee_{w(e)} \hat{C}_{i_-(e)} \otimes T^\vee_{w(e)} \hat{C}_{i_+(e)}, \quad e \in \on{Edge}_{< \infty}(\Gamma)$$
be the deformation parameters for the nodes. A collection of deformation parameters $ \ul{\gamma} = ( \gamma_e)_{e \in
\on{Edge}(\Gamma)}$ is {\em balanced} if the signed product
\begin{equation} \label{balanced} \prod_{e \in P} \gamma_e^{\pm 1} \end{equation}
of parameters corresponding to any non-self-crossing path $P$ between transition components is equal to $1$, where the sign is positive for edges pointing towards the root vertex and equal to $-1$ if the edge is oriented away from it. Let $Z_\Gamma$ denote the set of deformation parameters satisfying the condition \eqref{balanced}. Then there is a morphism
$$ \M_{n,1,\Gamma}(C) \times Z_\Gamma \to \overline{\M}_{n,1}(C) $$
described as follows. Choose local \'etale coordinates $z_e^\pm$ on the adjacent components to each node $w_e, \in \on{Edge}_{<
\infty}(\Gamma)$ and glue together the components using the identifications $z_e^+ \mapsto \gamma_e/ z_e^-$, see for example \cite[p. 176]{ar:alg2}, \cite[2.2]{ol:logtwist}. Set the scaling on the root component
$$ \delta = \prod_{e \in P} \gamma_e $$
where $P$ is a path of nodes from the root component to the transition component, independent of the choice of component by \eqref{balanced}. This gives a family $(\hat{C},u,\delta,\ul{z})$ of stable scaled curves over $\M_{n,1,\Gamma}(C) \times Z_\Gamma$ and hence a morphism to $\overline{\M}_{n,1}(C)$. The family $(\hat{C},\ul{z},u,\delta)$ defines a universal deformation of any curve of type $\Gamma$. Indeed, $(\hat{C},\ul{z})$ is a versal deformation of any of its prestable fibers by \cite{ar:alg2}, and it follows that the family $(\hat{C},\ul{z},u)$ is a versal deformation of any of its fibers since there is a unique extension of the stable map on the central fiber, up to automorphism. The equation \eqref{product} implies that any family of stable scaled curves satisfies the balanced relation \eqref{balanced} between the deformation parameters for any family of marked curves with scalings. This provides a cover of $\overline{\M}_{n,1}(C)$ by varieties. It follows that $\overline{\M}_{n,1}(C)$ is a variety.
The stack of prestable scaled curves $\overline{\MM}_{n,1}(C)$ is an Artin stack of locally finite type. Charts for the stack $\overline{\MM}_{n,1}(C)$, as in the case of prestable curves in \cite{be:gw}, are given by using forgetful morphisms $\overline{\M}_{n+k,1}(C) \to \overline{\MM}_{n,1}(C)$. Since these morphisms admit sections locally, they provide a smooth covering of $\overline{\MM}_{n,1}(C)$ by varieties.
We check the valuative criterion for properness for $\overline{\M}_{n,1}(C)$. Given a family of stable scaled marked curves over a punctured curve $S$ with finite scaling $\delta$
$$(\hat{C}, u: \hat{C} \to C, \ul{z},\delta) \to S = \overline{S} - \{ \infty \} $$
we wish to construct there exists an extension over $\overline{S}$. We consider only the case $\hat{C} \cong C \times S$; the general case is similar. After forgetting the scaling $\delta$ and stabilizing we obtain a family of stable maps to $C$ of degree $[C]$,
$$ (\hat{C}^{\on{st}}, u: \hat{C}^{\on{st}} \to C, \ul{z}^{\on{st}}) \to \overline{S} - \{ \infty \} .$$
By properness of the stack $\overline{\M}_n(C)$ of stable maps to $C$, this family extends over the central fiber $\infty$ to give a family over $\overline{S}$. The section $\delta$ of $\omega_{\hat{C}^{\on{st}}/C}$ defines an extension over $\overline{S}$ except possibly at the nodes. Here there are possible irremovable singularities corresponding to the following situation: suppose that $\hat{C}_0,\ldots \hat{C}_i$ is a chain of components in the curve at the central fiber, with $\hat{C}_0 \cong C$ the root component. Suppose that $\hat{C}_{i}, \hat{C}_{i+1}$ are adjacent component with $\delta$ infinite on $\hat{C}_i$ and zero on $\hat{C}_{i+1}$. Taking the closure of the graph of $\delta$ gives a family $\hat{C}$ of curves over $C$ given by replacing some of the nodes of $\hat{C}^{\on{st}}$ with fibers of $\mathbb{P}(\omega_{\hat{C}^{\on{st}}/C} \oplus \mO_{\hat{C}^{\on{st}}})$ over the node. The relative cotangent bundle of $\hat{C}$ is related to that of $\hat{C}^{\on{st}}$ by a twist at $D_0,D_\infty$: If $\pi: \hat{C} \to \hat{C}^{\on{st}}$ denotes the projection onto $\hat{C}$ then on the components of $\hat{C}$ collapsed by $\pi$ we have
$$ \omega_{\hat{C}/C} = \pi^* \omega_{\hat{C}^{\on{st}}/C} (-D_0 -D_\infty) $$
where $D_0,D_\infty$ are the inverse images of the sections at zero and infinity in $\mathbb{P}( \omega_{\hat{C}^{\on{st}}/C} \oplus \mO_{\hat{C}^{\on{st}}})$. Abusing notation $\omega_{\hat{C}_i^{\on{st}}/C} (-D_0 ) = \omega_{\hat{C}_i^{\on{st}}/C}$ resp. $\omega_{\hat{C}_i^{\on{st}}/C} (-D_\infty ) = \omega_{\hat{C}_i^{\on{st}}/C}$ on components $\hat{C}_i^{\on{st}}$ contained in $D_0$ resp. $D_\infty$. The extension of $\delta$ to a rational section of $\pi^* \omega_{\hat{C}^{\on{st}}/C}$ has, by definition a zero at $\delta^{-1}(D_0)$ and a pole at $\delta^{-1}(D_\infty)$. Hence the extension of $\delta$ to a section of $\pi^* \omega_{\hat{C}^{\on{st}}/C}(-D_0 - D_\infty)$ has no zeroes at $D_0$ and a double pole at $D_\infty$. This implies that $\delta$ extends uniquely as a section of $\mathbb{P}( \omega_{\hat{C}/C} \oplus \mO_{\hat{C}})$ to all of $\overline{S}$.
By the construction \eqref{oneform}, the extension of $\delta$ satisfies the monotonicity condition \eqref{monotone}. Indeed suppose that a component $\hat{C}_i$ is further away from a component $\hat{C}_j$ in the path of components from the root component $\hat{C}_0$. Since all deformation parameters $\gamma_{w_{i,k}}(z)$ are approaching zero, from \eqref{oneform}, at most one of the limits
$\delta | \hat{C}_i, \delta | \hat{C}_j$ can be finite, and
$$ \begin{cases} \delta | \hat{C}_i \ \text{finite} \ \implies \delta | \hat{C}_j \ \ \text{zero} \\
\delta | \hat{C}_j \ \text{finite} \ \implies \delta | \hat{C}_i \ \ \text{infinite} . \end{cases}. $$
The condition \eqref{monotone} follows. \end{proof}
\section{Mumford stability}
In this section we review the relationship between the stack-theoretic quotient and Mumford's geometric invariant theory quotient \cite{mu:ge}. First we introduce various Lie-theoretic notation. Let $G$ be a connected complex reductive group with Lie algebra $\lie{g}$. When $G$ is abelian (so a complex torus) we denote by
$$ \lie{g}_\mathbb{Z} = \{ D \phi(1) \in \lie{g} \ | \ \phi \in \on{Hom}(\mathbb{C}^\times,G) \}, \quad \lie{g}_\mathbb{Q} = \lie{g}_\mathbb{Z} \otimes_\mathbb{Z} \mathbb{Q}$$
the {\em coweight lattice} of derivatives of one-parameter subgroups resp. rational one-parameter subgroups. Dually denote by
$$ \lie{g}_\mathbb{Z}^\vee = \{ D \chi \in \lie{g}^\vee \ | \ \chi \in \on{Hom}(G,\mathbb{C}^\times) \}, \quad \lie{g}_\mathbb{Q}^\vee = \lie{g}_\mathbb{Z}^\vee \otimes_\mathbb{Z} \mathbb{Q}$$
the {\em weight lattice} of derivatives of characters of $G$ and the set of {\em rational weights}, respectively. If $G$ is non-abelian then we still denote by $\lie{g}_\mathbb{Q}$ the set of derivatives of rational one-parameter subgroups.
The targets of our maps are quotient stacks defined as follows. Let $X$ be a smooth projective $G$-variety. Let $X/G$ denote the quotient stack, that is, the category fibered in groupoids whose fiber over a scheme $S$ has objects pairs $v = (P,u)$ consisting of a principal $G$-bundle $P \to S$ and a section $u: S \to P \times_G X$; and whose morphisms are given by diagrams
$$ \begin{diagram} \node{P_1} \arrow{s} \arrow{e,t}{\phi} \node{P_2}
\arrow{s} \\ \node{S_1} \arrow{e,b}{\psi} \node{S_2} \end{diagram}, \quad \phi(X) \circ u_1 = u_2 \circ \psi $$
where $\phi(X) : P_1(X) \to P_2(X)$ denotes the map of associated fiber bundles \cite{dm:irr}, \href{http://stacks.math.columbia.edu/tag/04UV}{Tag 04UV} \cite{dejong:stacks}.
Mumford's geometric invariant theory quotient \cite{mu:ge} is traditionally defined as the projective variety associated to the graded ring of invariant sections of a linearization of the action in the previous paragraph. Let $\tilde{X} \to X$ be a linearization, that is, ample $G$-line bundle. Then
$$ X \qu G := \on{Proj} \left( \oplus_{k \ge 0} H^0(\tilde{X}^k)^G \right) .$$
Mumford \cite{mu:ge} realizes this projective variety as the quotient of a {\em semistable locus} by an equivalence relation. The semistable locus consists of points $x \in X$ such that some tensor power $\tilde{X}^k, k > 0 $ of $\tilde{X}$ has an invariant section non-vanishing at $x$, while the unstable locus is the complement of the semistable locus:
$$ X^{\on{ss}} = \{ x \in X \ | \ \exists k > 0, \sigma \in H^0(\tilde{X}^k)^G, \quad \sigma(x) \neq 0 \}, \quad X^{\on{us}} := X - X^{\on{ss}} .$$
A point $x \in X$ is {\em polystable} if its orbit is closed in the semistable locus $\overline{Gx \cap X^{\on{ss}}} = Gx \cap X^{\on{ss}}$. A point $x \in X$ is {\em stable} if it is polystable and the stabilizer $G_x$ of $x$ is finite. In Mumford's definition the git quotient is the quotient of the semistable locus by the {\em orbit equivalence
relation}
$$ (x_1 \sim x_2) \iff \overline{Gx_1}\cap \overline{Gx_2} \cap X^{\on{ss}} \neq \emptyset. $$
Each semistable point is then orbit-equivalent to a unique polystable point. However, here we define the git quotient as the stack-theoretic quotient
$$ X \qu G := X^{\on{ss}}/G.$$
We shall always assume that $X^{\on{ss}}/G$ is a Deligne-Mumford stack (that is, the stabilizers $G_x$ are finite) in which case the coarse moduli space of $X^{\on{ss}}/G$ is the git quotient in Mumford's sense. The Luna slice theorem \cite{luna:slice} implies that $X^{\on{ss}}/G$ is \'etale-locally the quotient of a smooth variety by a finite group, and so has finite diagonal. By the Keel-Mori theorem \cite{km:quot}, explicitly stated in \cite[Theorem 1.1]{conrad:kl}, the morphism from $X^{\on{ss}}/G$ to its coarse moduli space is proper. Since the coarse moduli space of $X^{\on{ss}}/G$ is projective by Mumford's construction, it is proper, hence $X^{\on{ss}}/G$ is proper as well.
Later we will need the following observation about the unstable locus. As the quotient $X \qu G$ is non-empty, there exists an ample divisor $D$ containing the unstable locus: take $D$ to be the vanishing locus of any non-zero invariant section of $\tilde{X}^k$ for some $k > 0$:
\begin{equation} \label{invsection} D = \sigma^{-1}(0), \quad \sigma \in H^0(\tilde{X}^k)^G - \{ 0 \} .\end{equation}
The Hilbert-Mumford numerical criterion \cite[Chapter 2]{mu:ge} provides a computational tool to determine the semistable locus: A point $x \in X$ is $G$-semistable if and only if it is $\mathbb{C}^\times$-semistable for all one-parameter subgroups $\mathbb{C}^\times \to G$. Given a rational element $\lambda\in \lie{g}_\mathbb{Z}$ denote the corresponding one-parameter subgroup $\mathbb{C}^\times \to G, \ z \mapsto z^\lambda$. Denote by
$$ x_\lambda := \lim_{z \to 0} z^\lambda x $$
the limit under the one-parameter subgroup. Let $\mu(x,\lambda) \in \mathbb{Z}$ be the weight of the linearization $\tilde{X}$ at $x_\lambda$ defined by
$$ z \tilde{x} = z^{\mu(x,\lambda)} \tilde{x}, \quad \forall z \in \mathbb{C}^\times, \tilde{x} \in \tilde{X}_{x_\lambda} .$$
By restricting to the case of a projective line one sees that the point $x \in X$ is semistable if and only if $ \mu(x,\lambda) \leq 0$ for all $\lambda \in \lie{g}_\mathbb{Z}.$ Polystability is equivalent to semistability and the additional condition $ \mu(x,\lambda) = 0 \iff \mu(x,-\lambda) =0 .$ Stability is the condition that $ \mu(x,\lambda) < 0$ for all $\lambda \in \lie{g}_\mathbb{Z} - \{ 0 \} .$
The Hilbert-Mumford numerical criterion \cite[Chapter 2]{mu:ge} can be applied explicitly to actions on projective spaces as follows. Suppose that $G$ is a torus and $X = \mathbb{P}(V)$ the projectivization of a vector space $V$. Let
$\tilde{X} = \mO_X(1) \otimes \mathbb{C}_\theta $
be the $G$-equivariant line bundle given by tensoring the hyperplane bundle $\mO_X(1)$ and the one-dimensional representation $\mathbb{C}_\theta$ corresponding to some weight $\theta \in \lie{g}_\mathbb{Z}^\vee$. Recall if $p \in X$ is represented by a line $l \subset V$ then the fiber of $\mO_X(1) \otimes \mathbb{C}_\theta$ at $p$ is $l^\vee\otimes \mathbb{C}_\theta$. In particular if $z^\lambda$ fixes $p$ then $z^\lambda$ scales $l$ by some $z^{\mu(\lambda)}$, so that $z^\lambda \tilde{x} = z^{-\mu(\lambda)+\theta(\lambda)}\tilde{x}$. Let $k = \dim(V)$ and decompose $V$ into weight spaces $V_1,\ldots, V_k$ with weights $\mu_1,\dots,\mu_k\in \lie{g}_\mathbb{Z}^\vee .$ Identify
$$H^2_G(X) \cong H^2_{\mathbb{C}^\times \times G}(V) \cong \mathbb{Z} \oplus \lie{g}_\mathbb{Z}^\vee $$
Under this splitting the first Chern class $c_1^G(\tilde{X})$ becomes identified up to positive scalar multiple with the pair
\begin{equation}\label{eqiv c_1}
c_1^G(\tilde{X}) \mapsto (1,\theta) \in \mathbb{Z} \oplus \lie{g}_\mathbb{Z}^\vee.
\end{equation}
The following is essentially \cite[Proposition 2.3]{mu:ge}.
\begin{lemma} The semistable locus for the action of a torus $G$ on the projective space $X =P(V)$ with weights $\mu_1,\ldots, \mu_k$ and linearization shifted
by $\theta$ is $ X^{\on{ss}} = \mathbb{P}(V)^{\on{ss}} = \{ [x_1,\ldots,x_k] \in
\mathbb{P}(V) \ | \ \operatorname{hull} ( \{ \mu_i | x_i \neq 0 \}) \ni \theta \} .$ A
point $x $ is polystable iff $\theta$ lies in the interior of the
hull above, and stable if in addition the hull is of maximal
dimension. \end{lemma}
\begin{proof} The Hilbert-Mumford weights are computed as follows. For any non-zero $\lambda \in \lie{g}_\mathbb{Z}$, let
$$\nu(x,\lambda) := \min_i \left\{ - \mu_i(\lambda), x_i \neq 0 \right\} .$$
Then
\begin{eqnarray*}
z^\lambda [ x_1,\ldots, x_k ] &=& [ z^{\mu_1(\lambda)} x_1,\ldots, z^{\mu_k(\lambda)}
x_k] \\ &=& [ z^{\mu_1(\lambda) + \nu(x,\lambda)} x_1,\ldots,
z^{\mu_k(\lambda) + \nu(x,\lambda)} x_k ] \end{eqnarray*}
and
$$ (-\mu_i(\lambda) \neq \nu(x,\lambda)) \ \implies \ \left(\lim_{z \to
0} z^{\mu_i(\lambda) + \nu(x,\lambda)} = 0 \right) .$$
Let
$$ x_\lambda := \lim_{z \to 0} z^{\lambda} x = \lim_{z \to 0} [z^{\mu_i(\lambda)} x_i ]_{i =1}^k \in X $$
Then
$$ x_\lambda = [x_{\lambda,1},\ldots, x_{\lambda,k}],\quad x_{\lambda,i} = \begin{cases} x_i & -\mu_i(\lambda) = \nu(x,\lambda)
\\ 0 & \text{otherwise} \end{cases} .$$
The Hilbert-Mumford weight is therefore
\begin{equation} \label{hm} \mu(x,\lambda) = \nu(x,\lambda) + (\theta,\lambda) .\end{equation}
By the Hilbert-Mumford criterion, the point $x$ is semistable if and only if
$$ \nu(x,\lambda) := \min \{ - \mu_i(\lambda) \ | \ x_i \neq 0\} \leq (- \theta,\lambda), \quad \forall \lambda \in \lie{g}_\mathbb{Z} - \{ 0 \} .$$
That is,
$$ (x \in X^{\on{ss}}) \iff ( \theta \in \operatorname{hull}
\{ \mu_i \ | \ x_i \neq 0\} ) .$$
This proves the claim about the semistable locus. To prove the claim about polystability, note that $\mu(x,\lambda) = 0 = \mu(x,-\lambda)$ implies that the minimum $\nu(x,\lambda)$ is also the maximum. Thus the only affine linear functions $\xi: \lie{g}^\vee \to \mathbb{R}$ which vanish at $\theta$ are those $\xi$ that are constant on the hull of $\mu_i$ with $x_i$ nonzero. This implies that the span of $\mu_i$ with $x_i$ non-zero contains $\theta$ in its relative interior. The stabilizer $G_x$ of $x$ has Lie algebra $\lie{g}_x$ the annihilator of the span of the hull of the $\mu_i$ with $x_i \neq 0$. So the stabilizer $G_x$ is finite if and only if the span of $\mu_i$ with $x_i \neq 0$ is of maximal dimension $\dim(G)$. This implies the claim on stability. \end{proof}
We introduce the following notation. As above $G$ is a connected complex reductive group with maximal torus $T$ and $\lie{g},\lie{t}$ are the Lie algebras of $G,T$ respectively. Fix an invariant inner product $( \ , \ ):\lie{g} \times \lie{g} \to \mathbb{C}$ on $\lie{g}$ inducing an identification $\lie{g} \to \lie{g}^\vee$. By taking a multiple of the basic inner product on each factor we may assume that the inner product induces an identification $\lie{t}_\mathbb{Q} \to \lie{t}_\mathbb{Q}^\vee$. Denote by
$$\Vert \cdot \Vert: \lie{q}_\mathbb{Q} \to \mathbb{R}_{\ge 0}, \quad \Vert \xi \Vert : = (\xi , \xi)^{1/2}$$
the norm with respect to the induced metric.
Next recall the theory of Levi decompositions of parabolic subgroups from Borel \cite[Section 11]{bo:lag}. A parabolic subgroup $Q$ of $G$ is one for which $G/Q$ is complete, or equivalently, containing a maximal solvable subgroup $B \subset G$. Any parabolic $Q$ admits a Levi decomposition $ Q = L(Q) U(Q) $ where $L(Q)$ denote a maximal reductive subgroup of $Q$ and $U(Q)$ is a maximal unipotent subgroup. Let $\lie{l}(Q),\lie{u}(Q)$ denote the Lie algebras of $L(Q), U(Q)$. Let $\lie{g} = \lie{t} \oplus \bigoplus_{ \alpha \in R(G)} \lie{g}_\alpha$ denote the root space decomposition of $\lie{g}$, where $R(G)$ is the set of roots. The Lie algebras $\lie{l}(Q),\lie{u}(Q)$ decompose into root spaces as
$$ \lie{q} = \lie{t} \oplus \bigoplus_{\alpha \in R(Q)} \lie{g}_\alpha, \quad \lie{l}(Q) = \lie{t} \oplus \bigoplus_{\alpha \in R(Q) \cap -R(Q)} \lie{g}_\alpha, \quad \lie{u}(Q) = \lie{q}/\lie{l}(Q) $$
where $R(Q) \subset R(G)$ is the set of roots for $\lie{l}(Q)$. Let $\lie{z}(Q)$ denote the center of $\lie{l}(Q)$ and
$$\lie{z}_+(Q) = \{ \xi \in \lie{z}(Q) \ | \ \alpha(\xi) \ge 0, \ \forall \alpha \in R(Q) \} $$
the {\em positive chamber} on which the roots of $Q$ are non-negative. The Levi decomposition induces a homomorphism
\begin{equation} \label{piq} \pi_Q: Q \to Q/U(Q) \cong L(Q) .\end{equation}
This homomorphism has the following alternative description as a limit. Let $\lambda \in \lie{z}_+(Q) \cap \lie{g}_\mathbb{Q}$ be a positive coweight and
$$ \phi_\lambda: \mathbb{C}^\times \to L(Q), \quad z \mapsto z^\lambda $$
the corresponding central one-parameter subgroup. Then
$$ \pi_Q(g) = \lim_{z \to 0} \on{Ad} (z^\lambda) g .$$
In the case of the general linear group in which the parabolic consists of block-upper-triangular matrices, this limit projects out the off-block-diagonal terms.
The unstable locus admits a stratification by maximally destabilizing subgroups, as in Hesselink \cite{hess:strat}, Kirwan \cite{ki:coh}, and Ness \cite{ne:st}. The stratification reads
\begin{equation} \label{kn} X = \bigcup_{\lambda \in \mathcal{C}(X)} X_\lambda, \quad X_\lambda = G \times_{Q_\lambda} Y_\lambda, \quad Y_\lambda \mapsto Z_\lambda \ \text{affine fibers} \end{equation}
where $Y_\lambda,Z_\lambda,Q_\lambda,\mathcal{C}(X)$ are defined as follows. For each fixed point component $\overline{Z}_\lambda$ of $z^{\lambda}$ there exist a weight $\mu(\lambda)$ so $z^{\lambda}$
acts on $\tilde{X} | Z_\lambda$ with weight $\mu(\lambda)$:
$$ z^\lambda \tilde{x} = z^{\mu(\lambda)} \tilde{x}, \quad \forall \tilde{x}
\in \tilde{X} | Z_\lambda .$$
The group $G_\lambda/\mathbb{C}^\times_\lambda$ acts on $\overline{Z}_\lambda$ and we denote by $Z_\lambda \subset \overline{Z}_\lambda$ the semistable locus. Define
\begin{equation} \label{CX} \mathcal{C}(X) = \{ \lambda \in \lie{t}_+ \ | \ \exists Z_\lambda, \ \mu(\lambda) = (\lambda, \lambda) \} \end{equation}
using the metric, where $\lie{t}_+$ is the closed positive Weyl chamber. The variety $Y_\lambda$ is the set of points that flow to $Z_\lambda$ under $z^{\lambda}, z \to 0$:
$$ Y_\lambda = \left\{ x \in X \ | \ \lim_{z \to 0} z^\lambda x \in Z_\lambda \right\} $$
The group $Q_\lambda$ is the parabolic of group elements that have a limit under $ \on{Ad} (z^{\lambda})$ as $z \to 0$:
$$ Q_\lambda = \left\{ g \in G \ | \ \exists \lim_{z \to 0} \on{Ad} (z^\lambda) g \in G \right\} .$$
Then $Y_\lambda$ is a $Q_\lambda$-variety; and $X_\lambda$ is the flow-out of $Y_\lambda$ under $G$. By taking quotients we obtain a stratification of the quotient stack by locally-closed substacks
$$ X / G = \bigcup_{\lambda \in \mathcal{C}(X)} X_\lambda / G .$$
This stratification was used in Teleman \cite{te:qu} to give a formula for the sheaf cohomology of bundles on the quotient stack.
\section{Kontsevich stability}
In this section we recall the definition of Kontsevich's moduli stacks of stable maps \cite{ko:lo} as generalized to orbifold quotients by Chen-Ruan \cite{cr:orb} and in the algebraic setting by Abramovich-Graber-Vistoli \cite{agv:gw}. Let $X$ be a smooth projective variety. Recall that a {\em prestable map} with target $X$ consists of a prestable curve $C \to S$, a morphism $u: C \to X$, and a collection $z_1,\ldots,z_n : S \to C$ of distinct non-singular points called {\em markings}. An automorphism of a prestable map $(C,u,\ul{z})$ is an automorphism
$$\varphi:C \to C, \quad \varphi \circ u = u, \quad \varphi(z_i) = z_i, \quad i = 1,\ldots, n .$$
A prestable map $(C,u,\ul{z})$ is {\em stable} if the number $\# \on{Aut} (C,u,\ul{z})$ of automorphisms is finite. For $d \in H_2(X,\mathbb{Z})$ we denote by $\overline{\M}_{g,n}(X,d)$ \label{mgn} the moduli stack of stable maps $(C,u,\ul{z})$ of genus $g = \operatorname{genus}(C)$ and class $d = v_*[C]$ with $n$ markings.
The notion of stable map generalizes to orbifolds \cite{cr:orb}, \cite{agv:gw} as follows. These definitions are needed for the construction of the moduli stack of affine gauged maps in the case that the git quotient is an orbifold, but not if the quotient is free. First we recall the notion of twisted curve:
\begin{definition} \label{twistedcurve} {\rm (Twisted curves)} Let $S$ be a scheme. An {\em $n$-marked twisted curve} over $S$ is a collection of data $(f: \mathcal{C} \to S, \{ {\mathcal z}_i \subset \mathcal{C} \}_{i=1}^n)$ such that
\begin{enumerate}
\item {\rm (Coarse moduli space)} $\mathcal{C}$ is a proper stack over $S$
whose geometric fibers are connected of dimension $1$, and such that
the coarse moduli space of $\mathcal{C}$ is a nodal curve $C$ over $S$.
\item {\rm (Markings)} The ${\mathcal z}_i \subset \mathcal{C}$ are closed
substacks that are gerbes over $S$, and whose images in $C$ are
contained in the smooth locus of the morphism $C \to S$.
\item {\rm (Automorphisms only at markings and nodes)} If $\mathcal{C}^{ns}
\subset \mathcal{C}$ denotes the {\em non-special locus} given as the
complement of the ${\mathcal z}_i$ and the singular locus of $\mathcal{C}
\to S$, then $\mathcal{C}^{ns} \to C$ is an open immersion.
\item {\rm (Local form at smooth points)} If $p \to C$ is a geometric
point mapping to a smooth point of $C$, then there exists an integer
$r$, equal to $1$ unless $p$ is in the image of some ${\mathcal
z}_i$, an \'etale neighborhood $\on{Spec}(R) \to C$ of $p$ and an
\'etale morphism $\on{Spec}(R) \to \on{Spec}_S(\mO_S[x])$ such that the
pull-back $\mathcal{C} \times_C \on{Spec}(R)$ is isomorphic to $ \on{Spec}(R[z]/z^r
= x )/\mu_r .$
\item {\rm (Local form at nodal points)} If $p \to C$ is a geometric
point mapping to a node of $C$, then there exists an integer $r$, an
\'etale neighborhood $\on{Spec}(R) \to C$ of $p$ and an \'etale morphism
$\on{Spec}(R) \to \on{Spec}_S(\mO_S[x,y]/(xy - t))$ for some $t \in \mO_S$
such that the pull-back $\mathcal{C} \times_C \on{Spec}(R)$ is isomorphic to $
\on{Spec}(R[z,w]/zw - t', z^r - x, w^r - y )/\mu_r $ for some $t' \in
\mO_S$. \end{enumerate} \end{definition}
Next we recall the notion of twisted stable maps. Let $\mathcal{X}$ be a proper Deligne-Mumford stack with projective coarse moduli space $X$. Algebraic definitions of twisted curve and twisted stable map to a $\mathcal{X}$ are given in Abramovich-Graber-Vistoli \cite{agv:gw}, Abramovich-Olsson-Vistoli \cite{aov:twisted}, and Olsson \cite{ol:logtwist}.
\begin{definition} A {\em twisted stable map} from an $n$-marked twisted curve $(\pi : \mathcal{C} \to S, ( {\mathcal z}_i \subset \mathcal{C} )_{i=1}^n )$ over $S$ to $\mathcal{X}$ is a representable morphism of $S$-stacks $ u: \mathcal{C} \to \mathcal{X} $ such that the induced morphism on coarse moduli spaces $ u_c: C \to X $ is a stable map in the sense of Kontsevich from the $n$-pointed curve $(C, \ul{z} = (z_1,\ldots, z_n ))$ to $X$, where $z_i$ is the image of ${\mathcal z}_i$. The {\em homology class} of a twisted stable curve is the homology class $u_* [ \mathcal{C}_s] \in H_2(X,\mathbb{Q})$ of any fiber $\mathcal{C}_s$. \end{definition}
\noindent Twisted stable maps naturally form a $2$-category. Every $2$-morphism is unique and invertible if it exists, and so this $2$-category is naturally equivalent to a $1$-category which forms a stack over schemes \cite{agv:gw}.
\begin{theorem} (\cite[4.2]{agv:gw}) The stack $\overline{\M}_{g,n}(\mathcal{X})$ of
twisted stable maps from $n$-pointed genus $g$ curves into $\mathcal{X}$ is
a Deligne-Mumford stack. If $\mathcal{X}$ is proper, then for any $c > 0$
the union of substacks $\overline{\M}_{g,n}(\mathcal{X},d)$ with homology class $d
\in H_2(\mathcal{X},\mathbb{Q})$ satisfying $(d, c_1(\tilde{X}))< c$ is proper. \end{theorem}
The Gromov-Witten invariants takes values in the cohomology of the {\em inertia stack}
$$ \cI_\mathcal{X} := \mathcal{X} \times_{\mathcal{X} \times \mathcal{X}} \mathcal{X} $$
where both maps are the diagonal. The objects of $\cI_\mathcal{X}$ may be identified with pairs $(x,g)$ where $x \in \mathcal{X}$ and $g \in \on{Aut} _\mathcal{X}(x)$. For example, if $\mathcal{X} = X/G$ is a global quotient by a finite group then
$$ \cI_\mathcal{X} = \bigcup_{[g] \in G/ \on{Ad} (G)} X^g/Z_g $$
where $G/ \on{Ad} (G)$ denotes the set of conjugacy classes in $X$ and $Z_g$ is the centralizer of $g$. Let $\mu_r = \mathbb{Z}/r\mathbb{Z}$ denote the group of $r$-th roots of unity. The inertia stack may also be written as a hom stack \cite[Section 3]{agv:gw}
$$ \cI_\mathcal{X} = \cup_{r > 0} \cI_{\mathcal{X},r}, \quad \cI_{\mathcal{X},r} := \on{Hom}^{\operatorname{rep}}(B\mu_r, \mathcal{X}) . $$
The classifying stack $B\mu_r$ is a Deligne-Mumford stack and if $\mathcal{X}$ is a Deligne-Mumford stack then
$$ \overline{\cI}_\mathcal{X} := \cup_{r > 0} \overline{\cI}_{\mathcal{X},r}, \quad \overline{\cI}_{\mathcal{X},r} := \cI_{\mathcal{X}/r}/ B\mu_r . $$
is the {\em rigidified inertia stack} of representable morphisms from $B \mu_r$ to $\mathcal{X}$, see \cite[Section 3]{agv:gw}. There is a canonical quotient cover $\pi: \cI_\mathcal{X} \to \overline{\cI}_\mathcal{X}$ which is $r$-fold over $\overline{\cI}_{\mathcal{X},r}$. Pullback acts on cohomology by an isomorphism
$$ \pi^* H^*(\overline{\cI}_\mathcal{X},\mathbb{Q}) \to H^*(\cI_\mathcal{X},\mathbb{Q}) .$$
For the purposes of defining orbifold Gromov-Witten invariants, $\overline{\cI}_\mathcal{X}$ can be replaced by $\cI_\mathcal{X}$ at the cost of additional factors of $r$ on the $r$-twisted sectors. If $\mathcal{X} = X/G$ is a global quotient of a scheme $X$ by a finite group $G$ then
$$ \overline{\cI}_{X/G} = \coprod_{(g)} X^{g}/ (Z_g/ \langle g \rangle) $$
where $\langle g \rangle \subset Z_g$ is the cyclic subgroup generated by $g$.
For example, suppose that $X$ is a polarized linearized projective $G$-variety such that $X \qu G$ is locally free. Then
$$ \cI_{X \qu G} = \coprod_{(g)} X^{\on{ss},g}/ Z_g $$
where $X^{\on{ss},g}$ is the fixed point set of $g \in G$ on $X^{\on{ss}}$, $Z_g$ is its centralizer, and the union is over all conjugacy classes,
$$ \overline{\cI}_{X \qu G} = \coprod_{(g)} X^{\on{ss},g}/ (Z_g/ \langle g \rangle) $$
where $\langle g \rangle$ is the (finite) group generated by $g$. The moduli stack of twisted stable maps admits evaluation maps to the rigidified inertia stack
$$ \ev: \overline{\M}_{g,n}(\mathcal{X}) \to \overline{\cI}_\mathcal{X}^n, \quad \overline{\ev}: \overline{\M}_{g,n}(\mathcal{X}) \to \overline{\cI}_\mathcal{X}^n,
$$
where the second is obtained by composing with the involution $\overline{\cI}_{\mathcal{X}} \to \overline{\cI}_{\mathcal{X}} $ induced by the map $\mu_r \to \mu_r, \zeta \mapsto \zeta^{-1}$.
Constructions of Behrend-Fantechi \cite{bf:in} provide the stack of stable maps with virtual fundamental classes. The virtual fundamental classes
$$[\overline{\M}_{g,n,\Gamma}(\mathcal{X},d)] \in H(\overline{\M}_{g,n}(\mathcal{X}),\mathbb{Q}) $$
(where the right-hand-side denotes the singular homology of the coarse
moduli space) satisfy the splitting axioms for morphisms of modular
graphs similar to those in the case that $X$ is a variety. Orbifold
Gromov-Witten invariants are defined by virtual integration of
pull-back classes using the evaluation maps above. The orbifold
Gromov-Witten invariants satisfy properties similar to those for
usual Gromov-Witten invariants, after replacing rescaling the inner
product on the cohomology of the inertia stack by the order of the
stabilizer. The definition of orbifold Gromov-Witten invariants
leads to the definition of orbifold quantum cohomology as follows.
\begin{definition} {\rm (Orbifold quantum cohomology)}
To each component $\mathcal{X}_k$ of ${\cI}_\mathcal{X}$ is assigned a rational
number $\age(\mathcal{X}_k)$ as follows. Let $(x,g)$ be an object in
$\mathcal{X}_k$. The element $g$ acts on $T_x \mathcal{X}$ with eigenvalues
$\alpha_1, \ldots,\alpha_n$ with $ n = \dim(\mathcal{X})$. Let $r$ be the
order of $g$ and define $s_j \in \{ 0,\ldots, r - 1 \}$ by $\alpha_j
= \exp( 2\pi i s_j / r)$. The {\em age} is defined by $ \age(\mathcal{X}_k)
= (1/r) \sum_{j=1}^n s_j .$ Let $ \Lambda_\mathcal{X} \subset
\on{Hom}(H_2(\mathcal{X},\mathbb{Q}),\mathbb{Q}) $ denote the Novikov field of linear
combinations of formal symbols $q^d, d \in H_2(\mathcal{X},\mathbb{Q})$ where for
each $c> 0$, only finitely many $q^d$ with $(d,c_1(\tilde{X})) < c$
have non-zero coefficient. Denote the quantum cohomology
$$ QH(\mathcal{X}) = \bigoplus QH^\bullet(\mathcal{X}), \quad QH^\bullet(\mathcal{X}) = \bigoplus_{\mathcal{X}_k \subset \cI_\mathcal{X}} H^{\bullet + 2 \age(\mathcal{X}_k)}(\mathcal{X}_k) \otimes \Lambda_\mathcal{X} .$$
\end{definition}
\noindent The genus zero Gromov-Witten invariants define on $QH(\mathcal{X})$ the structure of a Frobenius manifold \cite{cr:orb}, \cite{agv:gw}.
\section{Mundet stability}
In this section we explain the Ramanathan condition for semistability of principal bundles \cite{ra:th} and its generalization to maps to quotients stacks by Mundet \cite{mund:corr}, and the quot-scheme and stable-map compactification of the moduli stacks.
\subsection{Ramanathan stability}
Morphisms from a curve to a quotient of a point by a reductive group are by definition principal bundles over the curve. Bundles have a natural semistability condition introduced half a century ago by Mumford, Narasimhan-Seshadri, Ramanathan and others in terms of {\em
parabolic reductions} \cite{ra:th}. First we explain stability for vector bundles. A vector bundle $E \to C$ of degree zero over a smooth projective curve $C$ is semistable if there are no sub-bundles of positive degree:
$$ (E \ \text{semistable} ) \quad \text{iff} \quad ( \deg(F) \leq 0, \quad \forall F \subset E \ \text{sub-bundles}) .$$
A generalization of the notion of semistability to principal bundles is given by Ramanathan \cite{ra:th} in terms of {\em parabolic
reductions}. A parabolic reduction of $P$ consists of a pair
$$Q \subset G, \quad \sigma : C \to P/Q$$
of a parabolic subgroup of $G$, that is and a section $\sigma: C \to P/Q$. Denote by $\sigma^* P \subset P$ the pull-back of the $Q$-bundle $P \to P/Q$, that is, the reduction of structure group of $P$ to $Q$ corresponding to $\sigma$. Associated to the homomorphism $\pi_Q$ of \eqref{piq} is an {\em associated graded} bundle $\Gr(P) := \sigma^*P \times_Q L(Q) \to C$ with structure group $L(Q)$. In the case that $P$ is the frame bundle of a vector bundle $E \to C$ of rank $r$, that is,
$$ P = \cup_z P_z, \quad P_z = \{ (e_1,\ldots, e_r) \in E_z^r \ | \ e_1 \wedge \ldots \wedge e_r \neq 0 \} $$
a parabolic reduction of $P$ is equivalent to a flag of sub-vector-bundles of $E$
$$ \{0 \} \subset E_{i_1} \subset E_{i_2} \subset \ldots \subset E_{i_l} \subset E. $$
Explicitly the parabolic reduction $\sigma^* P$ given by frames adapted to the flag:
$$ \sigma(z) = \{ (e_1,\ldots,e_r) \in E_z^r \ | \ e_j \in E_{i_k,z}, \ \forall j \leq i_k, k = 1,\ldots, l \} .$$
Conversely, given a parabolic reduction the associated vector bundle has a canonical filtration.
An analog of the degree of a sub-bundle for parabolic reductions is the degree of a line bundle defined as follows. Given $\lambda \in \lie{g}_\mathbb{Z} - \{ 0 \}$ we obtain from the identification $\lie{g} \to \lie{g}^\vee$ a rational weight $\lambda^\vee$. Denote the corresponding characters
$ \chi_\lambda: L(Q) \to \mathbb{C}^\times$ and $ \chi_\lambda\circ \pi_Q: Q \to \mathbb{C}^\times .$
Consider the associated line bundle over $C$ defined by
$P(\mathbb{C}_{\lambda^\vee}) := \sigma^* P \times_Q \mathbb{C}_{\lambda^\vee} .$
The {\em Ramanathan weight} \cite{ra:th} is the degree of the line bundle $P(\mathbb{C}_{\lambda^\vee}) $, that is,
$$ \mu_{BG}(\sigma,\lambda) := ([C], (c_1(P(\mathbb{C}_{\lambda^\vee})) \in \mathbb{Z} .$$
The bundle $P \to C$ is {\em Ramanathan semistable} if
$$ \mu_{BG}(\sigma,\lambda) \leq 0 , \quad \forall (\sigma,\lambda) .$$
As in the case of vector bundle, it suffices to check semistability for all reductions to {\em maximal parabolic} subgroups. In fact, any dominant weight may be used in the definition of $\mu_{BG}(\sigma,\lambda)$, which shows that Ramanathan semistability is independent of the choice of invariant inner product on the Lie algebra and one obtains the definition given in Ramanathan \cite{ra:th}.
\subsection{Mundet semistability}
The Mundet semistability condition generalizes Ramanathan's condition to morphisms from a curve to the quotient stack \cite{mund:corr}, \cite{schmitt:univ}. Let
$$(p: P \to C, u: C \to P(X)) \in \operatorname{Obj}( \on{Hom}(C,X/G)) $$
be a gauged map. Let $(\sigma,\lambda)$ consist of a parabolic reduction $\sigma: C \to P/Q$ and a positive coweight $\lambda \in \lie{z}_+(Q)$. Consider the family of bundles $ P^\lambda \to S := \mathbb{C}^\times $ obtained by conjugating by $z^\lambda$. That is, if $P$ is given as a cocycle in nonabelian cohomology with respect to a covering $\{ U_i \to X \}$
$$ [P] = [\psi_{ij} : (U_i \cap U_j) \to G] \ \in \ H^1(C,G) $$
then the twisted bundle is given by
$$ [P^\lambda] = [ z^{\lambda} \psi_{ij} z^{-\lambda}: (U_i \cap U_j) \to G ] \ \in \ H^1(C \times S,G) .$$
Define a family of sections
$$ u^\lambda: S \times C \to P^\lambda(X) $$
by multiplying $u$ by $z^\lambda, z \in \mathbb{C}^\times$. This family has an extension over $s = \infty$ called the {\em associated graded} bundle and stable section
\begin{equation} \label{assocgrad} \Gr(P) \to C, \quad \Gr(u): \hat{C} \to \Gr(P)(X) \end{equation}
whose bundle $\Gr(P)$ agrees with the definition of associated graded above. Note that the associated graded section $\Gr(u)$ exists by compactness of the moduli space of stable maps to $\Gr(P)(X)$. The composition of $\Gr(u)$ with projection $\Gr(P)(X)\to C$ is a map of degree one; hence there is a unique component $\hat{C}_0$ of $\hat{C}$ that maps isomorphically onto $C$. The construction above is $\mathbb{C}^\times$-equivariant and in particular over the central fiber $z = 0$ the group element $z^\lambda$ acts by an automorphism of $\Gr(P)$ fixing $\Gr(u)$ up to automorphism of the domain.
For each pair of a parabolic reduction and one-parameter subgroup as above, the Mundet weight is a sum of {\em Ramanathan} and {\em
Hilbert-Mumford} weights. To define the Mundet weight, consider the action of the automorphism $z^\lambda$ on the associated graded $\Gr(P)$. The automorphism of $X$ by $z^\lambda$ is $L(Q)$-invariant and so defines an automorphism of the associated line bundle $\Gr(u)^* P(\tilde{X}) \to \Gr(C)$. The weight of the action of $z^\lambda$ on the fiber of $\Gr(u)^* P(\tilde{X})$ over the root component $\hat{C}_0$ is the {\em Hilbert-Mumford weight}
$$ \mu_X (\sigma,\lambda) \in \mathbb{Z}, \quad z^\lambda \tilde{x} = z^{\mu_X(\sigma,\lambda)} \tilde{x}, \quad \forall \tilde{x} \in
(\Gr(u)|_{\hat{C}_0})^* \Gr(P) \times_G \tilde{X} .$$
\begin{definition} {\rm (Mundet stability)} Let $(P,u)$ be a gauged map from a smooth projective curve $C$ to the quotient stack $X/G$. The {\em Mundet weight} of a parabolic reduction $\sigma$ and dominant coweight $\lambda$ is
$$ \mu(\sigma,\lambda) = \mu_{BG}(\sigma,\lambda) + \mu_X(\sigma,\lambda) \in \mathbb{Z} .$$
The gauged map $(P,u)$ is Mundet {\em semistable} resp. {\em stable} if and only if
$$ \mu(\sigma,\lambda) \leq 0, \ \text {resp.} \ < 0, \quad \forall
(\sigma,\lambda) .$$
A pair $(\sigma,\lambda)$ such that $ \mu(\sigma,\lambda ) \ge 0 $ is a {\em destabilizing pair}. A pair $(P,u)$ is {\em polystable} iff
\begin{equation} \label{polystable} \mu(\sigma,\lambda) = 0 \iff \mu(\sigma,-\lambda) = 0, \quad \forall (\sigma,\lambda) .\end{equation}
That is, a pair $(P,u)$ is polystable if for any destabilizing pair the opposite pair is also destabilizing. \end{definition}
More conceptually the semistability condition above is the Hilbert-Mumford stability condition adapted to one-parameter subgroups of the complexified gauge group, as explained in \cite{mund:corr}. Semistability is independent of the choice of invariant inner product as follows for example from the presentation of the semistable locus in Schmitt \cite[Section 2.3]{schmitt:git}.
We introduce notation for various moduli stacks. Let $\M^G(C,X)$ denote the moduli space of Mundet semistable pairs; in general, $\M^G(C,X)$ is an Artin stack as follows from the git construction given in Schmitt \cite{schmitt:univ,schmitt:git} or the more general construction of hom stacks in Lieblich \cite[2.3.4]{lieblich:rem}. For any $d \in H_2^G(X,\mathbb{Z})$, denote by $\M^G(C,X,d)$ the moduli stack of pairs $v = (P,u)$ with
$$v_* [C] := (\phi \times_G \operatorname{id}_X)_* u_* [C] = d \in H_2^G(X,\mathbb{Z}) $$
where $\phi:P \to EG$ is the classifying map.
\subsection{Compactification}
Schmitt \cite{schmitt:univ} constructs a Grothendieck-style compactification \label{quots} of the moduli space of Mundet-semistable obtained as follows. Suppose $X$ is projectively embedded in a projectivization of a representation $V$, that is $ X \subset \mathbb{P}(V)$. Any section $u: C \to P(X)$ gives rise to a line sub-bundle
$ L := u^* (\mO_{\mathbb{P}(V)} (-1) \to \mathbb{P}(V))$
of the associated vector bundle $P \times_G V$. From the inclusion $\iota:L \to P(V)$ we obtain by dualizing a surjective map
$$ j: P(V^\vee) := P \times_G V^\vee \to L^\vee .$$
A {\em bundle with generalized map} in the sense of Schmitt \cite{schmitt:git} is a pair $(P,j)$ as above where $j$ is allowed to have base points in the sense that
$$\zeta \in C \ \text{basepoint} \ \iff ( (\rank(j_\zeta): P(V)_\zeta^\vee \to L_\zeta^\vee) = 0) .$$
Schmitt \cite{schmitt:git} shows that the Mundet semistability condition extends naturally to the moduli stack of bundles with generalized map. Furthermore, the compactified moduli space $\overline{\M}^{\on{quot},G}(C,X,d)$ is projective, in particular proper.
Schmitt's construction of the moduli space of bundles with generalized maps uses geometric invariant theory. After twisting by a sufficiently positive bundle we may assume that $P(V^\vee)$ is generated by global sections. A collection of sections $s_1,\ldots, s_l$ generating $P(V^\vee)$ is called an {\em $l$-level structure}. Equivalently, an $l$-level structure is a surjective morphism $ q: \mO_C^{\oplus l} \to P(V^\vee) .$ Denote by $\M^{G,{\on{lev}}}(C,\mathbb{P}(V))$ \label{level} the stack of gauged maps to $\mathbb{P}(V)$ with level structure. The group $GL(l)$ acts on the stack of $l$-level structures, with quotient
\begin{equation} \label{schmittgit} \M^{G,{\on{lev}}}(C,\mathbb{P}(V)) / GL(l) = \M^G(C,\mathbb{P}(V)) .\end{equation}
Denote by $\M^{G,{\on{lev}}}(C,X) \subset \M^{G,{\on{lev}}}(C,\mathbb{P}(V))$ the substack whose sections $u: C \to \mathbb{P}(V)$ have image in $P(X) \subset P(\mathbb{P}(V))$. Then by restriction we obtain a quotient presentation
$$ \M^{G,{\on{lev}}}(C,X) / GL(l) = \M^G(C,X) .$$
Allowing the associated quotient $P \times_G V^\vee \to P \times_G L^\vee$ to develop base points gives a compactified moduli stack of gauged maps with level structure $\overline{\M}^{G,\on{quot},{\on{lev}}}(C,X)$. Schmitt \cite{schmitt:univ, schmitt:git} shows that the stack $\overline{\M}^{G,\on{quot},{\on{lev}}}(C,X)$ has a canonical linearization and the git quotient $\overline{\M}^{G,\on{quot},{\on{lev}}}(C,X) \qu GL(l)$ defines a compactification $\overline{\M}^{G,\on{quot}}(C,X)$ of $\M^G(C,X)$ independent of the choice of $l$ as long as $l$ is sufficiently large. A version of the quot-scheme compactification with markings is obtained by adding tuples of points to the data. That is,
$$ \overline{\M}^{G,\on{quot}}_n(C,X) := \overline{\M}^{G,\on{quot}}(C,X) \times \overline{\M}_n(C) $$
where we recall that $\overline{\M}_n(C)$ is the moduli stack of stable maps $p: \hat{C} \to C$ of class $[C]$ with $n$ markings and genus that of $C$. The orbit-equivalence relation in can be described more naturally in terms of {\em $S$-equivalence}: Given a family $(P_S,u_S)$ of semistable gauged maps over a scheme $S$, such that the generic fiber is isomorphic to some fixed $(P,u)$, then we declare $(P,u)$ to be $S$-equivalent to $(P_s,u_s)$ for any $s \in S$. Any equivalence class of semistable gauged maps has a unique representative that is polystable, by the git construction in Schmitt \cite[Remark 2.3.5.18]{schmitt:univ}. From the construction evaluation at the markings defines maps to the quotient stack
$$ \overline{\M}_n^{G,\on{quot}}(C,X,d) \to (V/\mathbb{C}^\times)^n, \quad ((p \circ z_i)^*L, j \circ p \circ z_i) $$
rather than to the git quotient $X^n \subset \mathbb{P}(V)^n$.\footnote{The
Ciocan-Fontanine-Kim-Maulik \cite{cf:st} moduli space of {\em stable
quotients} remedies this defect by imposing a stability condition
at the marked points $z_1,\ldots, z_n \in C$. The moduli stack then
admits a morphism to $\overline{\cI}_{X \qu G}^n$ by evaluation at the
markings.}
\begin{example}\label{toric example} {\rm (Mundet semistable maps in the toric case)} If $G$ is a torus and $X = \mathbb{P}(V)$ then we can given an explicit description of Schmitt's quot-scheme compactification $\overline{\M}^{G,\on{quot}}(C,X,d) $ of Mundet semistable maps \cite{schmitt:univ}.
Specifically let $X = \mathbb{P}(V)$ where $V$ is a $k$-dimensional vector space and
\begin{equation} \label{decompose} V = \bigoplus_{i=1}^k V_i \end{equation}
is the decomposition of $V$ into weight spaces $V_i$ with weight $\mu_i \in \lie{g}_\mathbb{Z}^\vee$.
A point of $\M^{G}(C,X,d)$ is a pair $(P,u)$: \[ P\to C \ \ \ \ \ \ u\colon C \to P(X), \] where $P$ is a $G$-bundle and $u$ is a section. We consider $u$ as a morphism $\widetilde{u} \colon L\to P(V)$ with $L\to C$ a line bundle. Via the decomposition of $V$, we can write $\widetilde{u}$ as a $k$-tuple: \[ (\widetilde{u}_1, \dotsc, \widetilde{u}_k) \in \bigoplus_{i=1}^k H^0(P(V_i)\otimes L^\vee). \] The compactification $\overline{\M}^{G,\on{quot}}(C,X,d)$ is obtained by allowing the $\widetilde{u}_i$ to have simultaneous zeros: \[ \widetilde{u}_1^{-1}(0) \cap \dotsb \cap \widetilde{u}_k^{-1}(0) \neq \emptyset \] We make use of this example later one so we collect a few results about $\overline{\M}^{G,\on{quot}}(C,X,d)$ below.
Recall \eqref{eqiv c_1} there is a projection $H^2_G(X)\to H^2(B G)=\lie{g}_\mathbb{Z}^\vee $ and similarly we have $H_2^G(X)\to H_2(B G) = \lie{g}_\mathbb{Z}$. Associated to $v=(P,u)$ is the discrete data:
\begin{itemize} \item[] $v_*[C]=d \in H_2^G(X,\mathbb{Z})$ and its image $d(P) \in H_2(BG)$ \item[] $c_1^G(\widetilde{X}) \in H^2_G(X)$ and its image $\theta \in H^2(B G)$ \item[] $d(u):= -c_1(L) \in H^2(C,\mathbb{Z}) \cong \mathbb{Z}$. \end{itemize}
Note $d(P)$ is the degree of $P$; that is, $d(P) = c_1(P) \in H^2(C,\lie{g}_\mathbb{Z})\cong \lie{g}_\mathbb{Z}$. We can now state the following.
\begin{lemma}\label{torus action on P(V)} Let $G$ be a
torus acting on a vector space $V$. Let $V = \bigoplus_{i=1}^kV_i$
be its decomposition into weight spaces with weights $\mu_1, \dotsc,
\mu_k$.
\begin{enumerate} \item \label{one} The Mundet semistable locus consists of pairs $(P,u)$ such
that
\begin{equation} \label{oneeq} \operatorname{hull} ( \{
- d(P)^\vee + \mu_i | \tilde{u}_i \neq 0 \}) \ni
\theta. \end{equation}
\item \label{two} let $W = \bigoplus_{i=1}^k H^0(P(V_i)\otimes L^\vee)$ and let $W^{ss}$ consist of $(\widetilde{u}_1, \dotsc, \widetilde{u}_k)$ such that \eqref{oneeq} holds. Then
$ \overline{\M}^{G,\on{quot}}(C,X,d) \cong W^{ss}/G. $
\item \label{three} If $\widetilde{u}_i \neq 0$ then $
(\mu_i,d(P)^\vee)+d(u) \geq 0$. If moreover \eqref{oneeq} holds
then
$$ (\theta-d(P)^\vee,d(P)) + d(u) \geq 0. $$
\item \label{four} $(v_*[C],c_1(P(\widetilde{X})) =
(\theta,d(P))+d(u)$. \end{enumerate} \end{lemma}
\begin{proof} Since $G$ is abelian, $\Gr(P) = P$ for any
pair $(\lambda,\sigma)$. It follows that for any $\lambda \in
\lie{g}_\mathbb{Q}$, the Mundet weight is
$$ \mu(\sigma,\lambda) := \{ \min_i (d(P)^\vee,\lambda) -
\mu_i(\lambda) + \theta(\lambda), \tilde{u}_i \neq 0\} .$$
Hence the semistable locus is the space of pairs $(P,u)$ where
$$ \operatorname{hull} ( \{ - d(P)^\vee + \mu_i | \tilde{u}_i \neq 0 \}) \ni \theta .$$
This proves \eqref{one}. The description \eqref{two} follows immediately. For \eqref{three}, if $\widetilde{u}_i \neq 0$ then $\deg \operatorname{div}(\widetilde{u}_i) \geq 0$. But we also have
\begin{equation}\label{threeeq}
(\mu_i,d(P)^\vee)+d(u) = c_1(P(V_i)\otimes L^\vee) =\deg \operatorname{div}(\widetilde{u}_i) \geq 0 . \end{equation}
In particular $- d(P)^\vee + \theta \in \operatorname{hull} ( \{ \mu_i | \tilde{u}_i \neq 0 \})$. Together with \eqref{threeeq} this shows $(\theta+d(P)^\vee,d(P)) \geq 0$.
To prove \eqref{four} we use that the sections $\widetilde{u}_i$ above are homotopic to the zero section
$$ \tilde{u}_0: C \to P(V)\otimes L^\vee, \quad z \mapsto (z,0) $$
and $\tilde{X}$ is induced from an equivariant line bundle on $V$ with character $\theta$ at the fixed point at zero. Therefore we have
\begin{equation} \label{dform}
(v_*[C],c_1(P(\tilde{X}))) = (u_{0,*}[C], c_1(P(\mathbb{C}_\theta)\otimes L^\vee)) =
(\theta,d(P)) + d(u) .\end{equation}
\end{proof}
For an explicit example, if $G = \mathbb{C}^\times $ and $V = \mathbb{C}^k$ then
$$\deg( P(V_i)\otimes L^\vee) = \deg(P(V_i)) - \deg(L) = d(P) + d(u), \quad i = 1,\ldots, k .$$
It follows that the moduli stack admits an isomorphism
$$ \overline{\M}^{G,\on{quot}}(C,X,d) \cong \mathbb{C}^{k(d(P) + d(u) + 1),\times} / \mathbb{C}^\times \cong \mathbb{P}^{k(d(P) + d(u) + 1) - 1}.$$
This moduli stack is substantially simpler in topology than the moduli space of stable maps to $C × X/G$, despite the dramatically more complicated stability condition. This ends the example. \end{example}
A {\em Kontsevich-style compactification} of the stack of Mundet-semistable gauged maps which admits evaluation maps as well as a Behrend-Fantechi virtual fundamental class \cite{cross} is defined as follows. The objects in this compactification allow {\em stable
sections}, that is, stable maps $u : \hat{C} \to P(X) $ whose composition with $P(X) \to C$ has class $[C]$. Thus objects of $\overline{\M}^G_n(C,X)$ \label{mss} are triples $(P, \hat{C}, u,\ul{z})$ consisting of a $G$-bundle $P \to C$, a projective nodal curve $(\hat{C},\ul{z})$, and a stable map $u: \hat{C} \to P \times_G X$ whose class projects to $[C] \in H_2(C,\mathbb{Z})$. Morphisms are the obvious diagrams. To see that this category forms an Artin stack, note that the moduli stack of bundles $ \on{Hom}(C,BG)$ has a universal bundle
$$U \to C \times \on{Hom}(C,BG) .$$
Consider the associated $X$-bundle
$$U \times_G X \to C \times \on{Hom}(C,BG) .$$
The stack $\overline{\M}_n^G(C,X)$ is a substack of the stack of stable maps to $U \times_G X$, and is an Artin stack by e.g. Lieblich \cite[2.3.4]{lieblich:rem}, see \cite{qk2} for more details. Note that hom-stacks are not in general algebraic \cite{bhatt}.
Properness of the Kontsevich-style compactification follows from a combination of Schmitt's construction and the Givental map. A proper relative Givental map is described in Popa-Roth \cite{po:stable}, and in this case gives a morphism
\begin{equation} \label{givmor} \overline{\M}^G(C,X,d)
\to\overline{\M}^{G,\on{quot}}(C,X,d).\end{equation}
For each fixed bundle this map collapses bubbles of the section $u$ and replaces them with base points with multiplicity given by the degree of the bubble tree. Since the Givental morphism \eqref{givmor}, the forgetful morphism $\overline{\M}^G_n(C,X,d) \to \overline{\M}^G(C,X,d)$ and the quot-scheme compactification $\overline{\M}_n^{G,\on{quot}}(C,X,d)$ are all proper, so is $\overline{\M}^G_n(C,X,d)$.
\subsection{Energy positivity}
A natural notion of {\em energy} of a gauged map is defined as follows. For a gauged map $v = (P,u)$ the energy is given by the pairing with the equivariant first Chern class of the linearization
$$ \cE(v): = (d, c_1(P(\tilde{X}))) \in \mathbb{Z},\quad d = v_* [C] \in H_2^G(X,\mathbb{Z}) .$$
From Mundet's correspondence \cite{mund:corr} it is immediate that the energy is non-negative, since in the symplectic definition the energy is defined as an integral of a non-negative function (the {\em energy
density}) over the domain curve. Here we give an algebraic proof:
\begin{lemma} \label{positivity} For any Mundet-semistable gauged map $v = (P,u)$ from a smooth projective genus zero curve $C$ with class $d = u_* [C] \in H_2^G(X,\mathbb{Z})$, the pairing $\cE(v) = (d,
c_1(P(\tilde{X}))) \in \mathbb{Z}$ is non-negative. The energy $\cE(v)$
vanishes only if the bundle $P$ is trivializable and $u$ constant in
some trivialization of $P(X)$ induced by a trivialization of $P$. \end{lemma}
\begin{proof} We give two proofs. By a special case of the Drinfeld-Simpson theorem \cite{ds:red}, \cite[Lemma 3.2.7]{cf:st}, $P$ admits a reduction to a Borel subgroup $B \subset G$. Let $\pi_B: B \to T$ be the projection \eqref{piq}, and $\Gr(P)$ the associated graded. Since the map $\pi_Q$ is $T$-equivariant, the section $u$ induces a section $\Gr(u): \mathcal{C} \to \Gr(P)(X)$ that is also $T$-semistable. Therefore it suffices to consider the case $G = T$. Let $k = \dim(V)$ and $ V = \bigoplus_{i=1}^k V_i $ the decomposition of $V$ into weight spaces $V_i$ with weight $\mu_i$.
We use the notation introduced in example \ref{toric example}. In particular the first Chern class $c_1^G(\tilde{X})$ becomes identified, up to positive scalar multiple with a pair
$$c_1^G(\tilde{X}) \mapsto (1,\theta) \in \mathbb{Z} \oplus \lie{g}_\mathbb{Z}^\vee .$$
The Mundet semistability criterion for one-parameter subgroups of $ \on{Aut} (P) \cong T$ has Mundet weights equal to
$$ \mu_{BG}(\sigma,\lambda) = (d(P)^\vee,\lambda), \quad \mu_M(\sigma,\lambda) = (d(P)^\vee,\lambda) + \mu_X(\sigma,\lambda) .$$
By lemma \ref{torus action on P(V)} \eqref{three} we have \[
(\theta-d(P)^\vee,d(P)) + d(u) \geq 0 . \] This implies
\begin{eqnarray}\label{dPP} (v_*[C],c_1(P(\tilde{X}))) &=& (\theta,d(P)) + d(u) \\ &=& (\theta - d(P)^\vee, d(P)) + d(u) + (d(P)^\vee, d(P)) \\ &\ge&
(\theta - d(P)^\vee, d(P)) + d(u) \ge 0 \end{eqnarray}
as claimed. If $(d(P),{\on{d}}(P)^\vee)$ is zero then we must have $d(P) = 0$, hence $P$ is trivializable. Hence
$$P(\tilde{X}) = C \times \tilde{X}, \quad (\pi \circ u)_* [C] = 0 \in H_2(X)$$
where $\pi: P(X) \cong C \times X \to X$ is the projection on the second factor. This implies that $u$ is constant.
In the second proof we evaluate the Mundet weight for a carefully chosen one-parameter subgroup. As before assume $G = T=(\mathbb{C}^\times)^r$ and $X = \mathbb{P}(V)$. Consider $v \colon C \to X/G$ as a pair $(P\to C, P\xrightarrow{\alpha} \mathbb{P}(V))$ with $\alpha$ a $T$-equivariant map. The energy of $v$ is the pairing
$$ (v_*[C], c_1^G(\tilde{X})), \quad \text{where} \ v_*[C] \in \mathbb{Z}\oplus \mathfrak{t}_\mathbb{Z}, \quad c_1^G(\tilde{X}) \in \mathbb{Z} \oplus \mathfrak{t}_\mathbb{Z}^\vee .$$
The latter is up to positive scalar equal to $(1, \theta)$; the former is equal to $(\lambda, d(P))$ for an appropriate integer $\lambda$. The energy is equivalently
$$\cE(v)= v^*c_1^G(\tilde{X}) \in H^2(C) = \mathbb{Z} .$$
Hence $\lambda = \deg(\alpha)$. The following is readily verified. If $d(P) = (d_1, \dotsc, d_r)$ then $P$ is the frame bundle of the vector bundle $E$ defined by
$$E:=\oplus_{i = 1}^r \mO_C(d_i) .$$
The map $\alpha$ is given by global sections
$u_0, \dotsc, u_m \in H^0(\alpha^*\mO_{\mathbb{P}(V)}(1))$
which are weight vectors for $G$. Consider the weight space decomposition $V = \oplus_i V_i$ where $G$ acts on $V_i$ with weight $\mu_i$. Equivariance implies that $-\mu_i$ is the weight of $u_i$. We claim $\deg(\alpha)\geq (-\mu_i,d(P))$. To see this let
$$ |E| = \underline{\on{Spec}}(\operatorname{\Sym}^*(E^\vee))$$
be the total space of $E$. Via the clutching construction $T$ is given by gluing of trivializations in coordinate charts near $[1,0],[0,1] \in \mathbb{P}^1$,
$$ |E| = \on{Spec} \mathbb{C}[z,x_1, \dotsc, x_r] \cup \on{Spec} \mathbb{C}[z^{-1},y_1, \dotsc, y_r]$$
with $y_i = z^{-d_i}x_i$. The space of global sections $H^0(\alpha^*\mO_{\mathbb{P}(V)}(1))$ has a basis of pairs $(z^j\prod_ix_i^{n_i},z^{-k}\prod_i y_i^{n_i})$ that transform as follows \[ z^j \prod_i x_i^{n_i} = z^j \prod_i z^{n_i d_i} y_i^{n_i} = z^{j+(-\mu_i,d(P)) - \deg(\alpha)) }\prod_i y_i^{n_i} \] That is
$-k = j+(-\mu_i,d(P)) - \deg(\alpha) \leq 0. $
As $j\geq 0$ we conclude
$(-\mu_i,d(P))\leq \deg(\alpha) .$
Mundet stability for the one-parameter subgroup generated by $\lambda$ is defined by a limiting equivariant map $\alpha^0 \colon P\to \mathbb{P}(V)$ given by sections $u_i^0$ whose image is fixed by $z^\lambda$. The stability condition is
\[ \min_{i, u_i^0\neq 0} (d(P)^\vee,\lambda)+(\theta-\mu_i, \lambda)\leq 0. \]
Substitute in $\lambda = - d(P)^\vee$ and multiply by $-1$ to obtain
$$(d(P),d(P)^\vee) + (\theta,d(P)^\vee)+(-\mu_i,d(P)^\vee)\geq 0 .$$
Therefore
$$ \cE(v)= (\theta,d(P)^\vee)+\deg(\alpha)\geq (\theta,d(P)^\vee)+(-\mu_i,d(P)^\vee)\geq 0 .$$
For equality to hold we need $d(P) = 0$ and $\deg(\alpha)=0$. The first condition together with equivariance says $\alpha$ factors through $C$; the second condition says $\alpha$ is constant. \end{proof}
\begin{remark} \label{alten} The following gives an alternative proof of non-negativity of the energy in the case that any Mundet semistable map $(P,u)$ has non-empty semistable locus $u^{-1}(P(X^{\on{ss}}))$, see Corollary \ref{large} below. In this case an invariant ample divisor $D \subset X$ is given by choosing an invariant section of the ample bundle $\tilde{X}^k$ for $k$ large, as in \eqref{invsection}. Let
$$D \qu G : = (X^{\on{ss}} \cap D)/G \subset X \qu G $$
denote the associated divisor in the git quotient. We may assume that the divisor $D \qu G \subset X \qu G \subset X/G$ does not contain $v(C)$, since $D \qu G$ is ample. Since the divisor $D$ is $G$-invariant and ample, $D$ contains the unstable locus, that is, $ D \supset (X - X^{\on{ss}}) .$ The divisor $D$ then induces a divisor
$$P(D) = P \times_G D \subset P(X) .$$
Let $u^{X \qu G}: C \to X \qu G$ denote the induced map to the symplectic quotient. Since $u(C)$ is not contained in $P(D)$, the pairing is the number of intersection points counted with multiplicity:
$$ (v_*[C], c_1(P \times_G \tilde{X})) = \# u^{-1} (P(D)) .$$
If the pairing is zero then the image of $u$ is contained in the semistable locus, and $u$ induces a constant map to $X \qu G$. Hence the bundle and section are trivializable. \end{remark}
\subsection{Convex targets} \label{convex}
The definition of Mundet semistability also gives good moduli spaces in the cases of some affine targets. A finite dimensional complex $G$-vector space $V$ is said to be {\em convex} if there exists a central one-parameter subgroup $\phi_\xi: \mathbb{C}^\times \to G$ such that $V$ has positive weights for the induced action of $\phi_\xi$,
$$ V = \bigoplus_\mu V_i, \quad \mu_i(\xi) > 0, \quad i =1,\ldots,k .$$
Given a convex $G$-vector space, the {\em projectivization} of $V$ is the quotient
$$ \overline{V} = ((V \times \mathbb{C})^\times - \{ (0 , 0) \})/\mathbb{C}_\xi^\times $$
where $\mathbb{C}^\times$ acts on $\mathbb{C}$ with weight one. Thus $\overline{V}$ is a weighted projective space (in a particular a smooth Deligne-Mumford stack) and contains $V$ as an open substack. A quasiprojective $G$-variety $X$ is {\em convex} if there exists a projective embedding $\pi: X \to V$ to a convex $G$-vector space $V$ whose image intersects the locus $V - \{ 0 \}$. The following is a simple application of the technique called {\em symplectic cutting} in the literature \cite{le:sy2}:
\begin{lemma} Any convex $G$-variety $X$ embeds as a dense open
substack of a Deligne-Mumford stack $\overline{X}$ with complement a prime
$\mathbb{C}^\times_\xi$-fixed divisor isomorphic to $(X - \{ 0
\})/\mathbb{C}_\xi^\times$. \end{lemma}
\begin{proof} Let $\tilde{X} \to X$ denote the given linearization on $X$ and $\tilde{X}(l)$ the linearization on $X \times \mathbb{C}$ obtained by twisting by the $\mathbb{C}^\times$-character with weight $l$. Consider the git quotient
$$ \overline{X} = (X \times \mathbb{C}) \qu \mathbb{C}^\times_\xi .$$
The inverse image of $ (0,0) \in V \times \mathbb{C}$ is unstable, for sufficiently large $d$. Thus the proper morphism $X \to V$ induces a proper morphism $\overline{X}$ to $\overline{V}$. In particular, this implies that $\overline{X}$ is also proper. The $G$ action on $X \times \mathbb{C}$ given by $g(x,z) = (gx,z)$ descends to a $G$-action on $\overline{X}$, and restricts to the given action on the open substack $X \subset \overline{X}$. \end{proof}
In the following we will refer to $\overline{\M}_n^G(C,\overline{X},d)$ allowing $\overline{X}$ to be a smooth Deligne-Mumford stack without further comment; we do not allow stacky structures on the domain curves since we are only interested in defining $\overline{\M}_n^G(C,X,d)$ in which case the target $X$ is a variety.
\begin{corollary} \label{disjoint} Let $d \in H^2_G(\overline{X})$ be a class that pairs
trivially with the divisor class $[\overline{X} - X] \in H_2^G(\overline{X})$.
Then there exists a constant $l(E)$ such that if the energy bound
$\mE(d) < E$ holds and $l \ge l(E)$ then the moduli stack
$\overline{\M}_n^G(C,\overline{X},d)$ consists of maps whose images are disjoint
from $(\overline{X} - X)/G$. \end{corollary}
\begin{proof} The intersection number of any curve $u: \mathbb{P}^1 \to
\overline{V}$ contained in $\overline{V} - V$ with $\overline{V} - V$ is non-negative.
Indeed $\overline{V} - V \cong \mathbb{P}[\mu_1,\ldots,\mu_k]$ has ample normal
bundle $\mO_{\mathbb{P}[\mu_1,\ldots,\mu_k]}(1)$. On the other hand, there
are no stable gauged maps $C \to X/G$ with image in $(\overline{V} - V)/G$
for sufficiently large $l > l(d)$. The trivial reduction $\sigma$
together with the generator $\xi$ of the one-parameter subgroup
$\mathbb{C}^\times$ has weight $\mu(\sigma,\xi) \to \infty $ as $d \to
\infty$, while \eqref{dPP} implies that $( d(P)^\vee, \lambda )$ is
bounded in terms of the energy. Combining these observations let
$v: \hat{C} \to \overline{V}/G$ be a stable gauged map intersecting
$(\overline{V} - V)/G$. The intersection number $\# u^{-1}(P(\overline{V}- V))
> 0 $ is positive and equal to the pairing $(d, [\overline{V} - V]) \in
\mathbb{Q}$ of $d \in H_2^G(X,\mathbb{Q})$ with $[\overline{V} - V] \in H^2_G(\overline{V},\mathbb{Q})$.
The latter vanishes by assumption, a contradiction. \end{proof}
\begin{lemma} Suppose that $(P,u)$ is a map from $C$ to $X/G$ with $X$
convex. Any destabilizing pair $(\sigma,\lambda)$ has associated
graded $(\Gr(P),\Gr(u))$ disjoint from the divisor at infinity
$P(\overline{X} - X)$, for $l$ sufficiently large. \end{lemma}
\begin{proof}
As in the proof of Lemma \ref{positivity}, by choosing a Borel
structure refining the parabolic structure and passing to the
associated graded we may assume that $G$ is a torus. Suppose that
$(\sigma,\lambda)$ has associated graded $\Gr(P),\Gr(u)$
intersecting $P(\overline{X}- X)$. By invariance of intersection number,
the limit $\Gr(u)$ must take values in $P(\overline{X} - X)$, since
otherwise the intersection number with $P(\overline{X} - X)$ would be
positive. We suppose that $\mu_1,\ldots, \mu_k$ are the weights of
$G$ on $V$, so that $u$ has components $u_1,\ldots, u_k$. From the
description of the associated graded, if $\lambda$ satisfies
$\mu_i(\lambda) \ge 0$ for all $i = 1,\ldots, k$ such that $u_i$ is
non-zero, then the associated graded takes values in $P(X)$. Hence
$\mu_i(\lambda) > 0$ for non-empty subset $I \subset \{ 1,\ldots, k
\}$ of indices such that $u_i$ is non-zero. Let
$$m = \min_{i \in I} \frac {\mu_i(\lambda)}{ \mu_i(\xi)}. $$
The minimum $m$ is negative since some $\mu_i(\lambda) < 0 $ and $\mu_i(\xi) > 0$ for all $i = 1,\ldots, k$. The associated graded section is then given by the collection of sections $u_i, i \in I$ with $\mu_i(\lambda)/\mu_i(\xi) = m$. The corresponding Hilbert-Mumford weight is the weight of the action of $\mathbb{C}^\times_\lambda$ on the fibers of $\tilde{X} / \mathbb{C}^\times_\xi$, and is equal to $ml$. Therefore, the weight $\mu(x,\lambda)$ is positive. For $l$ sufficiently large the pair is not destabilizing. \end{proof}
As a result, for convex target it suffices to check semistability for pairs such that the associated graded exists without compactification.
\section{Variation of polarization}
The moduli space of Mundet-semistable gauged maps depends on the linearization. Changing the linearization leads to wall-crossing in which loci of bundles with the same associated graded are flipped \cite{cross} as is standard in variation of git as explained in e.g Thaddeus \cite{th:fl}. Consider the family of linearizations $\tilde{X}^k$ given by the $k$-th tensor product of the given one $\tilde{X}$ for $k$ a positive integer. While taking tensor products does not change the definition of semistability for $X$, it does change the definition of Mundet semistability.
\begin{lemma} \label{finitelem} For any fixed degree $d \in H_2^G(X)$,
there are at most finitely many changes in the stability condition
as $k$ varies. That is, there exist
$$ -\infty = k_0 < k_1 < \ldots < k_l =\infty \in \mathbb{Q} \cup \{ -\infty,\infty \} $$
such that if $k',k'' \in (k_i,k_{i+1})$ then the semistable loci for $k',k''$ are equal. \end{lemma}
\begin{proof} Denote by $\mu_k(\sigma,\lambda)$ the
Mundet weight corresponding to $\tilde{X}^k$. Changes in the
definition of stability correspond to pairs $(P,u)$ such that for
some pair $(\sigma,\lambda)$ and $k_-,k_+ \in \mathbb{Q}$,
$$ \mu_{k_-}(\sigma,\lambda) < 0 , \quad \mu_{k_+}(\sigma,\lambda) > 0
$$
while for some $k\in (k_-,k_+)$, $ \mu_{k}(\sigma,\lambda ) = 0 $ so that the pair $(P,u)$ is semistable but not stable. As in \ref{torus action on P(V)}(1), the wall-crossings arise from pairs $(P,u)$ such that
\begin{equation} \label{dP2}
\dim( \operatorname{hull} ( \{ \mu_i | \tilde{u}_i \neq 0 \})) < \rank(G),\quad
\operatorname{hull} ( \{ \mu_i | \tilde{u}_i \neq 0 \}) \ni \theta + d(P)/k. \end{equation}
Suppose there are infinitely many wall-crossings. Let $(P_k,u_k)$ denote the corresponding reducible gauged maps for some $k$ in an unbounded set $\cW(d) \subset \mathbb{Q}$. The equation \eqref{dP2} implies that $ \Vert d(P_k) \Vert > c k$ for some positive constant $c$ and all $k \in \cW(d)$. On the other hand, as in \eqref{dPP}
\begin{eqnarray*}\label{dPP2} (u_{k,*}[C],c_1(P_k(\tilde{X}))) &=& (\theta,d(P_k))+ d(u_k) \\ &=& (\theta - d(P_k)/k, d(P_k))+ d(u_k) + (d(P_k)/k,d(P_k)) \\ & \ge& (d(P_k),d(P_k))/k \ge c^2 k \end{eqnarray*}
for $k \in \cW(d)$. Since $\cW(d)$ is unbounded, this implies that the homology class $ v_{k,*}[C] \in H_2^G(X)$ is also unbounded, a contradiction. \end{proof}
Denote by $\overline{\M}^{G}(C,X,d,k)$ the moduli space of Mundet semistable maps using the linearization $\tilde{X}^k$. By the finite-ness above in Lemma \ref{finitelem}, we have the following:
\begin{corollary} \label{large} For any $d \in H_2^G(X,\mathbb{Q})$ there exists $k(d)$ such that for $k \ge k(d)$, the stack $\overline{\M}^G(C,X,d,k)$
consists of those bundles that are Mundet semistable for all $k \ge
k(d)$, that is,
$$ ( k_1,k_2 \ge k(d)) \ \implies \ (\overline{\M}^G(C,X,d,k_1) = \overline{\M}^G(C,X,d,k_2) ) .$$
More precisely, an object $(\hat{C},P,u,\ul{z})$ is destabilized by $(\sigma,\lambda)$ for some $k \ge k(d)$ iff it is destabilized for all $k \ge k(d)$. \end{corollary}
The following describes the Mundet semistability condition for large $k$.
\begin{lemma} \label{genlem} For any $d \in H_2^G(X,\mathbb{Q})$ there exists
$k(d)$ such that for $k \ge k(d)$, the stack $\overline{\M}^G(C,X,d,k)$
has objects given by tuples $(P,\hat{C},u, \ul{z})$ for which $(u |
\hat{C}_0)^{-1}(X^{\on{ss}}/G)$ is non-empty. \end{lemma}
\begin{proof} It suffices to show that if $v: C \to X/G$ is a Mundet unstable gauged map with class $d$ for large $k$, then $v(C)$ is contained in some Kirwan-Ness stratum $X_\lambda/G$ and vice versa. Let $(\sigma,\lambda)$ be a pair destabilizing $u$:
$$ \mu(\sigma,\lambda) = \mu_{BG}(\sigma,\lambda) + k \mu_X(\sigma,\lambda) > 0 .$$
By Corollary \ref{large}, we may assume that $v = (P,u)$ is Mundet
destabilized by $(\sigma,\lambda)$ for all $k \ge k(d)$. Then the
Hilbert Mumford weight
$\mu_X(\sigma,\lambda) > 0 $
must be positive. The associated graded $\Gr(u)$ is contained in the fixed point set $Z_\lambda$, that is,
$ \Gr(u) ( \hat{C}_0) \subset P(Z_\lambda) $,
and $(\Gr(P),\Gr(u))$ has positive Hilbert-Mumford weight with respect to $(\sigma,\lambda)$. Thus $u$ is generically unstable.
Conversely, suppose that $v = (P,u): \hat{C} \to X/G$ takes values in some stratum $X_\mu/G$ generically on the root component $\hat{C}_0 \subset \hat{C}$. The stratum fibers
\begin{equation} \label{Qproj}X_\mu = G \times_{Q_\mu} Y_\mu \to G/Q_\mu\end{equation}
as in \eqref{kn}. By composition with the map $P(X) \to P(Q_\mu) = P/Q_\mu$ arising from \eqref{Qproj} we obtain a map
$ \sigma: (u | \hat{C}_0 \cap u^{-1}(X_\mu)) \to P/Q_\mu $.
Locally $P$ is trivial, and so in a neighborhood of any point in $u^{-1}(X_\mu)$ the map is given by a map to $G/Q_\mu$. By completeness of $G/Q_\mu$ this map extends to $\sigma: \hat{C}_0 \to P/Q_\mu$, by definition a parabolic reduction $\sigma$. Consider the one-parameter subgroup generated by a positive coweight $\mu$. The associated graded $\Gr(u)$ maps to $Z_\mu$ on the root component. The Hilbert-Mumford weight $\mu_X(\sigma,\lambda)$ is positive, by construction. Hence $\mu_{BG}(\sigma,\lambda) + k \mu_X(\sigma,\lambda)$ is positive for large $k$, and the pair $(P,u)$ is Mundet unstable for large $k$. \end{proof}
\section{Scaled gauged maps}
The Mundet semistable moduli spaces have a large linearization limit which includes both stable maps to the git quotient as well as what we called affine gauged maps. This is an algebraic version of a limit that was first studied in the symplectic context by Gaio-Salamon \cite{ga:gw}.
\begin{definition} {\rm (Scaled gauged maps)} \label{scaledgauged} A prestable {\em scaled gauged map} is a datum
$(P,\hat{\mathcal{C}},u, \delta,\ul{z})$ consisting of a prestable scaled
curve $(\hat{\mathcal{C}},\delta,\ul{z})$ and pair $(P \to C,u :\hat{\mathcal{C}}
\to P(X) )$ giving a map to the quotient stack $\hat{C} \to X/ G$.
In the case that $X \qu G$ is an orbifold, the domain $\hat{C}$ is
allowed to have a twisted stacky structure $\hat{\mathcal{C}}$ so that the
points with non-trivial automorphism are nodes with infinite scaling
and the data above gives a representable morphism $v: \hat{\mathcal{C}} \to
X \qu G$ as in \cite{agv:gw}. Denote by
$$D_\infty = \mathbb{P}(\omega_{\hat{\mathcal{C}}/C}) \subset \mathbb{P}(\omega_{\hat{\mathcal{C}}/C}
\oplus \mO_{\hat{\mathcal{C}}}), \quad \text{resp.} \quad D_0 =
\mathbb{P}(\mO_{\hat{\mathcal{C}}}) \subset \mathbb{P}(\omega_{\hat{\mathcal{C}}/C} \oplus
\mO_{\hat{\mathcal{C}}})$$
the divisor at infinity resp. the zero section. The datum $(P,\hat{\mathcal{C}},u,\delta)$ is {\em semistable} if either
\begin{enumerate}
\item the scaling $\delta | \hat{C}_0$ is finite, and the datum
$(P,\hat{\mathcal{C}},u)$ is Mundet semistable; here we are interested in
the chamber $k \ge k(d)$ from Lemma \ref{genlem}, or
\item the scaling $\delta | \hat{\mathcal{C}}_0$ on $\hat{\mathcal{C}}_0$ is infinite,
and $\delta^{-1}(D_\infty) \subset \hat{\mathcal{C}}$ maps to the semistable
locus in $X/G$,
\end{enumerate}
A semistable scaled gauged map is {\em stable} if it has finitely many automorphisms. \end{definition}
We introduce the following notation for moduli stacks. Denote by $\overline{\M}^G_{n,1}(C,X)$ the moduli of stable marked scaled gauged maps. The existence of a universal scaled curve implies that again, $\overline{\M}^G_{n,1}(C,X)$ is a hom stack from a Deligne-Mumford stack to a quotient stack of a variety by a reductive group, and so Artin by \cite[Proposition 2.3.4]{lieblich:rem}.
The moduli stack of stable scaled curves defines a cobordism using the following forgetful morphism. Forgetting everything besides the map $\delta$ defines a morphism
$$ \rho: \overline{\M}_{n,1}^G(C,X,d) \to \overline{\M}_{0,1} \cong \mathbb{P}^1 , \quad (C,v,\delta,\ul{z} = (z_1,\ldots,z_n)) \mapsto \delta
|_{\hat{C}_0 \cong C} .$$
The fiber of $\rho$ over any non-infinite point $\alpha \in \mathbb{P}^1 - \{ \infty \}$ is
$$ \rho^{-1}(\alpha) \cong \overline{\M}_n^G(C,X,d) $$
the space of Mundet semistable gauged maps in the chamber $k \ge k(d)$. On the other hand, the fiber over infinity consists of stable maps to $C \times X \qu G$ of degree $(1,d)$ together with bubble trees which call {\em affine gauged maps} because of the affine structure given by the one-form. Affine gauged maps were introduced first in a symplectic context by Ziltener \cite{zilt:qk}; a Narasimhan-Seshadri correspondence which relates that viewpoint with the one given here is in Venugopalan-Woodward \cite{venuwood:class}.
\begin{definition} \label{affinegauged} An affine gauged map is a datum
$$ (C,\delta,\ul{z} = (z_0,\ldots,z_n),v)$$
where $(C,\delta,\ul{z})$ is an affine scaled marked curve from Definition \ref{affine}, and $v = (P,u): C \to X/ G$ is a morphism to the quotient stack such that
\begin{enumerate}
\item $v( \delta^{-1}(D_\infty)) \subset X^{\on{ss}}/G$. In other words,
on the locus $u^{-1}(\delta^{-1}(D_\infty))$, the map has image in
the $X$-semistable locus; and
\item on the locus $v^{-1}(\delta^{-1}(D_0))$, the bundle is trivial. \end{enumerate}
In the case that $X \qu G$ is an orbifold, $C$ is equipped with an twisted stacky structure $\mathcal{C}$ with non-trivial automorphism groups only at the nodes and marking with infinite scaling as in \cite{agv:gw} and data above defines a representable morphism $v: \mathcal{C} \to X/G$. Such a datum is {\em stable} if there exist only finitely many automorphisms $\varphi \in \on{Aut} (C,v,\delta,\ul{z})$, or in other words, if there each component $C_i$ on which the map $v_* [C_i] = 0 \in H_2^G(X,\mathbb{Q})$ the scaling is finite and non-zero (resp. zero or infinity) has at least two (resp. three) special points. \end{definition}
We introduce the following notation for moduli spaces of affine gauged maps. Let $\overline{\M}^G_{n,1}(\mathbb{A},X)$ denote the moduli space of affine gauged maps with group $G$ and target $X$ with $n$ markings in addition to the marking at infinity. For each $d \in H_2^G(X,\mathbb{Q})$, let $\overline{\M}^G_{n,1}(\mathbb{A},X,d)$ denote the locus of maps with $v_*[C] = d \in H_2^G(X,\mathbb{Q})$. The moduli stack admits natural evaluation maps
$$ \ev \times \ev_\infty : \overline{\M}^G_{n,1}(\mathbb{A},X) \to (X/G)^n \times \overline{\cI}_{X \qu G} $$
given by evaluation at the markings $z_i, i = 1,\ldots, n$ and $z_0$. Also define, for ease of notation,
$$ \overline{\M}_{n}(C,X \qu G,d) = \overline{\M}_{g,n}(C \times X \qu G,(1,d)) $$
the so called {\em graph space} of stable maps to $X \qu G \times C$ of degree $(1,d)$. Generalizing the positivity of energy of stable gauged maps in Lemma \ref{positivity} we have the following:
\begin{proposition} \label{posen}
Any object $(P,\hat{\mathcal{C}},u,\ul{z})$ of $\overline{\M}_{n,1}^G(C,X)$ or
object $(\mathcal{C}, \ul{z},\delta,P,u)$ of $\overline{\M}_{n,1}^G(\mathbb{A},X)$ has
non-negative energy, and vanishing energy only if the bundle and
section are trivializable on each component. \end{proposition}
\begin{proof} Each irreducible component of the domain carries either a Mundet-semistable map, a map to $X$, a map to $X \qu G$, or an affine gauged map which is necessarily generically semistable. The statement of the proposition follows from applying Lemma \ref{positivity} and Remark \ref{alten} to each component. \end{proof}
Later we will need a bound on the number of irreducible components of the domain in terms of the energy.
\begin{corollary} \label{kcor} Let $k$ be an integer such that if $x$ is any object of $ \on{Hom}(\pt,X
\qu G)$, then the order of the automorphism group $| \on{Aut} (x)|$ of $x$ divides $k$. Any object $(P,\hat{\mathcal{C}},u,\ul{z},\delta)$ of $\overline{\M}_{n,1}^G(C,X,d)$ or object $(P,\mathcal{C}, u, \ul{z},\delta)$ of $\overline{\M}_{n,1}^G(\mathbb{A},X,d)$ with non-zero energy has energy at least $1/k$. \end{corollary}
\begin{proof}
By Proposition \ref{posen}, any component with non-zero energy has positive energy defined as the pairing of $u^* c_1(P(\tilde{X}))$ with $[\mathcal{C}]$. By \cite[Theorem 1.187 part (iii)]{behrend:review}, $k u^* c_1(P(\tilde{X}))$ is represented by an integral divisor, and so an integral cohomology class. The statement of the Corollary follows. \end{proof}
\section{Properness for trivial actions}
In this section we show properness of the moduli stack of gauged
scaled maps in the case that the group acting is trivial.
\begin{proposition} \label{trivaction} Let $\mathcal{X}$ be a smooth proper
Deligne-Mumford stack with projective coarse moduli space $X$ and
ample line bundle on the coarse moduli space $\tilde{X} \to X$. For
any $n > 0, E > 0$, the union of the stacks
$\overline{\M}_{n,1}(C,\mathcal{X},d)$ for $(d, c_1(\tilde{X})) < E$ is proper. \end{proposition}
\begin{proof} By properness of moduli stacks of stable maps to Deligne-Mumford stacks \cite[Section 6]{abramovich:compactifying}, the union of the components
$$ \bigcup_d \overline{\M}_n(C,\mathcal{X},d) := \overline{\M}_{g,n}( C \times \mathcal{X}, (1,d)) , \quad (d, c_1(\tilde{X})) < E$$
is proper. Therefore it suffices to show that the forgetful morphism
$$ f: \overline{\M}_{n,1}(C,\mathcal{X},d) \to \overline{\M}_n(C,\mathcal{X},d) := \overline{\M}_{g,n}( C \times \mathcal{X}, (1,d)) $$
obtained by forgetting $\delta$ and collapsing unstable components is proper. Let $[u] \in \overline{\M}_n(C,\mathcal{X},d)$ with representative $u: \hat{C} \to \mathcal{X}$. Since $\mathcal{X}$ is projective, Bertini implies that there exists a divisor $D \subset X$ transverse $u$ and meeting each non-constant component of $u$ transversally and disjoint from the markings and images of unstable components of the domain. Let $\mathcal{D} = D \times_X \mathcal{X}$ and $U \subset \overline{\M}_{n,1}(C,\mathcal{X},d)$ be the open substack of maps such that each component meets $\mathcal{D}$ transversally and in a set of distinct points disjoint from the markings and ghost components. By taking a divisor $D$ of sufficiently large degree, we may assume that for each component $\hat{C}_i \subset \hat{C}$, the map $u$ restricted to $\hat{C}_i$ meets the divisor in at least three points:
$$ \# (u |_{\hat{C}_i})^{-1}(\mathcal{D}) \ge 3, \quad \forall \hat{C}_i \subset \hat{C} .$$
Choose an ordering of the additional points $u^{-1}(\mathcal{D})$ meeting $u$. Let $U$ denote the substack of $\overline{\M}_{n+k,1}(C,\mathcal{X},d)$ so that the last $k$ points represent transverse intersections with $\mathcal{D}$. The map forgetting the last $k$ points gives an \'etale morphism from $U$ to $\overline{\M}_n(C,\mathcal{X},d)$, see for example \cite[Proposition 4]{fu:st}. The map
$$\overline{\M}_{n,1}(C,\mathcal{X},d) \supset U \to \overline{\M}_{n+k,1}(C), \quad (u: \hat{\mathcal{C}} \to \mathcal{X}, \ul{z}) \mapsto (u: \hat{\mathcal{C}} \to \mathcal{X}, \ul{z} \cup u^{-1}(\mathcal{D}) ) $$
fits into a Cartesian diagram
$$ \begin{diagram} \node{ \overline{\M}_{n,1}(C,\mathcal{X},d)} \arrow{s} \node{U} \arrow{e} \arrow{w}
\arrow{s} \node{\overline{\M}_{n+k,1}(C,\mathcal{X},d)} \arrow{e} \arrow{s} \node{
\overline{\M}_{n+k,1}(C)} \arrow{s} \\ \node{ \overline{\M}_{n}(C,\mathcal{X},d)} \node{ f(U)}
\arrow{e} \arrow{w} \node{\overline{\M}_{n+k}(C,\mathcal{X},d)} \arrow{e}
\node{ \overline{\M}_{n+k}(C)} \end{diagram} $$
where the right-hand vertical arrow is proper. Since the pull-back of proper morphisms is proper and properness is \'etale local in the target, the left-hand-arrow is also proper. Since $\overline{\M}_{n}(C,\mathcal{X},d)$ is proper, $\overline{\M}_{n,1}(C,\mathcal{X},d)$ is proper as well.\end{proof}
\begin{proposition} \label{trivaction2} For any $E, n >0 $ the union of moduli stacks $\overline{\M}_{n,1}(\mathbb{A},\mathcal{X},d)$ with $(d, c_1(\tilde{\mathcal{X}})) < E$ is proper. \end{proposition}
\begin{proof} Consider the forgetful map
$ f: \overline{\M}_{n,1}(\mathbb{A},\mathcal{X},d) \to \overline{\M}_{0,n+1}(\mathcal{X},d) $
defined by composing $\overline{\M}_{n,1}(\mathbb{A},\mathcal{X},d) \to \overline{\MM}_{0,n+1}(\mathcal{X},d)$ with the stabilization map $\overline{\MM}_{0,n+1}(\mathcal{X},d) \to \overline{\M}_{0,n+1}(\mathcal{X},d)$ \cite[Proposition
3.10]{bm:gw}, \cite[Proposition 9.1.1]{abramovich:compactifying}. As before, choose $[u] \in \overline{\M}_{0,n+1}(\mathcal{X},d)$ with representative $u: \mathcal{C} \to \mathcal{X}$ and a divisor $\mathcal{D} \subset \mathcal{X}$ meeting $u$ transversally away from the markings and ghost components. Properness of $\M_{n+k,1}(\mathbb{A})$ and the Cartesian diagram
$$ \begin{diagram} \node{ \overline{\M}_{n,1}(\mathbb{A},\mathcal{X},d)}\arrow{s} \node{ U } \arrow{w} \arrow{e} \arrow{s} \node{ \overline{\M}_{n+k,1}(\mathbb{A})} \arrow{s} \\ \node{ \overline{\M}_{0,n}(\mathcal{X},d)} \node{ f(U)} \arrow{w} \arrow{e} \node{ \overline{\M}_{0,n+k}} \end{diagram} $$
imply that $\overline{\M}_{n,1}(\mathbb{A},\mathcal{X},d)$ is proper over $\overline{\M}_{0,n}(\mathcal{X},d)$. Since $\overline{\M}_{0,n}(\mathcal{X},d)$ is itself proper by \cite[Theorem 1.4.1]{abramovich:compactifying}, $\overline{\M}_{n,1}(\mathbb{A},\mathcal{X},d)$ is itself proper. \end{proof}
\section{Boundedness}
In this section we show that the moduli space of gauged scaled maps with fixed numerical invariants is finite type. The results of Ciocan-Fontanine-Kim-Maulik \cite[Section 3.2]{cf:st} imply such a result in the case of a vector space target. We extend the argument here to the case of projective spaces.
\begin{theorem} \label{bounded} (c.f. \cite[Theorem 3.2.5]{cf:st}) Let
$E > 0 $ and $\mathcal{C}$ a twisted prestable curve. Let $V$ be a
finite-dimensional complex vector space with an action of $G$ via a
representation $G \rightarrow GL(V)$ with finite kernel and $X = \mathbb{P}(V)$.
Suppose that the semistable locus $X^{\on{ss}}$ is non-empty and equal
to the stable locus. Then the following family of gauged maps is
bounded: pairs $v=(P,u)$ consisting of a principal $G$-bundle $P \to
\mathcal{C}$ and representable section $u: \mathcal{C} \to P(X)$ such that the
energy $\cE(v) < E$ and the section $u$ sends the generic point of
$\mathcal{C}$ to $P(X^{\on{ss}})$. \end{theorem}
\begin{proof}
A similar theorem with $\mathbb{P}(V)$ replaced by $V$ is given by
Ciocan-Fontanine-Kim-Maulik \cite[Theorem 3.2.5]{cf:st}.
First we assume that $\mathcal{C}$ is an ordinary curve and $G$ is a torus. By lemma \ref{torus action on P(V)}(4) we have
\begin{equation} \label{dform2}
\cE(v) = (u_*[C],c_1(P(\tilde{X}))) = (\theta,d(P)) + d(u) \in [0,E]
.\end{equation}
By the Hilbert-Mumford criterion the semistable locus in $\mathbb{P}(X)$ is
$$ X^{\on{ss}} = \mathbb{P}(V)^{\on{ss}} = \left\{ [x_1,\ldots,x_k] \in \mathbb{P}(V) \ |
\ \operatorname{hull} ( \{ \mu_i | x_i \neq 0 \}) \ni \theta \right\} .$$
Let $u: C \to P(X)$ be a section that is generically semistable. Recall example \ref{toric example} and that $u$ is given by a $k$-tuple $(\tilde{u}_1, \dotsc, \tilde{u}_k)$. The condition that $u$ is generically semistable means each $\tilde{u}_i \neq 0$ hence by lemma \ref{torus action on P(V)}(3)
$$ (\mu_i, d(P)^\vee) + d(u) \ge 0, \quad \forall i = 1,\ldots, k.$$
The same holds for any vector near $\theta$, since the condition of lying in the convex hull is open in in the interior. Choose a basis
$$ \xi_1,\ldots, \xi_r \in \lie{g}^\vee_\mathbb{Q}, \quad \operatorname{hull}(\xi_1,\ldots, \xi_r) \ni \theta $$
of points near $\theta$ so that
\begin{equation} \label{xiform}
(\xi_i,d(P)^\vee) - d(u) \ge 0, \quad i = 1,\ldots, r .\end{equation}
Combining \eqref{xiform} and \eqref{dform2} shows that the possible degrees $d(P)$ lie in a finite set.
Next we consider the case that $\mathcal{C}$ is an ordinary curve and $G$ is an arbitrary compact connected reductive complex group. By a simple case of the Drinfeld-Simpson theorem \cite{ds:red}, \cite[Lemma 3.2.7]{cf:st}, the bundle $P$ admits a reduction to a Borel subgroup $B \subset G$. Let $\pi_B: B \to T$ be the projection \eqref{piq}, and $\Gr(P)$ the associated graded. Since the map $\pi_Q$ is $T$-equivariant, the section $u$ induces a section $\Gr(u): \mathcal{C} \to \Gr(P)(X)$ that is also $T$-semistable. A $G$-bundle corresponds via a faithful representation $ G \to GL(r)$ to a vector bundle $F \to C$ together with a reduction of the structure group, given by a section of an affine bundle $GL(r)/G$. Equation \eqref{xiform} shows that splitting type of the associated graded $\Gr(F)$ is uniformly bounded given a fixed $d \in H_2^G(X,\mathbb{Z})$, and furthermore the first Chern class $d(P)$ has bounded pairing with $c_1^G(\tilde{X})$.
Given this bound on the splitting type of the associated graded, a standard argument (see for example the boundedness arguments in \cite[3.3]{hl:mod}) shows that after twisting by a sufficiently positive bundle depending only on a bound on the splitting type, any vector bundle $F \to C$ as above is generated by their global sections and has no higher cohomology. Indeed the long exact sequence in cohomology shows that, for any locally free subsheaf $F' \subset F$ appearing as a summand in the associated graded $\Gr(F)$ we have an exact sequence
$$ H^0(C,F') \to H^0(C,F) \to H^0(C,F/F') \to H^1(C,F') .$$
From this and the corresponding sequence for the twist $F(-z), z \in C$ one obtains that if $F',F/F'$ are generated by their global sections and have no higher cohomology then $F$ has the same property. An induction shows that $F \to C$ is a quotient of a fixed trivial bundle $ \mO_C^{\oplus l}$ for $l \ge k(E)$ where $k(E)$ is a constant depending only on the energy bound $E$. Thus the family of bundles is bounded.
To show that the families of sections are bounded, note that any two sections $u_0,u_1: \hat{C} \to P(X)$ have homology classes that differ by an element of $H_2(X,\mathbb{Q}) \subset H_2^G(X,\mathbb{Q})$. The homomorphism
$$( \cdot, c_1^G(\tilde{X})) \in \on{Hom}(H_2^G(X,\mathbb{Q}),\mathbb{Q})$$
restricts to the standard pairing on $H_2(X,\mathbb{Q})$ corresponding to the hyperplane class, the energy bound \eqref{dform2} implies that $d(u)$ is bounded from above and below. Now the difference of homology classes of any sections of $P(\mathbb{P}(V))$ lies in the kernel of the map $H_2(P(\mathbb{P}(V)) \to H_2(C)$ and so, since $\mathbb{P}(V)$ is simply-connected, lie in the image of the inclusion $H_2(\mathbb{P}(V)) \to H_2(P(\mathbb{P}(V))$ of a fiber. It follows that the degree also classifies homology classes of sections:
\begin{equation} \label{iff} (u_1 \cong u_2 ) \iff (d(u_1) = d(u_2)) .\end{equation}
By \eqref{iff} the possible homology classes of the sections are bounded as well. This shows that the family of maps to $X/G$ is bounded.
Finally suppose that $\mathcal{C}$ is a twisted curve and $G$ is complex connected reductive. Let $\hat{\mathcal{C}} \to S$ be a family of twisted curves, $P \to \hat{\mathcal{C}}$ a family of bundles and $u: \mathcal{C} \to P(X)$ a family of sections as above. By \cite[Theorem 1.14]{ol:logtwist}, after \'etale cover there exists a finite flat morphism $\pi: Z \to \hat{\mathcal{C}}$ from a projective scheme $Z \to S$ to $\hat{\mathcal{C}}$; the proof in fact shows that $\pi$ is surjective. By faithfully flat descent, sheaves on $\mathcal{C}$ may be described in terms of descent data as sheaves $E$ on $ Z \lefttwoarrow Z \times_{\mathcal{C}} Z. $ Such data consists of the bundle $Z$ together with isomorphisms $\varphi: \pi_1^* E \to \pi_2^* E,$ where $\pi_1,\pi_2$ are the projections onto the factors of $Z \times_{\mathcal{C}} Z$ see \href{http://stacks.math.columbia.edu/tag/03O6}{Tag 03O6} in \cite{dejong:stacks}. A principal $G$-bundle is given via an embedding $GL(r)$ as descent data for a locally free sheaf of rank $r$ together with a reduction given by a section of the associated $GL(r)/G$ bundle and an isomorphism $\varphi: \pi_1^* E \to \pi_2^* E$ preserving the $G$-reduction. Any such substack may be realized via quot scheme techniques as a quotient of a variety by a reductive group action as above. \end{proof}
\begin{corollary}
\label{mainfin} For any real $E > 0$, the union of components $\overline{\M}_{n,1}^G(C,X,d)$, $\overline{\M}_n^G(C,X,d)$, and
$\overline{\M}_{n,1}^G(\mathbb{A},X,d)$ with $(d, c_1^G(\tilde{X})) < E$ is finite
type. \end{corollary}
\begin{proof}
We consider only $\overline{\M}_{n,1}^G(C,X,d)$; the proof for the other
moduli spaces is similar. We first show that only finitely many
combinatorial types are possible for a given energy bound. Let $v =
(P \to C, u: \hat{\mathcal{C}} \to P(X))$ be a gauged map of class $d$ and
energy $\cE(v) = ( d, c_1^G(\tilde{X}) )$. The energy of any component
with non-zero energy $ \langle v_*[\mathcal{C}_i], c_1^G(\tilde{X}) \rangle$ is at
least $1/k$ for some integer $k$, by Corollary \ref{kcor}. It
follows that the number of irreducible components $\hat{\mathcal{C}}_i$ of
the domain $\hat{\mathcal{C}}$ with positive energy is bounded by $ k \cE(v)
+ n$.
The bound on the number of components with positive energy gives a
bound on the total number of components as follows. Any component
$\hat{\mathcal{C}}_i$ of $\hat{\mathcal{C}}$ on which the map $v: \hat{\mathcal{C}} \to X/G$
is trivial and has trivial scaling has at least three special
points. Removing the component $\hat{\mathcal{C}}_i$ defines a curve
$\hat{\mathcal{C}} - \hat{\mathcal{C}}_i$ with at least three connected components,
and so a partition of the markings $\{ z_1,\ldots, z_n \}$ and
irreducible components of $\hat{\mathcal{C}}$ with non-trivial energy into
three non-empty subsets. Thus the number of components of the
domain with trivial scaling is also bounded by the number of
partitions of $ k \mE(d) + n$. On the other hand, by the
monotonicity condition the number of components with scaling is at
most the number of terminal components, since there is at most one
component with finite, non-zero scaling on the any path from a root
component to the terminal component. That is, the number of
vertices $\on{Vert}(\Gamma)$ of the combinatorial type $\Gamma$ of
$\hat{\mathcal{C}}$ is bounded by an integer $v = v(d)$. There are finitely
many trees $\Gamma$ satisfying this bound, hence finitely many
possibilities for $\Gamma$.
Given an energy bound, the possible homology classes of each
component form a finite set by Theorem \ref{bounded} and the
requirement that $d(P) \in H_2(\mathcal{C},\mathbb{Z}/k)$. It follows that there
are only finitely many possible labellings $\on{Vert}(\Gamma) \to
H_2^G(X,\mathbb{Q})$ of the given graphs by degree two homology classes with
the given energy bound; hence finitely many combinatorial types as
claimed. It follows that the image of $\overline{\M}_{n,1}^G(C,X,d) \to
\overline{\MM}_{n,1}(C)$ is contained in an Artin stack of finite type for
each $d \in H_2^G(X,\mathbb{Z})$.
We now use boundedness of the splitting type to prove that the
moduli stack is finite type. As in the proof of Theorem
\ref{bounded}, we describe bundles on stacky curves in terms of
descent data. Let $\hat{\mathcal{C}} \to S$ be a family of stacky curves,
$P \to \hat{\mathcal{C}}$ a family of bundles and $u: \mathcal{C} \to P(X)$ a family
of sections as above. By \cite[Theorem 1.14]{ol:logtwist}, after
\'etale cover there exists a finite flat morphism $\pi: Z \to
\hat{\mathcal{C}}$ from a projective scheme $Z \to S$ to $\hat{\mathcal{C}}$; the
proof in fact shows that $\pi$ is surjective. By faithfully flat
descent, sheaves on $\mathcal{C}$ may be described in terms of descent data
as sheaves $E$ on $ Z \lefttwoarrow Z \times_{\mathcal{C}} Z ;$ that is, $Z$
together with isomorphisms $\varphi: \pi_1^* E \to \pi_2^* E,$ where
$\pi_1,\pi_2$ are the projections onto the factors of $Z
\times_{\mathcal{C}} Z$ see
\href{http://stacks.math.columbia.edu/tag/03O6}{Tag 03O6} in
\cite{dejong:stacks}. A principal $G$-bundle is given via an
embedding $GL(r)$ as descent data for a locally free sheaf of rank
$r$ together with a reduction given by a section of the associated
$GL(r)/G$ bundle and an isomorphism $\varphi: \pi_1^* E \to \pi_2^*
E$ preserving the $G$-reduction. Because the splitting type of $P
\to \mathcal{C}$ is bounded, the splitting type of $E$ is bounded as well,
and the image of $P$ in $ \on{Hom}(Z,BG)$ is contained in a substack of
finite type. Any such substack may be realized by standard
constructions via quot scheme techniques as a quotient of a variety
by a reductive group action as in \cite[2.3.4]{lieblich:rem}. Since
the homology class $d \in H_2^G(X,\mathbb{Q})$ is bounded, the image of $S$
in $ \on{Hom}(Z,X/G)$ consists of sections with bounded homology class,
and so also lies in a substack of finite type by \cite[Theorem
1.4.1]{abramovich:compactifying}, see also Lieblich
\cite{lieblich:rem}. It follows that $\overline{\M}_{n,1}^G(C,X)$ is
covered by finitely many stacks of finite type, and so is itself
finite type. \end{proof}
\section{Universal closure}
In this section we show that the moduli stack of scaled gauged maps is universally closed using the valuative criterion and Schmitt's git construction \cite{schmitt:univ}.
\subsection{Removal of singularities for bundles on surfaces}
We begin with the following theorem of J.-L. Colliot-Th\'el\`ene and J.-J. Sansuc, \cite{ciollot} describes extensions of bundles on complements of finite subsets of surfaces, see also Ciocan-Fontanine-Kim-Maulik \cite{cf:st}: For any scheme $X$ and reductive group $G$ a {\em principal $G$-bundle} is a scheme $P$ with a free right action of $G$ that is locally trivial in the fpqc topology on $X$. If $X$ is smooth, then this is equivalent to local triviality in the \'etale topology.
\begin{theorem} \label{extend} Let $X$ be a smooth complex variety of dimension two and $G$ a connected reductive group. If $U \subset X$ is the complement
of a finite set of non-singular points on $X$, then any principal
$G$-bundle on $U$ is the restriction of a principal $G$-bundle on
$X$, unique up to isomorphism. \end{theorem}
We briefly recall the main point of the proof. First, we prove the corresponding result for vector bundles. let $F \to U$ be a locally free sheaf and $i: F \to X$ the inclusion. The sheaf $i_* F \cong i_*F^{\vee \vee}$ is reflexive by e.g. Hartshorne \cite[Corollary
1.7]{har:ref}, and reflexive sheaves on surfaces are locally free by e.g. Hartshorne \cite[Corollary 1.4]{har:ref}. This shows that $F$ has an extension. The extension is unique up to isomorphism by Hartog's theorem. If $F_1,F_2 \to X$ are two such locally free extensions then the given isomorphism $\varphi \in H^0(U, \on{Hom}(F_1,F_2))$ extends over $X$ since $X - U$ is a finite set.
Next we consider the case of arbitrary reductive groups. Fix an embedding $G \to GL(r)$ for some $r \ge 1$ and let $P \to U$ be a principal $G$-bundle. Using the embedding, we obtain a vector bundle $E$, which extends uniquely to $X$. Let $Q$ denote the frame bundle of $F$. The bundle $P$ corresponds to a reduction of structure group $\sigma: U \to Q/P$. Since $G$ is connected reductive, it follows from Matsushima's criterion \cite{mats}, \cite{bb} that the homogeneous space $GL(r)/G$ is a smooth affine variety. Moreover, by assumption, $Q/G$ admits a section $\sigma$ over $U$. Thus, by Hartogs' theorem, $\sigma$ extends over $X$.
\subsection{Existence of limits for families with finite scaling}
First we show properness over the space of finite scalings using Schmitt's git construction \cite{schmitt:univ} the Keel-Mori theorem \cite{km:quot}.
\begin{lemma} \label{forprop} The forgetful morphism to the moduli space of finite scalings
$$ \overline{\M}_{n,1}^G(C,X,d) \supset \rho^{-1}(\M_{0,1}) \to {\M}_{0,1} \subset \overline{\M}_{0,1}, \quad [\hat{\mathcal{C}},u, P, \ul{z}, \delta] \mapsto \delta $$
is proper. \end{lemma}
\begin{proof}
This follows an argument given with Gonz\'alez \cite{cross}: there
is a proper relative Givental map described in Popa-Roth
\cite{po:stable}
$$\overline{\M}^G(C,X,d) \to\overline{\M}^{G,\on{quot}}(C,X,d).$$
For each fixed bundle, this map collapses bubbles of the section $u$
and replaces them with base points with multiplicity given by the
degree of the bubble tree. On the other hand,
$\overline{\M}^{G,\on{quot}}(C,X,d)$ has a git construction given in
\eqref{schmittgit} and so has a proper coarse moduli space. Finally
$\overline{\M}_n^G(C,X,d) \to \overline{\M}^G(C,X,d)$ is proper, each forgetful
map being isomorphic to a universal curve. Under the
stable=semistable assumption, the Luna slice theorem
\cite{luna:slice} implies that $\overline{\M}_n^{G,\on{quot}}(C,X,d)$ is
\'etale-locally the quotient of a smooth variety by a finite group
and so has finite inertia stack. By the Keel-Mori theorem
\cite{km:quot}, explicitly stated in \cite[Theorem 1.1]{conrad:kl},
the morphism from $\overline{\M}_n^{G,\on{quot}}(C,X,d)$ to its coarse moduli
space is proper, so $\overline{\M}_n^G(C,X,d)$ is proper as well. Hence
$$ \rho^{-1}(\mathbb{C}) \cong \overline{\M}_n^G(C,X,d) \times \mathbb{C} \to \mathbb{C} $$
is proper. Schmitt's construction \cite[Section 2.7.2]{schmitt:git} implies that if stable=semistable then the automorphism groups of objects in $\overline{\M}^{G,\on{quot}}_n(C,X,d)$ are finite, and so the moduli stack is Deligne-Mumford. Since quot schemes are projective, and moduli spaces of stable maps to projective schemes are projective, the moduli spaces $\overline{\M}^{G,\on{quot}}_n(C,X,d)$ have projective coarse moduli spaces and so are proper. \end{proof}
Next we show the valuative criterion for universal closure in the case that the scalings go to infinity. This is a combination of properness for stable maps to the targets, its quotient, and removal of singularities for bundles on surfaces.
\begin{theorem} \label{finite} Given a family of scaled Mundet-semistable gauged maps over a punctured curve $S$ with finite scaling $\delta$
$$(P,\hat{C}, u, \ul{z},\delta) \to S = \overline{S} - \{ \infty \} $$
there exists an extension over $\overline{S}$, after \'etale cover. \end{theorem}
\begin{proof} It suffices, by the Lemma \ref{forprop}, to consider the case that the scaling $\delta$ becomes infinite. We first consider the case that $\hat{\mathcal{C}} \cong C$. There are three steps, in which we construct the central fiber curve and a scaled gauged map by stages. First we construct a limit
$$\hat{\mathcal{C}}_\infty^{X \qu G} \to C, \quad v_\infty^{X \qu G} : \hat{\mathcal{C}}_\infty^{X \qu G} \to X \qu G, \quad \ul{z}_\infty \subset \hat{\mathcal{C}}_\infty^{X \qu G} $$
by properness of the moduli space of stable maps to $X \qu G$. However, the limiting domain $\hat{\mathcal{C}}_\infty^{X \qu G}$ is not the one we want because there may be bubbling in $X$ that is not captured by bubbling in $X \qu G$. Forgetting some of the components of $\hat{C}_\infty^{X \qu G}$ and using removal of singularities for bundles on surfaces gives a curve and map
$$\hat{\mathcal{C}}_{BG} \to C, \quad \phi: \hat{\mathcal{C}}_{BG} \to BG $$
where the map $\phi$ is a classifying map for an extension of the bundle $P$ over $\hat{\mathcal{C}}_{BG}$. Then we apply properness of the moduli stack of sections of $P(X)$ to obtain the desired limiting curve
$$\hat{\mathcal{C}}_\infty \to C, \quad u_\infty: \hat{\mathcal{C}}_\infty \to P(X), \quad \delta_\infty: \hat{\mathcal{C}}_\infty \to \mathbb{P}(\omega_{\hat{\mathcal{C}}_\infty/C} \oplus \mO_{\hat{\mathcal{C}}_\infty}), \quad \ul{z}_\infty \subset \hat{\mathcal{C}}_\infty, .$$
Here are the details:
\vskip .1in \noindent {\em Step 1: Construct the part with infinite
scaling.} We first introduce the following notation for the maps to the git quotient. By definition of $k(d)$ and Lemma \ref{genlem}, the maps $u: C \to P(X)$ are generically semistable, and so defines a curve and map
$$C^{X \qu G} := u^{-1} P(X^{\on{ss}}) \neq \emptyset, \quad u^{X \qu G} = (u^{X \qu G}: C^{X \qu G} \to X \qu G) .$$
By properness of $X \qu G$, $u^{X \qu G}$ extends to a family of stable maps with domain $C \times S$. Order the base points so that they give sections
$$ \zeta_i: S \to C, \quad \zeta_i(S) \subset P(X^{\on{us}}), i = 1,\ldots, l.$$
Denote by $\ul{z} \cup \ul{\zeta}: S \to C^{n+ l}$ the family of sections obtained by adding the base points and removing duplicates; that is, after restricting to an open subvariety we may assume that any two sections that coincide in one fiber, coincide everywhere; then we just remove one of the duplicate sections. Because the domain is irreducible, the datum $ (C, u^{X \qu G}, \ul{z} \cup \ul{\zeta},\delta )$ is a stable scaled map to the smooth Deligne-Mumford stack $X \qu G$.
Properness for the moduli space of scaled maps with trivial group action in Proposition \ref{trivaction} implies that the family extends over the central fiber: Since $\overline{\M}_{n,1}(C, X \qu G,d)$ is proper, the map $u^{X \qu G}$ extends over the central fiber to a stable scaled map
$$ \left( \hat{\mathcal{C}}_\infty^{X \qu G} \to C, \quad u^{X \qu G}_\infty: \hat{\mathcal{C}}_\infty^{X \qu G} \to X \qu G, \quad \ul{z}_\infty \cup \ul{\zeta}_\infty, \quad \delta_\infty \right) .$$
In particular, the markings $\ul{\zeta}_\infty$ lie on the locus of $\hat{\mathcal{C}}_\infty^{X \qu G}$ with finite scaling $\hat{\mathcal{C}}_\infty^{X
\qu G} - \delta_\infty^{-1}(D_\infty)$.
\vskip .1in \noindent {\em Step 2: Construct the part with finite
scaling.} Let $\Gamma$ be the combinatorial type of the limit $\hat{\mathcal{C}}_\infty^{X \qu G}$ in the previous step and $\Gamma'$ the combinatorial type obtained by forgetting the components of
$\hat{\mathcal{C}}_\infty^{X \qu G}$ on which the scale $\delta | \hat{\mathcal{C}}_\infty^{X \qu G}$ is zero. More precisely, choose a family of sections $ S \to (\hat{\mathcal{C}}_\infty^{X \qu G})^k $ taking values in the locus with non-zero scaling with the property that the components with non-zero scaling become stable. By e.g. Behrend-Manin \cite[Lemma 3.12]{bm:gw}, there exists a proper family $\hat{\mathcal{C}}_\infty^{BG}$ of stable curves with a morphism from $\hat{\mathcal{C}}_\infty^{X\qu G}$ collapsing the components with zero scaling. The family $\hat{\mathcal{C}}_\infty^{BG}$ consists of a collection of components on which the scaling is finite and non-zero, or infinite, with a morphism
$$ \varphi: \hat{\mathcal{C}}_\infty^{X \qu G}\to \hat{\mathcal{C}}_\infty^{BG} .$$
The scaling $\delta_\infty$ is finite at the base points $\ul{\zeta}_\infty$ and markings $\ul{z}_\infty$. The image of the base points $\ul{\zeta}_\infty$ under the morphism $\varphi$ are denoted
$\ul{\zeta}_\infty^{BG}= \varphi(\ul{\zeta}_\infty) .$
The points $\ul{\zeta}_\infty^{BG}$ are no longer necessarily distinct from each other and the markings. Because the scalings $\delta^{X \qu
G}_\infty$ are finite at $\ul{\zeta}_\infty$, the scalings $\delta^{BG}_\infty$ are finite at $\ul{\zeta}_\infty^{BG}$, that is, $ \delta^{BG}_\infty(\ul{\zeta}_\infty^{BG}) < \infty .$ In particular, all of the points $\ul{\zeta}_\infty^{BG}$ are non-singular, since the only nodes in $\hat{\mathcal{C}}_\infty^{BG}$ are contained in $(\delta^{BG}_\infty)^{-1}(\infty)$.
Removal of singularities for bundles on surfaces implies that the bundle extends over the central fiber. The morphism $u^{X \qu G}$ induces an extension of the bundle $P_\infty^{BG}$ over the complement of the base points $\ul{\zeta}^{BG}_{\infty}$, given by pull-back of
$$ P_\infty^{BG} \to \hat{\mathcal{C}}_\infty^{BG} -
\ul{\zeta}_\infty^{BG}, \quad P_\infty^{BG} := (u^{ X\qu G}_\infty | \hat{\mathcal{C}}_\infty^{BG} - \ul{\zeta}_\infty^{BG})^* (X^{\on{ss}} \to X^{\on{ss}}/G)$$
under $u^{X \qu G}_\infty$. By construction, the points $\zeta_{i,\infty}$ are non-singular points in $\hat{C}^{BG}_\infty$. By removal of singularities for bundles Theorem \ref{extend}, the bundle $P^{BG} \to \hat{\mathcal{C}}^{BG}$ given by $P_\infty^{BG}$ over the central fiber has a unique extension over the points $\zeta_{i,\infty}$. This implies the existence of a limiting bundle $P_\infty \to \hat{\mathcal{C}}_\infty^{BG}$ with classifying map
$$ \phi_\infty : \hat{\mathcal{C}}^{BG}_\infty \to BG, \quad P_\infty := \phi_\infty^*(EG \to BG). $$
Denote by $\hat{C}^{BG}$ the resulting family over $\mathbb{C}^\times$, and $P$ the resulting bundle over $\hat{\mathcal{C}}^{BG}$.
\vskip .1in \noindent {\em Step 3: Construct the full limit.} In the last step we apply properness for the moduli stack of stable sections. The associated fiber bundle $P(X) \to \hat{\mathcal{C}}^{BG}$ is projective, since $X$ is projective and $\hat{\mathcal{C}}^{BG}$ is projective. Then it follows from properness of stable maps to $P(X)$ that there exists a limit $u_\infty : \hat{\mathcal{C}}_\infty \to P(X)$ extending $u$. The scaling naturally extends to a scaling $\delta_\infty$, possibly after adding additional components with finite scaling and trivial maps.
We check that the limit constructed above satisfies the axioms of a stable scaled gauged map. The monotonicity condition on the scalings is guaranteed by the description of the one-form in \eqref{oneform}. Furthermore, on the locus $\delta^{-1}(D_\infty)$, the map $u_\infty $ agrees with the pull-back of $u_\infty^{X \qu G}$ and so takes values in the semistable locus. The locus $\delta^{-1}(D_0)$ is a union of components that map to points in $\hat{\mathcal{C}}_\infty^{BG}$. This implies that bundle $P$ is trivial on $\delta^{-1}(D_0)$. Finally, the inequality $\delta(z_{i, \infty}) < \infty$ is automatically satisfied since the scaling on $\hat{\mathcal{C}}_\infty^{X \qu G}$ is finite at the markings, and the forgetful map maps all components with zero scaling to loci where the scaling is finite. Each component on which the scaling and gauged are trivial has at least three special points, since the limit $u_\infty$ is a stable section. Each component with finite, non-zero scaling has at least two special points contains either the limit of a marking or a base point. If trivial such a component is attached to a component with trivial scaling, and so has at least two special points. Finally each component with infinite scaling occurs in the domain $u^{X \qu G}$ and so has at least three special points.
Finally we consider the general case that domains of the family are nodal. That is, we have a family of gauged maps $(P,\hat{\mathcal{C}}, u, \ul{z},\delta) \to S = \overline{S} - \{ \infty \} $ such that every $\hat{\mathcal{C}}_s$ is a nodal projective curve. In this case we repeat the first two steps for the family obtained by restricting to the root component $\hat{\mathcal{C}}_0 \subset \hat{\mathcal{C}}$. In the last stage, properness for stable maps to $P(X)$ implies the existence of a limit of $u: \hat{\mathcal{C}} \to P(X)$. \end{proof}
Almost exactly the same argument shows that the moduli stack of affine gauged maps is universally closed:
\begin{lemma} \label{infinite} Given a family of stable affine gauged maps over a punctured curve $S$,
$$(P,\hat{\mathcal{C}}, u, \ul{z},\delta) \to S := \overline{S} - \{ \infty \} $$
there exists an extension over $\overline{S}$, after \'etale cover. \end{lemma}
We use this and the statement of the Lemma above, which dealt with irreducible domain, to prove universal closure of the moduli space of scaled gauged maps:
\begin{theorem} The moduli stack $\overline{\M}_n^G(C,X)$ is universally closed. \end{theorem}
\begin{proof} Let $(P,\hat{C}, v, \ul{z},\delta) \to S = \overline{S} - \{ \infty \}$ be a family of scaled gauged maps over a curve $S$. We may assume that either $\delta$ is finite or infinite on the root component $\hat{C}_0$ for all $s \in S$, after possibly replacing $S$ with an open subscheme. In the finite case, the existence of a central extension over the central fiber follows from \ref{finite}. In the infinite case, the central fiber is a collection of affine gauged maps and maps to $X \qu G$ by gluing at the nodes. That is, the curve $\hat{C}$ is a union of components $\hat{C} = \hat{C}_0 \cup \hat{C}_1 \cup \ldots \hat{C}_r $ where $\hat{C}_0$ is a family of curves with infinite scaling and $\hat{C}_1,\ldots, \hat{C}_r$ are families of affine scaled curves. By Lemma \ref{infinite}, the restriction of the families $(P,\hat{C}, v, \ul{z},\delta)$ to $\hat{C}_1,\ldots, \hat{C}_r$ have extension over the central fiber. Similarly properness of the moduli stack of stable maps to $X \qu G$ implies the existence of a limit of the restriction of the family to $\hat{C}_0$. By closure of the diagonal, these families glue together to a scaled gauged map on the central fiber.
\end{proof}
\section{Separation}
In this section we check the valuative criterion for separatedness. This is again a combination of separatedness of the moduli stack of stable maps to the target, its quotient, and uniqueness of extensions on bundles on surfaces.
\begin{proposition} For $i = 0,1$ let $v^i := (\hat{C}^i \to \overline{S}, P^i \to \hat{C}^i, u^i: \hat{C}^i \to X / G, \delta^i, \ul{z}^i: S \to \hat{C}^{i,n}) $ be families of stable scaled gauged maps over a curve $\overline{S}$ that are isomorphic over the punctured curve $S = \overline{S} - \{ \infty \}$. Then $v^0$ is isomorphic to $v^1$ over $\overline{S}$. \end{proposition}
\begin{proof} Again this follows from a three-step process, in which we show that the maps agree on parts of the limit corresponding to infinite, finite, and zero scaling.
\vskip .1in \noindent {\em Step 1: The maps agree on the part with
non-zero scaling.} First we introduce for notation the induced map to the git quotient. Let $ \hat{\mathcal{C}}_\infty^{i,X \qu G}$ denote the union of components $\hat{\mathcal{C}}^i_{\infty} $ with infinite or finite, non-zero scaling $\delta_\infty^i | \hat{\mathcal{C}}^i_{\infty}$. The inverse image of the semistable locus $X \qu G$ is dense in $\hat{\mathcal{C}}_\infty^{i,X \qu G}$ by Lemma \ref{genlem}. By properness of the stack $X \qu G$ we obtain maps
$$ u_\infty^{i,X \qu G} : \hat{\mathcal{C}}_\infty^{i,X \qu G} \to X \qu G .$$
We would like to apply separatedness for the moduli stack of stable scaled maps to the git quotient to show that these maps are isomorphic. However, the maps $u_\infty^{i, X \qu G}$ may not be stable since they may contain unstable components. To remedy this, choose an ample invariant divisor $D \subset X$ as in \eqref{invsection}. Let $\ul{\zeta}^i = u^{-1}(P(D)) $ denote the points mapping to $P(D)$. Consider the scaled curve to $X \qu G$ given by $ (u^{i,X \qu G} , \delta^i, \ul{z}^i \cup \ul{\zeta}^i) $ where $\ul{z}^i \cup \ul{\zeta}^i$ denotes the union obtained by adding the intersection points with the divisor and deleting duplicates and nodes. After restricting to an open subscheme of $S$ containing the central fiber, we may assume that $\ul{z}^i \cup \ul{\zeta}^i$ are distinct, non-singular points.
We claim that the tuples constructed in the previous paragraph are stable scaled maps. Any component of $\hat{\mathcal{C}}^i$ with finite, non-zero scaling that becomes a unstable component after passing to $X \qu G$ must either be stable, or correspond to a map to $X/ G$ that lies generically in a fiber of $X^{\on{ss}} \to X^{\on{ss}}/G$. Such maps always intersect $D$ since $D$ is ample. Hence any such component has at least two special points and a non-trivial scaling, and so is stable.
Separation for the moduli stack of stable scaled maps to the git quotient implies that the central fibers are isomorphic: By Proposition \ref{trivaction} there exists an isomorphism
\begin{multline} (u_\infty^{0,X \qu G} : \hat{\mathcal{C}}_\infty^{0,X \qu G} \to X \qu G , \quad
\delta^0, \ul{z}^0 \cup \ul{\zeta}^0 ) \\ \cong (u_\infty^{1,X \qu
G} : \hat{\mathcal{C}}_\infty^{1,X \qu G} \to X \qu G , \quad \delta^1,
\ul{z}^1 \cup \ul{\zeta}^1 ) .\end{multline}
\vskip .1in \noindent {\em Step 2: The limits agree on the part with
non-zero scaling.} This step is an application of uniqueness of removal of singularities for bundles on surfaces in Theorem \ref{extend}. Let $\hat{\mathcal{C}}_{\infty}^{i,X \qu G} $ denote the curves from Step 1. The classifying maps
$ \phi_i: \hat{\mathcal{C}}_{\infty}^{i,X \qu G} \to BG $
are isomorphic (that is, the corresponding bundles are isomorphic) except at finitely many non-singular points, the base points, since the maps $u_i^{X \qu G}$ agree. Furthermore, the base points $\ul{\zeta}_i$ are contained in components of $\hat{\mathcal{C}}_\infty^{i,X
\qu G}$ with infinite scaling, since $D$ contains the unstable locus $X^{\on{us}}$. By uniqueness of the completion of bundles on surfaces in Theorem \ref{extend}, the bundles $P_0,P_1$ are isomorphic over $\hat{\mathcal{C}}^{i,X \qu G} $.
\vskip .1in \noindent {\em Step 3: The limits agree entirely.} Finally we apply separatedness for families of stable sections to show that the limiting sections are isomorphic. Separation of stable maps to $P_\infty(X)$ where
$ P_\infty : = P_i |_{\hat{\mathcal{C}}_{\infty}^{i,X \qu G} } $
implies that there exists an isomorphism,
$$ (u^0 : \hat{\mathcal{C}}^{0} \to P_\infty(X) , \ul{z}^0 ) \cong (u^1 : \hat{\mathcal{C}}^{1} \to P_\infty(X), \ul{z}^1 ) .$$
Since the scaled curves appearing in the limit already agree, this implies that the stable scaled maps
$$(\hat{C}^i \to \overline{S}, P_i \to \hat{C}^i, u^i: \hat{C}^i \to P_i(X), \delta^i, \ul{z}^i: \overline{S} \to \hat{C}^{i,n}) , \quad i = 0,1$$
are isomorphic.
The existence of a unique limit in the case of a family with infinite scaling is similar and left to the reader. \end{proof}
This proves the valuative criterion for separatedness. Universal closure and of finite-type was shown in previous sections. This completes the proof of properness of $\overline{\M}_{n,1}^G(C,X,d)$ in Theorem \ref{main}. Finally we complete the proof of the properness of moduli stacks of affine gauged maps.
\begin{corollary} For any $E > 0$, the union of moduli stacks of affine gauged maps $\overline{\M}^G_{n,1}(\mathbb{A},X,d)$ with $(d, c_1^G(\tilde{X})) \leq E$ is proper. \end{corollary}
\begin{proof} The proof is by an embedding argument. The stack
$\overline{\M}^G_{n,1}(\mathbb{A},X,d)$ embeds in $\overline{\M}^G_n(C,X,y)$ as
follows: Given an affine gauged map $(C_0,\delta,\ul{z},u)$ and a
point $z \in C$ define $\hat{\mathcal{C}} := (C_0 \sqcup C)/(z_0 \sim z)$
and extend the map $u$ so that it is constant on the root component
$\hat{\mathcal{C}}_0 \cong C$. Since any closed substack of a proper stack
is proper, $\overline{\M}^G_{n,1}(\mathbb{A},X,d)$ is proper. \end{proof}
\begin{remark} {\rm (Convex targets)} We conclude by describing results for convex targets: For the moduli stack of gauged maps to a convex variety $X$ defined in Section \ref{convex}, the conclusion of Theorem \ref{main} also holds. As explained in Corollary \ref{disjoint}, Mundet stable maps to $X$ are equivalent to maps to $\overline{X}$ as long as the linearization $\tilde{X}$ is chosen so that the linearization is obtained from $\tilde{X}(l)$ for $l$ sufficiently large.
Conversely, for any class $d$ which pairs trivially with the class of the divisor at infinity, affine gauged maps to $\overline{X} \qu G$ and maps to $X \qu G$ of class $d$ are equivalent: the intersection number between the map and divisor is zero and any such map cannot have a component mapping entirely to the divisor. In the case of maps to the quotient, any point in $\overline{X}$ is unstable for $\tilde{X}(l)$ for $l$ sufficiently large, and so $\overline{X} \qu G$ is isomorphic to $X \qu G$. It follows that the inclusion $\overline{\M}_{n,1}^G(C,X) \to \overline{\M}_{n,1}^G(C,\overline{X})$ is an isomorphism, as claimed, and in particular the union of components $\overline{\M}_{n,1}^G(C,X,d)$ with $\mE(d) < E$ is proper for any energy bound $E > 0$. \end{remark}
\section{Table of notation} \label{table}
This section contains a table of the notations for the different moduli stacks of stable maps to quotient stacks used in the paper.
\begin{center}
\begin{tabular}{l|l|l} Notation & Moduli stack & Page Number \\ \hline
$\overline{\M}_{g,n}(X)$ & Stable maps of genus $g$ with $n$ markings & \pageref{mgn} \\ $\overline{\M}^G_n(C,X)$ & Mundet-semistable gauged maps with $n$ markings & \pageref{mss} \\ $ \overline{\M}_n(C, X \qu G)$ & Stable sections of $C \times X \qu G \to C$ & \pageref{graphs} \\ $\overline{\M}^{G,\on{quot},{\on{lev}}}_n(C,X)$ & Gauged maps with level structure & \pageref{level} \\ $\overline{\M}^{G,\on{quot}}_n(C,X)$ & Quot-scheme compactification of gauged maps & \pageref{quots} \\ $\overline{\M}^G_{n,1}(\mathbb{A},X)$ & Scaled affine gauged maps & \pageref{affinegauged} \\ $\overline{\M}^G_{n,1}(C,X)$ & Scaled gauged maps with domain $C$, $n$ markings & \pageref{scaledgauged} \\ \end{tabular} \end{center}
\def$'${$'$} \def$'${$'$} \def$'${$'$} \def$'${$'$} \def$'${$'$} \def$'${$'$} \def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
\lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}} \def$'${$'$} \def$'${$'$}
\end{document}
\end{document} | arXiv |
\begin{document}
\date{} \title{A closed formula for the generating function of $p$-Bernoulli numbers} \author{Levent Karg\i n$^{1}$ and Mourad Rahmani$^{2}$\\$^{1}$Akseki Vocational School, Alanya Alaaddin Keykubat University,\\Antalya TR-07630, Turkey\\$^{2}$USTHB, Faculty of Mathematics, P. O. Box32, El Alia, \\Bab Ezzouar, 16111, Algiers, Algeria\\[email protected] and [email protected]} \maketitle
\begin{abstract} In this paper, using geometric polynomials, we obtain a generating function of $p$-Bernoulli numbers. As a consequences this generating function, we derive closed formulas for the finite summation of Bernoulli and harmonic numbers involving Stirling numbers of the second kind.
\textbf{2000 Mathematics Subject Classification: }11B68, 11B83
\textbf{Key words: }$p$-Bernoulli number, geometric polynomial, harmonic number.
\end{abstract}
\section{Introduction}
Rahmani \cite{Rahmani} introduced $p$-Bernoulli numbers by constructing an infinite matrix as follows:
The first row of the matrix $B_{0,p}=1$ and each entry $B_{n,p}$ is given recursively by \[ B_{n+1,p}=pB_{n,p}-\frac{\left( p+1\right) ^{2}}{p+2}B_{n,p+1}. \] The first column of the matrix $B_{n,0}=B_{n}$. Here, $B_{n}$ is the $n$th Bernoulli number.
For every integer $p\geq-1$, these numbers have an explicit formula \begin{equation} B_{n,p}=\sum_{k=0}^{n}\left( -1\right) ^{k}
\genfrac{\{}{\}}{0pt}{}{n}{k}
\binom{k+p+1}{k}^{-1}k!, \label{1} \end{equation} and are closely related to Bernoulli numbers by the following formula \begin{equation} B_{n,p}=\frac{p+1}{p!}\sum_{j=0}^{p}\left( -1\right) ^{j}
\genfrac{[}{]}{0pt}{}{p}{j}
B_{n+j}, \label{11} \end{equation} where $
\genfrac{[}{]}{0pt}{}{n}{k}
$ is the Stirling number of the first kind \cite{Graham}. The first few generating functions for $B_{n,p}$ ($p=1,2$) are \begin{align*} \sum_{n=0}^{\infty}B_{n,1}\frac{t^{n}}{n!} & =\frac{2\left[ \left( t-1\right) e^{t}+1\right] }{\left( e^{t}-1\right) ^{2}},\\ \sum_{n=0}^{\infty}B_{n,2}\frac{t^{n}}{n!} & =\frac{3\left[ \left( 2t-3\right) e^{2t}+4e^{t}-1\right] }{2\left( e^{t}-1\right) ^{3}}. \end{align*}
The main purpose of this study is to give a close form of the above results as \begin{equation} \sum_{n=0}^{\infty}B_{n,p}\frac{t^{n}}{n!}=\frac{\left( p+1\right) \left( t-H_{p}\right) e^{pt}}{\left( e^{t}-1\right) ^{p+1}}+\left( p+1\right) \sum_{k=1}^{p}\binom{p}{k}\frac{H_{k}}{\left( e^{t}-1\right) ^{k+1}}, \label{4} \end{equation} where $H_{n}$ is the harmonic numbers, defined by \cite[p. 258]{Graham} \[ H_{n}=\sum_{j=1}^{n}\frac{1}{j}. \] As a consequences of (\ref{4}), we have closed formulas for the finite summation of Bernoulli and harmonic numbers.
For the proof of (\ref{4}), we use some properties of geometric polynomials. The geometric polynomials are defined by means of the following generating function \cite{T} \begin{equation} \frac{1}{1-x\left( e^{t}-1\right) }=\sum_{n=0}^{\infty}w_{n}\left( x\right) \frac{t^{n}}{n!}\text{,} \label{10} \end{equation} and have the explicit formula \begin{equation} w_{n}\left( x\right) =\sum_{k=0}^{n}
\genfrac{\{}{\}}{0pt}{}{n}{k}
k!x^{k}, \label{2} \end{equation} where $
\genfrac{\{}{\}}{0pt}{}{n}{k}
$ is the Stirling number of the second kind \cite{Graham}. The Stirling numbers of the second kind are defined by means of the following generating function \begin{equation} \sum_{n=0}^{\infty}
\genfrac{\{}{\}}{0pt}{}{n}{k}
\frac{t^{n}}{n!}=\frac{\left( e^{t}-1\right) ^{k}}{k!}. \label{6} \end{equation}
Some other properties of geometric polynomials can be found in \cite{B, B3, B4, BD, Diletal, Kargin}.
\section{A new generating function for $p$-Bernoulli numbers}
In this section, the main theorem and its applications are given.
Now, we give the main theorem of this paper.
\begin{theorem} \label{teo1}For $p\geq0$, the following generating function holds true: \begin{equation} \sum_{n=0}^{\infty}B_{n,p}\frac{t^{n}}{n!}=\frac{\left( p+1\right) \left( t-H_{p}\right) e^{pt}}{\left( e^{t}-1\right) ^{p+1}}+\left( p+1\right) \sum_{k=1}^{p}\binom{p}{k}\frac{H_{k}}{\left( e^{t}-1\right) ^{k+1}}. \label{5} \end{equation}
\end{theorem}
For the proof of main theorem, we first need the following proposition.
\begin{proposition} For $n>p\geq0$, we have \begin{equation} \frac{\left( -1\right) ^{p}}{\left( p+1\right) !}B_{n,p} =\underset{p+1\text{ times}}{\underbrace{
{\displaystyle\int\limits_{-1}^{0}}
{\displaystyle\int\limits_{0}^{x_{p}}}
\ldots
{\displaystyle\int\limits_{0}^{x_{3}}}
{\displaystyle\int\limits_{0}^{x_{2}}}
}}w_{n}\left( x_{1}\right) dx_{1}dx_{2}\ldots dx_{p-1}dx_{p}. \label{3} \end{equation}
\end{proposition}
\begin{proof} If we integrate both sides of (\ref{2}) with respect to $x_{1}$ from $0$ to $x_{2}$, we have \[
{\displaystyle\int\limits_{0}^{x_{2}}}
w_{n}\left( x_{1}\right) dx_{1}=\sum_{k=0}^{n}
\genfrac{\{}{\}}{0pt}{}{n}{k}
k!\frac{x_{2}^{k+1}}{k+1}. \] Integrating both sides of the above equation with respect to $x_{2}$ from $0$ to $x_{3}$, we obtain \[
{\displaystyle\int\limits_{0}^{x_{3}}}
{\displaystyle\int\limits_{0}^{x_{2}}}
w_{n}\left( x_{1}\right) dx_{1}dx_{2}=\sum_{k=0}^{n}
\genfrac{\{}{\}}{0pt}{}{n}{k}
k!\frac{x_{3}^{k+2}}{\left( k+1\right) \left( k+2\right) }. \] Applying the same procedure for $p$ times yields \[
{\displaystyle\int\limits_{0}^{x_{p}}}
\ldots
{\displaystyle\int\limits_{0}^{x_{3}}}
{\displaystyle\int\limits_{0}^{x_{2}}}
w_{n}\left( x_{1}\right) dx_{1}dx_{2}\ldots dx_{p-1}=\sum_{k=0}^{n}
\genfrac{\{}{\}}{0pt}{}{n}{k}
k!\frac{x_{p}^{k+p}}{\left( k+1\right) \cdots\left( k+p\right) }. \] Finally, integrating both sides of the above equation with respect to $x_{p}$ from $-1$ to $0$ and using (\ref{1}) gives the desired equation. \end{proof}
We note that taking $p=0$ in (\ref{3}) gives \cite[Theorem 1.2]{KELLER}.
Now, we are ready to give the proof of the main theorem.
\begin{proof} [Proof of Theorem \ref{teo1}]Multiplying both sides of (\ref{3}) with $\frac{t^{n}}{n!}$ and summing over $n$ from $0$ to infinitive, we have \begin{align*} \frac{\left( -1\right) ^{p}}{\left( p+1\right) !}\sum_{n=0}^{\infty }B_{n,p}\frac{t^{n}}{n!} & =
{\displaystyle\int\limits_{-1}^{0}}
{\displaystyle\int\limits_{0}^{x_{p}}}
\ldots
{\displaystyle\int\limits_{0}^{x_{3}}}
\left[
{\displaystyle\int\limits_{0}^{x_{2}}}
\left( \sum_{n=0}^{\infty}w_{n}\left( x_{1}\right) \frac{t^{n}}{n!}\right) dx_{1}\right] dx_{2}\ldots dx_{p-1}dx_{p}\\ & =
{\displaystyle\int\limits_{-1}^{0}}
{\displaystyle\int\limits_{0}^{x_{p}}}
\ldots
{\displaystyle\int\limits_{0}^{x_{3}}}
\left[
{\displaystyle\int\limits_{0}^{x_{2}}}
\frac{1}{1-x_{1}\left( e^{t}-1\right) }dx_{1}\right] dx_{2}\ldots dx_{p-1}dx_{p}. \end{align*} If we evaluate the first integral, we obtain \[
{\displaystyle\int\limits_{0}^{x_{2}}}
\frac{1}{1-x_{1}\left( e^{t}-1\right) }dx_{1}=\frac{-1}{e^{t}-1}\ln\left( 1-x_{2}\left( e^{t}-1\right) \right) . \] For the second time, we evaluate \begin{align*} & \frac{-1}{e^{t}-1}
{\displaystyle\int\limits_{0}^{x_{3}}}
\ln\left( 1-x_{2}\left( e^{t}-1\right) \right) dx_{2}\\ & \quad=\frac{1}{\left( e^{t}-1\right) ^{2}}\left[ \left( 1-x_{3}\left( e^{t}-1\right) \right) \ln\left( 1-x_{3}\left( e^{t}-1\right) \right) -\left( 1-x_{3}\left( e^{t}-1\right) \right) +1\right] . \end{align*} By induction on $p$, let us assume that the following equation holds \begin{align} &
{\displaystyle\int\limits_{0}^{x_{p}}}
\ldots
{\displaystyle\int\limits_{0}^{x_{3}}}
{\displaystyle\int\limits_{0}^{x_{2}}}
w_{n}\left( x_{1}\right) dx_{1}dx_{2}\ldots dx_{p-1}\label{12}\\ & \quad=\frac{\left( -1\right) ^{p}}{\left( e^{t}-1\right) ^{p}\left( p-1\right) !}\left( 1-x_{p}\left( e^{t}-1\right) \right) ^{p-1}\ln\left( 1-x_{p}\left( e^{t}-1\right) \right) \nonumber\\ & \quad\quad\quad-\frac{\left( -1\right) ^{p}}{\left( e^{t}-1\right) ^{p}\left( p-1\right) !}\left[ H_{p-1}\left( 1-x_{p}\left( e^{t} -1\right) \right) ^{p-1}+H_{p-1}\right] \nonumber\\ & \quad\quad\quad+\sum_{k=1}^{p-2}\frac{\left( -1\right) ^{p-k} H_{p-1-k}x_{p}^{k}}{\left( e^{t}-1\right) ^{p-k}\left( p-1-k\right) !k!}.\nonumber \end{align} Now, we want to prove that (\ref{12}) holds for the case $p+1.$ Let us integrate both sides of (\ref{12}) with respect to $x_{p}$ from $0$ to $y.$ Then we have \begin{align*} &
{\displaystyle\int\limits_{0}^{y}}
{\displaystyle\int\limits_{0}^{x_{p}}}
\ldots
{\displaystyle\int\limits_{0}^{x_{3}}}
{\displaystyle\int\limits_{0}^{x_{2}}}
w_{n}\left( x_{1}\right) dx_{1}dx_{2}\ldots dx_{p-1}dx_{p}\\ & \quad=\frac{\left( -1\right) ^{p}}{\left( e^{t}-1\right) ^{p}\left( p-1\right) !}
{\displaystyle\int\limits_{0}^{y}}
\left( 1-x_{p}\left( e^{t}-1\right) \right) ^{p-1}\ln\left( 1-x_{p}\left( e^{t}-1\right) \right) dx_{p}\\ & \quad\quad-\frac{\left( -1\right) ^{p}H_{p-1}}{\left( e^{t}-1\right) ^{p}\left( p-1\right) !}
{\displaystyle\int\limits_{0}^{y}}
\left( 1-x_{p}\left( e^{t}-1\right) \right) ^{p-1}dx_{p}+\frac{\left( -1\right) ^{p}H_{p-1}}{\left( e^{t}-1\right) ^{p}\left( p-1\right) !}
{\displaystyle\int\limits_{0}^{y}}
dx_{p}\\ & \quad\quad+\sum_{k=1}^{p-2}\frac{\left( -1\right) ^{p-k}H_{p-1-k} }{\left( e^{t}-1\right) ^{p-k}\left( p-1-k\right) !k!}
{\displaystyle\int\limits_{0}^{y}}
x_{p}^{k}dx_{p}. \end{align*} The first integral in the right hand-side equals \begin{align} & \frac{\left( -1\right) ^{p}}{\left( e^{t}-1\right) ^{p}\left( p-1\right) !}
{\displaystyle\int\limits_{0}^{y}}
\left( 1-x_{p}\left( e^{t}-1\right) \right) ^{p-1}\ln\left( 1-x_{p}\left( e^{t}-1\right) \right) dx_{p}\label{13}\\ & \quad=\frac{\left( -1\right) ^{p+1}}{\left( e^{t}-1\right) ^{p+1} p!}\left[ \left( 1-y\left( e^{t}-1\right) \right) ^{p}\ln\left( 1-y\left( e^{t}-1\right) \right) -\frac{\left( 1-y\left( e^{t}-1\right) \right) ^{p}}{p}+\frac{1}{p}\right] .\nonumber \end{align} For the second integral in the right hand-side, we obtain \begin{align} & \frac{\left( -1\right) ^{p}H_{p-1}}{\left( e^{t}-1\right) ^{p}\left( p-1\right) !}
{\displaystyle\int\limits_{0}^{y}}
\left( 1-x_{p}\left( e^{t}-1\right) \right) ^{p-1}dx_{p}\label{14}\\ & \quad=\frac{\left( -1\right) ^{p+1}H_{p-1}}{\left( e^{t}-1\right) ^{p+1}p!}\left( 1-y\left( e^{t}-1\right) \right) ^{p}-\frac{\left( -1\right) ^{p+1}H_{p-1}}{\left( e^{t}-1\right) ^{p+1}p!}.\nonumber \end{align} For the third and fourth integrals, we find \begin{equation} \frac{\left( -1\right) ^{p}H_{p-1}}{\left( e^{t}-1\right) ^{p}\left( p-1\right) !}
{\displaystyle\int\limits_{0}^{y}}
dx_{p}=\frac{\left( -1\right) ^{p+1}H_{p-1}}{\left( e^{t}-1\right) ^{p}\left( p-1\right) !}y \label{15} \end{equation} and \begin{equation} \sum_{k=1}^{p-2}\frac{\left( -1\right) ^{p-k}H_{p-1-k}}{\left( e^{t}-1\right) ^{p-k}\left( p-1-k\right) !k!}
{\displaystyle\int\limits_{0}^{y}}
x_{p}^{k}dx_{p}=\sum_{k=2}^{p-1}\frac{\left( -1\right) ^{p+1-k}H_{p-k}y^{k} }{\left( e^{t}-1\right) ^{p+1-k}\left( p-k\right) !k!}, \label{16} \end{equation} respectively. Combining (\ref{13}), (\ref{14}), (\ref{15}) and (\ref{16}), we achieve that \begin{align*} &
{\displaystyle\int\limits_{0}^{y}}
{\displaystyle\int\limits_{0}^{x_{p}}}
\ldots
{\displaystyle\int\limits_{0}^{x_{3}}}
{\displaystyle\int\limits_{0}^{x_{2}}}
w_{n}\left( x_{1}\right) dx_{1}dx_{2}\ldots dx_{p-1}dx_{p}\\ & \quad=\frac{\left( -1\right) ^{p+1}}{\left( e^{t}-1\right) ^{p+1} p!}\left( 1-y\left( e^{t}-1\right) \right) ^{p}\ln\left( 1-y\left( e^{t}-1\right) \right) \\ & \quad-\frac{\left( -1\right) ^{p+1}}{\left( e^{t}-1\right) ^{p+1} p!}\left[ H_{p}\left( 1-y\left( e^{t}-1\right) \right) ^{p}+H_{p}\right] \\ & \quad+\sum_{k=1}^{p-1}\frac{\left( -1\right) ^{p+1-k}H_{p-k}y^{k} }{\left( e^{t}-1\right) ^{p+1-k}\left( p-k\right) !k!}. \end{align*} Finally, setting $y=-1$ in the above equation and using (\ref{3}), we arrive at the desired equation. \end{proof}
As an application of Theorem \ref{teo1}, we give the following theorem.
\begin{theorem} \label{teo2}For $n>p\geq0$, we have \begin{equation} \sum_{k=p+1}^{n}\binom{n}{k}
\genfrac{\{}{\}}{0pt}{}{k}{p+1}
B_{n-k,p}=\frac{p^{n-1}\left( n-pH_{p}\right) }{p!}+\sum_{j=1}^{p}
\genfrac{\{}{\}}{0pt}{}{n}{p-j}
\frac{H_{j}}{j!}. \label{9} \end{equation}
\end{theorem}
\begin{proof} Multiplying both sides of (\ref{5}) with $\frac{\left( e^{t}-1\right) ^{p+1}}{\left( p+1\right) !}$ and using (\ref{6}), the left hand side of (\ref{5}) becomes \begin{align} \frac{\left( e^{t}-1\right) ^{p+1}}{\left( p+1\right) !}\sum_{n=0} ^{\infty}B_{n,p}\frac{t^{n}}{n!} & =\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}
\genfrac{\{}{\}}{0pt}{}{k}{p+1}
\frac{B_{n,p}}{k!n!}t^{n+k}\label{8}\\ & =\sum_{n=0}^{\infty}\left[ \sum_{k=0}^{n}\binom{n}{k}
\genfrac{\{}{\}}{0pt}{}{k}{p+1}
B_{n-k,p}\right] \frac{t^{n}}{n!}.\nonumber \end{align} For the right hand side of (\ref{5}), we obtain \begin{align} & \frac{te^{pt}}{p!}-\frac{H_{p}e^{pt}}{p!}+\sum_{k=1}^{p}\frac{H_{k}} {k!}\frac{\left( e^{t}-1\right) ^{p-k}}{\left( p-k\right) !}\label{7}\\ & \qquad\qquad=\sum_{n=1}^{\infty}\left[ \frac{p^{n-1}\left( n-pH_{p} \right) }{p!}+\sum_{k=1}^{p}\frac{H_{k}}{k!}
\genfrac{\{}{\}}{0pt}{}{n}{p-k}
\right] \frac{t^{n}}{n!}.\nonumber \end{align} Finally, comparing the coefficients of $\frac{t^{n}}{n!}$ in (\ref{8}) and (\ref{7}) completes the proof. \end{proof}
Using (\ref{11}) in Theorem \ref{teo2} gives the following corollary.
\begin{corollary} \label{cor1}For $n>p\geq0$, \[ \sum_{k=p+1}^{n}\sum_{j=0}^{p}\binom{n}{k}
\genfrac{\{}{\}}{0pt}{}{k}{p+1}
\genfrac{[}{]}{0pt}{}{p}{j}
\left( -1\right) ^{j}B_{n+j-k}=\frac{p^{n-1}\left( n-pH_{p}\right) } {p+1}+\frac{p!}{p+1}\sum_{j=1}^{p}
\genfrac{\{}{\}}{0pt}{}{n}{p-j}
\frac{H_{j}}{j!}. \]
\end{corollary}
As a consequences of Corollary \ref{cor1}, the following sums are obtained $\left( p=1,2\right) :$
\begin{align*} \sum_{k=2}^{n}\binom{n}{k}
\genfrac{\{}{\}}{0pt}{}{k}{2}
B_{n+1-k} & =\frac{-\left( n-1\right) }{2},\\ \sum_{k=3}^{n}\binom{n}{k}
\genfrac{\{}{\}}{0pt}{}{k}{3}
\left( B_{n+2-k}-B_{n+1-k}\right) & =\frac{2^{n-1}\left( n-3\right) +2}{3}. \end{align*}
Setting $n=p+1$ in Theorem \ref{teo2} and using $B_{0,p}=1$, we arrive at the following corollary.
\begin{corollary} For $p\geq1$, we obtain a new closed formula for the finite summation of harmonic numbers \[ \sum_{j=1}^{p}
\genfrac{\{}{\}}{0pt}{}{p+1}{p-j}
\frac{H_{j}}{j!}=\frac{p!-\left( p+1\right) p^{p}+p^{p+1}H_{p}}{p!}. \]
\end{corollary}
\end{document} | arXiv |
Prof. Dr. Camillo De Lellis
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PDE, Geometrische Mass Theorie
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Zurich Colloquium in Mathematics
KO2F150
Rémi Abgrall, Joseph Ayoub, Peter Bühlmann, Marc Burger, Camillo De Lellis, Horst Knörrer
BOOKS and LECTURE NOTES
C. De Lellis
Almgren's center manifold in a simple setting
Lectures held at Park City 9-13 July 2018 PDF
Il teorema di Schlaefli: un invito alla quarta dimensione
The paper, in italian, has appeared first in the journal "il Volterriano". The file which can be downloaded here contains minor modifications and will be published by Rivista dell'UMI. PDF
Il teorema di Liouville ovvero perche' ``non esiste'' la primitiva di exp(x^2)
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Allard's interior regularity theorem: an invitation to stationary varifolds
To appear in the Collections of the CMSA Harvard PDF
Rectifiable sets, densities and tangent measures
Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich. PDF
EMS Publishing House
Errata to ``Lecture Notes on Rectifiable Sets, Densities, and Tangent Measures''
M. Barsanti, F. Conti, C. De Lellis, T. Franzoni
Le Olimpiadi della Matematica. Seconda Edizione
Zanichelli
GEOMETRIC MEASURE THEORY
C. De Lellis, G. De Philippis, J. Hirsch, A. Massaccesi
On the boundary behavior of mass-minimizing integral currents
Boundary regularity of mass-minimizing integral currents and a question of Almgren
C. De Lellis, A. Marchese, E. Spadaro, D. Valtorta
Rectifiability and upper Minkowski bounds for singularities of harmonic Q-valued maps
C. De Lellis, J. Ramic
Min-max theory for minimal hypersurfaces with boundary
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$2$-dimensional almost area minimizing currents
Boll. Unione Mat. Ital. 9 (2016), no. 1, 3–67. PDF
C. De Lellis, E. Spadaro, L. Spolaor
Regularity theory for 2-dimensional almost minimal currents III: blowup
To appear in Jour. Diff. Geom. PDF PDF
Regularity theory for 2-dimensional almost minimal currents II: branched center manifold
Ann. PDE 3 (2017), no. 2, Art. 18, 85 pp. PDF PDF
Regularity theory for 2-dimensional almost minimal currents I: Lipschitz approximation
Trans. Amer. Math. Soc. 370 (2018), no. 3, 1783–1801 PDF PDF
Uniqueness of tangent cones for 2-dimensional almost minimizing currents
Comm. Pure Appl. Math. 70, 1402-1421 PDF PDF
The size of the singular set of area-minimizing currents
Surveys in differential geometry 2016. Advances in geometry and mathematical physics, 1–83, Surv. Differ. Geom., 21, Int. Press, Somerville, MA, 2016. PDF PDF
The regularity of minimal surfaces in higher codimension
Current developments in mathematics 2014, 153–229, Int. Press, Somerville, MA, 2016. PDF PDF
C. De Lellis, F. Ghiraldin, F. Maggi
A direct approach to Plateau's problem
J. Eur. Math. Soc. (JEMS) 19 (2017), no. 8, 2219–2240 PDF
C. De Lellis, M. Focardi, B. Ruffini
A note on the Hausdorff dimension of the singular set for minimizers of the Mumford-Shah functional
Adv. Calc. Var. 7, pp. 539-545, 2014 PDF
C. De Lellis, E. Spadaro
Regularity of area-minimizing currents III: blow-up
Ann. of Math. (2) 183 (2016), no. 2, 577–617. PDF
Regularity of area-minimizing currents II: center manifold
Regularity of area-minimizing currents I: L^p gradient estimates
Geom. Funct. Anal. 24 (2014), no. 6, 1831–1884. PDF
Errata to "Regularity of area-minimizing currents I: L^p gradient estimates"
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Multiple valued functions and integral currents
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 14 (2015), no. 4, 1239–1269. PDF
L. Ambrosio, C. De Lellis, T. Schmidt
Partial regularity for mass-minimizing currents in Hilbert spaces.
J. Reine Angew. Math. 734, 99-144 PDF
C. De Lellis, M. Focardi
Density lower bound estimates for local minimizers of the 2d Mumford-Shah energy
Manuscripta Math. 142 (2013), 215-232. PDF
Higher integrability of the gradient for minimizers of the 2d Mumford-Shah energ
J. Math. Pures Appl. (9) 100 (2013), 391-409. PDF
Hyperbolic equations and SBV functions
Journées équations aux dérivées partielles (2010), Exp. No. 6, 10 p. PDF
Journées équations aux dérivées partielles
Center manifold: a case study
To appear in the Proceedings for the 2009 Conference in honor of De Giorgi and Stampacchia, held in Erice
Discrete and Continuous Dynamical Systems Volume 31, Issue 4, Pages : 1249 - 1272, 2011 PDF
Discrete and Continuous Dynamical Systems
Errata to "Center manifold: a case study"
C. De Lellis, M. Focardi, E. Spadaro
Lower semicontinuous functionals for Almgren's multiple valued functions
Annales Academiae Scientiarum Fennicae vol. 36 pp. 393-410 (2011) PDF
Annales Academiae Scientiarum Fennicae
C. De Lellis, D. Tasnady
The existence of embedded minimal hypersurfaces
J. Differential Geom. 95 (2013), no. 3, 355–388. PDF
Journal of Differential Geometry
Errata to ``The existence of embedded minimal hypersurfaces''
S. Bianchini, C. De Lellis, R. Robyr
SBV regularity for Hamilton-Jacobi equations in R^n.
Arch. Ration. Mech. Anal. 200 (2011) 1003-1021 PDF
Errata to ``SBV regularity for Hamilton-Jacobi equations in R^n''
Q-valued functions revisited
Memoirs of the AMS 211 (2011), no. 991. PDF
Memoirs of the AMS
Errata to "Q-valued functions revisited"
Almgren's Q-valued functions revisited
Proceedings of the International Congress of Mathematicians Hyderabad, India, 2010 PDF
C. De Lellis, F. Pellandini
Genus bounds for minimal surfaces arising from min-max constructions.
J. Reine Angew. Math. 644 (2010), 47–99. PDF
Crelle
A note on Alberti's rank-one theorem.
Transport equations and multi-D hyperbolic conservation laws, 61–74, Lect. Notes Unione Mat. Ital., 5, Springer, Berlin, 2008. PDF
Errata to ``A note to Alberti's rank-one theorem''
C. De Lellis, C. R. Grisanti, P. Tilli
Regular selections for multiple-valued functions.
Ann. Mat. Pura Appl. (4) 183 (2004), no. 1, 79–95. PDF
Ann. Mat. Pura Appl.
T. H. Colding, C. De Lellis
The min-max construction of minimal surfaces.
Surveys in differential geometry, Vol. VIII (Boston, MA, 2002), 75–107, Int. Press, Somerville, MA, 2003. PDF
Errata to "The min-max construction of minimal surfaces"
Some fine properties of currents and applications to distributional Jacobians.
Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), no. 4, 815–842. PDF
Proc. Roy. Soc. Edinburgh
Errata to "Some fine properties of currents and applications to distributional jacobians"
EULER and NAVIER-STOKES EQUATIONS
C. De Lellis, L. Székelyhidi Jr.
On turbulence and geometry: from Nash to Onsager
To appear in the Notices of the AMS PDF
The Onsager theorem
Surveys in differential geometry 2017. PDF
M. Colombo, C. De Lellis, A. Massaccesi
The generalized Caffarelli-Kohn-Nirenberg Theorem for the hyperdissipative Navier-Stokes system
M. Colombo, C. De Lellis, L. De Rosa
Ill-posedness of Leray solutions for the ipodissipative Navier--Stokes equations
T. Buckmaster, C. De Lellis, L. Székelyhidi Jr., V. Vicol
Onsager's conjecture for admissible weak solutions
To appear in CPAM PDF
High dimensionality and h-principle in PDE
Bull. Amer. Math. Soc. (N.S.) 54 (2017), no. 2, 247–282. PDF PDF
The $h$-principle and Onsager's conjecture
Eur. Math. Soc. Newsl. No. 95 (2015), 19–24. PDF
T. Buckmaster, C. De Lellis, P. Isett, L. Székelyhidi Jr.
Anomalous dissipation for 1/5-Hoelder Euler flows
Ann. of Math.
Errata to "Anomalous dissipation for 1/5-Hoelder Euler flows"
T. Buckmaster, C. De Lellis, L. Székelyhidi Jr.
Dissipative Euler flows with Onsager-critical spatial regularity
Comm. Pure Appl. Math. 69 (2016), no. 9, 1613–1670. PDF
Transporting microstructure and dissipative Euler flows
Dissipative Euler Flows and Onsager's Conjecture
J. Eur. Math. Soc. (JEMS) 16 (2014), no. 7, 1467–1505. PDF PDF
Errata to "Dissipative Euler flows and Onsager's conjecture"
A. Choffrut, C. De Lellis, L. Székelyhidi Jr.
Dissipative continuous Euler flows in two and three dimensions
Continuous dissipative Euler flows and a conjecture of Onsager.
European Congress of Mathematics, 13–29, Eur. Math. Soc., Zürich, 2013. PDF
Errata to ``Continuous dissipative Euler flows and a conjecture of Onsager''
Dissipative continuous Euler flows
Inventiones Mathematicae 193, Issue 2 (2013), Page 377-407 PDF
Inventiones
The h-principle and the equations of fluid dynamics.
Bull. Amer. Math. Soc. 49, 347-375, 2012 PDF
Bull. AMS
Errata to "The h-principle and the equations of fluid dynamics"
Y. Brenier, C. De Lellis, L. Székelyhidi Jr.
Weak-strong uniqueness for measure-valued Solutions
Comm. Math. Phys. 305 (2011), 351-361 PDF
Comm. Math. Phys.
On admissibility criteria for weak solutions of the Euler equations.
Arch. Ration. Mech. Anal. 195 (2010), no. 1, 225–260. PDF
Arch. Ration. Mech. Anal.
Errata to "On admissibility criteria for weak solutions of the Euler equations"
Le equazioni di Eulero dal punto di vista delle inclusioni differenziali (Italian)
Boll. Unione Mat. Ital. (9) 1 (2008), no. 3, 873–879 PDF
Boll. UMI
The Euler equations as a differential inclusion.
Ann. of Math. (2) 170 (2009), no. 3, 1417–1436. PDF
C. De Lellis, D. Inauen
$C^{1,\alpha}$ isometric embeddings of polar caps
A. Carlotto, C. De Lellis
Min-max embedded geodesic lines on asymptotically conical surfaces
C. De Lellis, D. Inauen, L. Székelyhidi Jr.
A Nash-Kuiper theorem for $C^{1,\sfrac{1}{5}-\delta}$ immersions of surfaces in $3$ dimensions
To appear in Revista matemática Iberoamericana PDF
C. De Lellis, P. M. Topping
Almost Schur Lemma
Calc. Var. 43 (2012) 347-354 PDF
Calc. Var.
S. Conti, C. De Lellis, L. Székelyhidi Jr.
h-principle and rigidity for C^{1,\alpha} isometric embeddings
Nonlinear Partial Differential Equations
Abel Symposia Volume 7, 2012, pp 83-116 PDF
Proceedings of the Abel Symposium 2010
T. H. Colding, C. De Lellis, W. P. Minicozzi II
Three circles theorems for Schrödinger operators on cylindrical ends and geometric applications.
Comm. Pure Appl. Math. 61 (2008), no. 11, 1540–1602. PDF
C. De Lellis, S. Müller
A $C^0$ estimate for nearly umbilical surfaces.
Calc. Var. Partial Differential Equations 26 (2006), no. 3, 283–296. PDF
Optimal rigidity estimates for nearly umbilical surfaces.
J. Differential Geom. 69 (2005), no. 1, 75–110. PDF
Singular limit laminations, Morse index, and positive scalar curvature.
Topology 44 (2005), no. 1, 25–45. PDF
TRANSPORT EQUATIONS
C. De Lellis, P. Gwiazda, A. Swierczewska-Gwiazda
Transport equation with integral terms
Calc. Var. Partial Differential Equations 55 (2016), no. 5, Paper No. 128, 17 pp. PDF
ODEs with Sobolev coefficients: the Eulerian and the Lagrangian approach.
Discrete Contin. Dyn. Syst. Ser. S 1 (2008), no. 3, 405–426. PDF
DCDS
Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio [after Ambrosio, DiPerna, Lions].
Séminaire Bourbaki. Vol. 2006/2007. Astérisque No. 317 (2008), Exp. No. 972, viii, 175–203. PDF
Séminaire Bourbaki.
G. Crippa, C. De Lellis
Estimates and regularity results for the DiPerna-Lions flow.
Notes on hyperbolic systems of conservation laws and transport equations.
Handbook of differential equations: evolutionary equations. Vol. III, 277–382, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007.
WARNING. THIS PDF IS NOT THE MOST UPDATED VERSION: PDF
Handbook of EDE
Errata to "Notes on hyperbolic systems of conservation laws and transport equations."
L. Ambrosio, C. De Lellis, J. Maly
On the chain rule for the divergence of BV-like vector fields: applications, partial results, open problems.
Perspectives in nonlinear partial differential equations, 31–67, Contemp. Math., 446, Amer. Math. Soc., Providence, RI, 2007. PDF
The chain rule for the divergence of BV-like vector fields.
Hyperbolic problems: theory, numerics and applications. I, 105–112, Yokohama Publ., Yokohama, 2006 PDF
Oscillatory solutions to transport equations.
Indiana Univ. Math. J. 55 (2006), no. 1, 1–13. PDF
Indiana Univ. Math. J.
HYPERBOLIC CONSERVATION LAWS
C. De Lellis, Radu Ignat
A regularizing property of the 2d-eikonal equation
Comm. Partial Differential Equations 40 (2015), no. 8, 1543–1557. PDF
E. Chiodaroli, C. De Lellis, O. Kreml
Surprising solutions to the isentropic Euler system of gas dynamics
Hyperbolic problems: theory, numerics, applications, 1–10, AIMS Ser. Appl. Math., 8, Am. Inst. Math. Sci. (AIMS), Springfield, MO, 2014. PDF
Global ill-posedness of the isentropic system of gas dynamics
Ill-posedness for bounded admissible solutions of the 2-dimensional p--system.
Hyperbolic problems: theory, numerics and applications, 269–278, Proc. Sympos. Appl. Math., 67, Part 1, Amer. Math. Soc., Providence, RI, 2009. PDF
C. De Lellis, F. Golse
A quantitative compactness estimate for scalar conservation laws.
Comm. Pure Appl. Math. 58 (2005), no. 7, 989–998. PDF
Blowup of the BV norm in the multidimensional Keyfitz and Kranzer system.
Duke Math. J. 127 (2005), no. 2, 313–339. PDF
Duke Math. J.
L. Ambrosio, F. Bouchut, C. De Lellis
Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions.
Comm. Partial Differential Equations 29 (2004), no. 9-10, 1635–1651. PDF
Comm. PDE
Errata to "Well-posedness for the for a class of hyperbolic systems of conservation laws in several space dimensions."
L. Ambrosio, C. De Lellis
A note on admissible solutions of 1D scalar conservation laws and 2D Hamilton-Jacobi equations.
J. Hyperbolic Differ. Equ. 1 (2004), no. 4, 813–826. PDF
JHDE
C. De Lellis, F. Otto, M. Westdickenberg
Minimal entropy conditions for Burgers equation.
Quart. Appl. Math. 62 (2004), no. 4, 687–700. PDF
Quart. Appl. Math.
C. De Lellis, T. Rivière
The rectifiability of entropy measures in one space dimension.
J. Math. Pures Appl. (9) 82 (2003), no. 10, 1343–1367. PDF
J. Math. Pures Appl.
Errata to "The rectifiability of entropy measures in one space dimension".
Structure of entropy solutions for multi-dimensional scalar conservation laws.
Arch. Ration. Mech. Anal. 170 (2003), no. 2, 137–184 PDF
C. De Lellis, M. Westdickenberg
On the optimality of velocity averaging lemmas.
Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), no. 6, 1075–1085. PDF
Ann. IHP
Existence of solutions for a class of hyperbolic systems of conservation laws in several space dimensions.
Int. Math. Res. Not. 2003, no. 41, 2205–2220. PDF
CALCULUS OF VARIATIONS
C. De Lellis, F. Ghiraldin
An extension of the identity Det=det.
C. R. Math. Acad. Sci. Paris 348 (2010), no. 17-18, 973–976 PDF
C. R. Math. Acad. Sci. Paris
S. Conti, C. De Lellis
Sharp upper bounds for a variational problem with singular perturbation.
Math. Ann. 338 (2007), no. 1, 119–146. PDF
Math. Ann.
Simple proof of two-well rigidity.
C. R. Math. Acad. Sci. Paris 343 (2006), no. 5, 367–370 PDF
Some remarks on the theory of elasticity for compressible Neohookean materials.
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 3, 521–549. PDF
Annali SNS
Errata to "Some remarks on the theory of elasticity for compressible Neohookean materials"
S. Conti, C. De Lellis, S. Müller, M. Romeo
Polyconvexity equals rank-one convexity for connected isotropic sets in $\Bbb M^{2\times 2}$.
C. R. Math. Acad. Sci. Paris 337 (2003), no. 4, 233–238. PDF
C. De Lellis, F. Otto
Structure of entropy solutions to the eikonal equation.
J. Eur. Math. Soc. (JEMS) 5 (2003), no. 2, 107–145. PDF
Some remarks on the distributional Jacobian.
Nonlinear Anal. 53 (2003), no. 7-8, 1101–1114. PDF
Nonlinear Anal.
An example in the gradient theory of phase transitions.
ESAIM Control Optim. Calc. Var. 7 (2002), 285–289 PDF
ESAIM COCV
L. Ambrosio, C. De Lellis, C. Mantegazza
Line energies for gradient vector fields in the plane
Calc. Var. Partial Differential Equations 9 (1999), no. 4, 327–255. PDF
Fractional Sobolev regularity for the Brouwer degree
Comm. Partial Differential Equations 42 (2017), no. 10, 1510–1523. PDF
John Nash's nonlinear iteration
Forthcoming in Memorial Volume for Professor John Nash, eds. Joseph Kohn and Hong Jun, World Scientific Publishers, 2017 PDF
The masterpieces of John Forbes Nash Jr.
To appear in H. Holden and R. Piene (editors): The Abel Prize 2013–2017. Springer Verlag. PDF
C. De Lellis, R. Robyr
Hamilton-Jacobi equations with obstacles
A. Bressan, C. De Lellis
Existence of optimal strategies for a fire confinement problem.
C. De Lellis, T. Kappeler, P. Topalov
Low-regularity solutions of the periodic Camassa-Holm equation.
Comm. Partial Differential Equations 32 (2007), no. 1-3, 87–126. PDF
C. De Lellis, G. Royer-Carfagni
Interaction of fractures in tensile bars with non-local spatial dependence.
J. Elasticity 65 (2001), no. 1-3, 1–31 (2002). PDF
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Gäste Aufenthalt
Prof. Dr. Simon Brendle, Department of Mathematics, Columbia University (New York)
Talk: A boundary value problem for minimal Lagrangian graphs
20.12.10 - 22.12.10 De Lellis, Camillo
Prof. Dr. Edriss Titi, University of California Irvine (California)
Talk: On the Question of Global Regularity for Three-dimensional Navier-Stokes Equations and Relevant Geophysical Models
Prof. Dr. Franco Maddalena, Politecnico di Bari (Bari)
Talk: Mass Transport Minimizing Branched Structures
Prof. Dr. Miles Simon, Albert-Ludwigs-Universität Freiburg (Freiburg im Breisgau)
Talk: Expanding solitons with non-negative curvature operator coming out of cones
Prof. Dr. Mario Pulvirenti, University La Sapienza (Roma)
Talk: The Cauchy Problem for the 3-D Vlasov-Poisson System with Point Charges
Prof. Dr. Pertti Mattila, University of Helsinki, Dept. of Mathematics (Helsinki)
Dr. Emanuele Spadaro, Max Planck Institute, Leipzig (Leipzig)
Prof. Dr. Iskander Taimanov, Sobolev Institute of Mathematics, Novosibirsk State University (Novosibirsk State University)
Talk: Periodic magnetic geodesics on almost every energy level via variational methods
Luca Granieri,
Prof. Dr. Giovanni Alberti, Department of Mathematics, University of Genova (Genova)
Prof. Dr. William K. Allard, (Duke University)
Talk: Total Variation Regularization for Image Denoising
Prof. Dr. Luigi Ambrosio, Scuola Normale di Pisa (Italien)
Prof. Dr. Matteo Focardi, Università di Firenze (Florenz)
Prof. Dr. Bernd Kirchheim, Universitaet Leipzig (Leipzig)
Talk: Rank-one Convexity and Ornstein's L1-Noninequalities
© Universität Zürich | May 3, 2019 | CommonCrawl |
\begin{document}
\title{Maximal displacement of a branching random walk in time-inhomogeneous environment}
\begin{abstract} Consider a branching random walk evolving in a macroscopic time-inhomogeneous environment, that scales with the length $n$ of the process under study. We compute the first two terms of the asymptotic of the maximal displacement at time $n$. The coefficient of the first (ballistic) order is obtained as the solution of an optimization problem, while the second term, of order $n^{1/3}$, comes from time-inhomogeneous random walk estimates, that may be of independent interest. This result partially answers a conjecture of Fang and Zeitouni. Same techniques are used to obtain the asymptotic of other quantities, such as the consistent maximal displacement. \end{abstract}
\section{Introduction} \label{sec:introduction}
A \textit{time-inhomogeneous branching random walk on $\mathbb{R}$} is a process evolving as follows: it starts with one individual located at the origin at time 0. At each time $k \in \mathbb{N}$, every individual alive at generation $k$ dies, giving birth to a certain number of children, which are positioned around their parent according to independent versions of a point process, whose law may depend on the time. If the law of the point process governing the reproduction does not depend on the time, the process is simply called a \textit{branching random walk}.
We write $M_n$ for the maximal displacement at time $n$ in the branching random walk. The asymptotic of $M_n$, when the reproduction law does not depend on the time, is well understood. The first results come from early works of Hammersley \cite{Ham74}, Kingman \cite{Kin75} and Biggins \cite{Big76}, who obtained the existence of an explicit constant $v$ such that $\frac{M_n}{n} \conv{n}{+\infty} v$; Hu and Shi \cite{HuS09} and Addario-Berry and Reed \cite{ABR09} exhibited the logarithmic second term in the asymptotic $c_0 \log n$ for some explicit constant $c_0$; finally Aidekon proved in \cite{Aid13} the convergence of $M_n - nv - c_0 \log n$ to a random shift of a Gumble distribution.
A model of time-inhomogeneous branching random walk has been introduced by Fang and Zeitouni in \cite{FaZ11}. In this process, the reproduction law of the individuals evolves at macroscopic scale: every individual has two children, each of which moves independently according to a centred Gaussian random variable, with variance $\sigma_1^2$ (respectively $\sigma_2^2$) if the child is alive before time $n/2$ (resp. between times $n/2$ and $n$). In this branching random walk, the asymptotic of $M_n$ is still given by a ballistic first order, plus a negative logarithmic correction and bounded fluctuations of order one. However, first and second order strongly depend on the sign of $\sigma^2_1-\sigma^2_2$, and the coefficient of the logarithmic correction exhibits a phase transition as $\sigma^2_1$ grows bigger than $\sigma^2_2$. This result has been generalised to more general reproduction laws in \cite{Mal14}.
The study of the maximal displacement in a time-inhomogeneous branching Brownian motion, the continuous time counterpart of the branching random walk, with smoothly varying environment has been started in \cite{FaZ12}. In this process individuals split into 2 children at rate 1, and move using independent Gaussian diffusion with variance $\sigma^2_{t/n}$ at time $t \in [0,n]$. In \cite{FaZ12}, Fang and Zeitouni conjectured that, under mild hypotheses, for an explicit $v^*$, the sequence $(M_n - nv^* - g(n), n \geq 1)$ is tensed for a given function $g$ such that \[ -\infty < \liminf_{n \to +\infty} \frac{g(n)}{n^{1/3}} \leq \limsup_{n \to +\infty} \frac{g(n)}{n^{1/3}} \leq 0.\] They proved this result for smoothly decreasing variance. Using PDE techniques, Nolen, Roquejoffre and Ryzhik \cite{NRR13} established, again in the case of decreasing variances, that $g(n) = l^* n^{1/3} + O(\log n)$ for some explicit constant $l^*$. Maillard and Zeitouni \cite{MaZ13} obtained, independently from our result, the more precise result $g(n) = l^* n^{1/3} - c_1 \log n$, for some explicit $c_1$, using proofs similar to the ones presented here, based on first and second moment particles computation and the study of a particular PDE.
We prove in this article that for a large class of time-inhomogeneous branching random walks, $M_n - nv^* \sim_{n \to +\infty} l^* n^{1/3}$ for some explicit constants $v^*$ and $l^*$. Conversely to previous articles in the domain, the displacements we authorize are non necessarily Gaussian. Moreover, the law of the number of children is correlated with the displacement and depends on the time. More importantly, we do not restrict ourselves to --a hypothesis similar to-- decreasing variance. The decreasing variance case remains interesting, as in this case quantities such as $v^*$ and $l^*$ admit a closed expression. We do not prove in this article there exists a function $g$ such that $M_n-n v^* -g(n)$ is tight, therefore we do not exactly answer to the conjecture of Fang and Zeitouni. However, Fang \cite{Fan12} proved that the sequence $M_n$ shifted by its median is tight for a large class of \textit{generalized branching random walks}. This class does not exactly covers the class of time-inhomogeneous branching random walks we consider, but there is a non-trivial intersection. On this subset, the conjecture is then proved applying Theorem \ref{thm:main}.
To address the fact the displacements are non Gaussian, we use the Sakhanenko estimate \cite{Sak84}. This theorem couples sums of independent random variables with a Brownian motion. Consequently we use Brownian estimates to compute our branching random walk asymptotics. The non-monotonicity of the variance leads to additional concerns. We discuss in Section \ref{subsec:heuristic} a formula for $\lim_{n \to +\infty} \frac{M_n}{n}$, given as the solution of an optimization problem under constraints \eqref{eqn:existence_max}. This equation is solved in Section \ref{sec:optimization}, using some analytical tools such as the existence of Lagrange multipliers in Banach spaces described in \cite{Kur76}. When we solve this problem, an increasing function appears naturally, that replaces the inverse of the variance in computations of \cite{NRR13} and \cite{MaZ13}.
\paragraph*{Notation} In this article, $c,C$ stand for two positive constants, respectively small enough and large enough, which may change from line to line, and depend only on the law of the random variables we consider. We always assume the convention $\max \emptyset = - \infty$ and $\min \emptyset = +\infty$. For $x \in \mathbb{R}$, we write $x_+ = \max(x,0)$, $x_- = \max(-x,0)$ and $\log_+(x) = (\log x)_+$. For all function $f : [0,1] \to \mathbb{R}$, we say that $f$ is \textit{Riemann-integrable} if \[
\liminf_{n \to +\infty} \frac{1}{n} \sum_{k=0}^{n-1} \min_{s \in [\frac{k-1}{n},\frac{k+2}{n}]} f_s = \limsup_{n \to +\infty} \frac{1}{n} \sum_{k=0}^{n-1} \max_{s \in [\frac{k-1}{n},\frac{k+2}{n}]} f_s, \] and this common value is written $\int_0^1 f_s ds$. In particular, a Riemann-integrable function is bounded. A subset $F$ of $[0,1]$ is said to be \textit{Riemann-integrable} if and only if $\mathbf{1}_F$ is Riemann-integrable. For example, an open subset of $(0,1)$ of Lebesgue measure 1/2 that contains all rational numbers is not Riemann-integrable. Finally, if $A$ is a measurable event, we write $\E(\cdot ; A)$ for $\E(\cdot \mathbf{1}_A)$. An index of notations is present in Appendix \ref{app:notation}
The rest of the introduction is organised as follows. We start with some branching random walk notation in Section \ref{subsec:notation}. We describe in Section \ref{subsec:heuristic} the optimization problem that gives the speed of the time-inhomogeneous branching random walk. In Section \ref{subsec:main}, we state the main result of this article: the asymptotic of the maximal displacement in a time-inhomogeneous branching random walk. We also introduce another quantity of interest for the branching random walk: the consistent maximal displacement with respect to the optimal path in Section \ref{subsec:cmbintro}. Finally, in Section \ref{subsec:introrw}, we introduce some of the random walk estimates that are used to estimate moments of the branching random walk.
\subsection{Branching random walk notation} \label{subsec:notation}
Let $\mathbf{T}$ be a plane rooted tree --which can be encoded according to the Ulam-Harris notation, for example-- and $V : \mathbf{T} \to \mathbb{R}$. We call $(\mathbf{T},V)$ a (plane, rooted) \textit{marked tree}. For a given individual $u \in \mathbf{T}$, we write $|u|$ for the generation to which $u$ belongs, i.e. its distance to the root $\emptyset$ according to the graph distance in the tree $\mathbf{T}$. If $u$ is not the root, we denote by $\pi u$ the parent of $u$. For $k \leq |u|$, we write $u_k$ its ancestor alive at generation $k$, with $u_0 = \emptyset$. We call $V(u)$ the position of $u$ and $(V(u_0), V(u_1), \ldots V(u))$ the path followed by $u$.
For any $n \in \mathbb{N}$, we call $\{ u \in \mathbf{T}, |u|=n\}$ the \textit{$n^\mathrm{th}$ generation of the tree $\mathbf{T}$}, which is often abbreviated as $\{|u|=n\}$ if the notation is clear in the context. For $u \in \mathbf{T}$, the set of children of $u$ is written $\Omega(u) = \{ u' \in \mathbf{T} : \pi u' = u\}$, and let $L^u = (V(u')-V(u), u' \in \Omega(u))$ be the point process of the displacement of the children with respect to their parent. In this article, the point processes we consider are a finite or infinite collection of real numbers --repetitions are allowed-- that admit a maximal element and no accumulation point.
We add a piece of notation on random point processes. Let $L$ be a point process with law $\mathcal{L}$, i.e. a random variable taking values in $\cup_{k \in \mathbb{Z}_+ \cup \{+\infty\}} \mathbb{R}^k$. As the point processes we consider have a maximal element and no accumulation point, we can write $L = (\ell_1,\ldots \ell_N)$, where $N \in \mathbb{Z}_+ \cup \{+\infty\}$ is the random variable of the number of points in the process, and $\ell_1 \geq \ell_2 \geq \cdots \geq \ell_N$ is the set of points in $L$, listed in the decreasing order, with the convention $\ell_{+\infty} = -\infty$.
We now consider $(\mathcal{L}_t, t \in [0,1])$ a family of laws of point processes. For $t \in [0,1]$, we write $L_t$ for a point process with law $\mathcal{L}_t$. For $\theta \geq 0$, we denote by $\kappa_t(\theta) = \log \E\left[ \sum_{\ell \in L_t} e^{\theta \ell}\right]$ the log-Laplace transform of $\theta$ and by $\kappa^*_t(a) = \sup_{\theta > 0} [\theta a - \kappa_t(\theta)]$ its Fenchel-Legendre transform. We recall the following elementary fact: if $\kappa_t^*$ is differentiable at $a$, then putting $\theta = \partial_a \kappa^*_t(a)$, we have \begin{equation}
\label{eqn:legendreestimate}
\theta a - \kappa_t(\theta) = \kappa^*_t(a). \end{equation}
The \textit{branching random walk of length $n$ in the time-inhomogeneous environment $(\mathcal{L}_t, t \in [0,1])$} (BRWtie) is the marked tree $(\mathbf{T}^{(n)},V^{(n)})$ such that $\{L^u, u \in \mathbf{T}^{(n)}\}$ forms a family of independent point processes, where $L^u$ has law $\mathcal{L}_\frac{|u|+1}{n}$ if $|u|<n$, and is empty otherwise. In particular, $\mathbf{T}^{(n)}$ is the (time-inhomogeneous) Galton-Watson tree of the genealogy of this process. When the value of $n$ is clear in the context, we often omit the superscript, to lighten notations.
It is often easier to consider processes that never get extinct, and have supercritical offspring above a given straight line with slope $p$. We introduce the (strong) supercritical assumption \begin{equation}
\label{eqn:breeding}
\forall t \in [0,1], \mathbf{P}(L_t = \emptyset) = 0 \quad \mathrm{and} \quad \exists p \in \mathbb{R} : \inf_{t \in [0,1]} \mathbf{P}(\# \{ \ell \in L_t : \ell \geq p \} \geq 2)>0. \end{equation} Such a strong supercritical assumption is not always necessary, but is technically convenient to obtain concentration inequalities for the maximal displacement. It is also helpful to guarantee the existence of a solution of the optimization problem that defines $v^*$. The second part of the assumption is used only once in this article: to prove Lemma 5.4 (which is used to prove the second equation of Lemma 5.6).
In this article, we assume that the function $t \to \mathcal{L}_t$ satisfies some strong regularity assumptions. We write \begin{equation}
\label{eqn:defDomain} D = \{ (t,\theta) \in [0,1] \times [0,+\infty) : \kappa_t(\theta)< +\infty\} \quad \mathrm{and} \quad D^* = \{ (t,a) : \kappa^*_t(a)<+\infty \}, \end{equation} and we assume that $D$ and $D^*$ are non-empty, that $D$ (resp. $D^*$) is open in $[0,1] \times [0,+\infty)$ (resp. $[0,1] \times \mathbb{R}$) and that \begin{equation}
\label{eqn:regularity}
\kappa \in \mathcal{C}^{1,2}\left(D\right) \text{ and } \kappa^* \in \mathcal{C}^{1,2}\left(D^*\right). \end{equation} These regularity assumptions are used to ensure that the solution of the optimization problem defining $v^*$ is a regular point in some sense. If we know, by some other way, that the solution is regular, then these assumptions are no longer needed. These assumptions imply that the maximum of the point processes we consider has at least exponential tails, and that the probability that this maximum is equal to the essential supremum is zero. They are not optimal, but sufficient to define a quantity that we prove to be the speed of the BRWtie. For example, a random number of i.i.d. random variables with exponential left tails satisfy the above condition. Conversely, heavy tailed random variables, or if the maximum of the point process verifies \[
\mathbf{P}(\max\{ \ell \in L \} \geq x ) \sim_{x \to +\infty} x^{-1 - \epsilon} e^{-x} \] will not satisfy \eqref{eqn:defDomain}.
\subsection{The optimization problem} \label{subsec:heuristic}
We write $\mathcal{C}$ for the set of continuous functions, and $\mathcal{D}$ for the set of càdlàg --right-continuous with left limits at each point-- functions on $[0,1]$ which are continuous at point $1$. To a function $b \in \mathcal{D}$, we associate a path --i.e. a sequence-- of length $n$ defined for $k \leq n$ by $\bar{b}^{(n)}_k = \sum_{j=1}^k b_{j/n}$. We say that $b$ is the \textit{speed profile} of the path $\bar{b}^{(n)}$, and we introduce \[
K^* :
\begin{array}{rcl}
\mathcal{D} & \longrightarrow & \mathcal{C}\\
b & \longmapsto & \left(\int_0^t \kappa^*_s(b_s) ds, t \in [0,1]\right).
\end{array} \] By standard computations on branching random walks (see, e.g. \cite{BigBO}), for all $t \in [0,1]$, the mean number of individuals that follow the path $\bar{b}^{(n)}$ --i.e. that stay at all time within distance $\sqrt{n}$ from the path-- until time $tn$ verifies
\[ \frac{1}{n} \log \E\left[ \sum_{|u|=\floor{nt}} \ind{ \left|V(u_k) - \bar{b}^{(n)}_{k/n} \right| < \sqrt{n}, k \leq nt} \right] \approx_{n \to +\infty} -K^*(b)_t.\] Therefore, $e^{-n K^*(b)_t}$ is a good approximation of the number of individuals that stay close to the path $\bar{b}^{(n)}$ until time $tn$.
If there exists $t_0 \in (0,1)$ such that $K^*(b)_{t_0}>0$, then with high probability, there is no individual who stayed close to this path until time $n t_0$. Conversely, if for all $t \in [0,1]$, $K^*(b)_t \leq 0$, one would expect to find with positive probability at least one individual at time $n$ to the right of $\bar{b}^{(n)}$. Following this heuristic, we introduce \begin{equation}
\label{eqn:defv}
v^* = \sup \left\{ \int_0^1 b_s ds, b \in \mathcal{D} : \forall t \in [0,1], K^*(b)_t \leq 0 \right\}. \end{equation} We expect $nv^*$ to be the highest terminal point in the set of paths that are followed with positive probability by individuals in the branching random walk. Therefore we expect that $\lim_{n \to +\infty} \frac{M_n}{n} = v^*$ in probability.
We are interested in the path that realises the maximum in \eqref{eqn:defv}. We define the optimization problem under constraints \begin{equation}
\label{eqn:existence_max}
\exists a \in \mathcal{D} : v^* = \int_0^1 a_s ds \quad \mathrm{and} \quad \forall t \in [0,1], K^*(a)_t \leq 0. \end{equation} We say that $a$ is a solution to \eqref{eqn:existence_max} if $\int_0^1 a_s ds = 0$ and $K^*(a)$ is non-positive. Such a solution $a$ is called an \textit{optimal speed profile}, and $\bar{a}^{(n)}$ an \textit{optimal path} for the branching random walk. The path followed by the rightmost individual at time $n$ is an optimal path, thus describing such a path is interesting to obtain the second order correction. In effect, as highlighted in the time-homogeneous branching random walk in \cite{AiS10}, the second order of the asymptotic of $M_n$ is linked to the difficulty for a random walk to follow the optimal path. \begin{proposition} \label{prop:regularity} Let $a \in \mathcal{D}$. Under the assumptions \eqref{eqn:breeding} and \eqref{eqn:regularity}, $a$ is a solution to \eqref{eqn:existence_max}, i.e. \[
v^* = \int_0^1 a_s ds \quad \mathrm{and} \quad \forall t \in [0,1], K^*(a)_t \leq 0, \] if and only if, setting $\theta_t = \partial_a \kappa^*_t(a_t)$, we have \begin{enumerate}
\item $\theta$ is positive and non-decreasing ;
\item $K^*(a)_1=0$ ;
\item $\int_0^1 K^*(a)_s d\theta^{-1}_s=0$. \end{enumerate} There exists a unique solution $a$ to \eqref{eqn:existence_max}, and $a$ (and $\theta$) are Lipschitz. \end{proposition}
Consequence of this proposition, we now call $a$ the optimal speed profile, and $\bar{a}$ the optimal path. This result is proved in Section \ref{sec:optimization}. The optimization problem \eqref{eqn:existence_max} is similar to the one solved for the GREM by Bovier and Kurkova \cite{BoK07}.
\subsection{Asymptotic of the maximal displacement} \label{subsec:main}
Under the assumptions \eqref{eqn:regularity} and \eqref{eqn:existence_max}, let $a$ be the optimal speed profile characterised by Proposition \ref{prop:regularity}. For $t \in [0,1]$ we denote by \begin{equation}
\label{eqn:thetadef}
\theta_t = \partial_a \kappa^*_t(a_t) \quad \mathrm{and} \quad \sigma_t^2 = \partial^2_\theta \kappa_t(\theta_t). \end{equation} To obtain the asymptotic of the maximal displacement, we introduce the following regularity assumptions: \begin{equation}
\label{eqn:regularitytheta}
\theta \text{ is absolutely continuous, with a Riemann-integrable derivative } \dot{\theta}, \end{equation} \begin{equation}
\label{eqn:regularityenergy}
\{ t \in [0,1] : K^*_t(a)=0 \} \text{ is Riemann-integrable}. \end{equation} Finally, we make the following second order integrability assumption: \begin{equation}
\label{eqn:integrability2}
\sup_{t \in [0,1]} \E\left[ \left( \sum_{\ell \in L_t} e^{\theta_t \ell_t} \right)^2 \right] < +\infty. \end{equation}
\begin{remark} This last integrability condition is not optimal. Using the spinal decomposition as well as estimates on random walks enriched by random variables depending only on the last step, as in \cite{Mal14} would lead to an integrability condition of the form $\E(X (\log X)^2) < +\infty$ instead of \eqref{eqn:integrability2}. However, this assumption considerably simplifies the proofs. \end{remark}
The main result of this article is the following. \begin{theorem}[Maximal displacement in the BRWtie] \label{thm:main} We assume \eqref{eqn:breeding}, \eqref{eqn:regularity}, \eqref{eqn:regularitytheta}, \eqref{eqn:regularityenergy} and \eqref{eqn:integrability2} are verified. We write $\alpha_1$ for the largest zero of the Airy function of first kind --recall that $\alpha_1 \approx -2.3381...$-- and we set \begin{equation}
\label{eqn:definel}
l^* = \frac{\alpha_1}{2^{1/3}} \int_0^1 \frac{(\dot{\theta}_s \sigma_s)^{2/3}}{\theta_s} ds \leq 0. \end{equation} Then we have for all $l>0$, \[ \lim_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\left(M_n \geq nv^* + (l^*+l)n^{1/3}\right) = -\theta_0 l,\] and for all $\epsilon>0$,
\[ \limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\left( \left|M_n - nv^* -l^* n^{1/3}\right| \geq \epsilon n^{1/3}\right) < 0.\] \end{theorem}
This theorem is proved in Section \ref{sec:maxdis}. The presence of the largest zero of the Airy function of first kind is closely related to the asymptotic of the Laplace transform of the area under a Brownian motion staying positive, \[\E\left( e^{-\int_0^t B_s ds} ; B_s \geq 0, s \leq t\right) \approx_{t \to +\infty} e^{\frac{\alpha_1}{2^{1/3}} t + o(t)}.\] The fact that the second order of $M_n$ is $n^{1/3}$ can be explained as follows: when $\theta$ is strictly increasing at time $t$, the optimal path has to stay very close to the boundary of the branching random walk at time $nt$. In particular, if $\theta$ is strictly increasing on $[0,1]$, the optimal path has to stay close to the boundary. The $n^{1/3}$ second order is then similar to the asymptotic of the consistent minimal displacement for the time-homogeneous branching random walk, which is of order $n^{1/3}$, as proved in \cite{FaZ10,FHS12}.
\subsection{Consistent maximal displacement} \label{subsec:cmbintro} The arguments we develop for the proof of Theorem \ref{thm:main} can easily be extended to obtain the asymptotic of the consistent maximal displacement with respect to the optimal path in the branching random walk, which we define now. For $n \in \mathbb{N}$ and $u \in \mathbf{T}^{(n)}$, we denote by \[
\Lambda(u) = \max_{k \leq |u|} \left[ \bar{a}^{(n)}_k - V(u_k) \right], \] the maximal delay an ancestor of the individual $u$ has with respect to the optimal path. The consistent maximal displacement with respect to the optimal path is defined by \begin{equation}
\label{eqn:defineCMD}
\Lambda_n = \min_{u \in \mathbf{T}^{(n)}, |u|=n} \Lambda(u). \end{equation} This quantity corresponds to the smallest distance from the optimal path $\bar{a}$ at which one can put a barrier below which individuals get killed such that the global system still survives. The consistent maximal displacement has been studied for time-homogeneous branching random walks in \cite{FaZ10}. We obtain the following asymptotic for the consistent maximal displacement in the BRWtie. \begin{theorem} \label{thm:cmd} Under the assumptions \eqref{eqn:breeding}, \eqref{eqn:regularity}, \eqref{eqn:existence_max}, \eqref{eqn:regularitytheta}, \eqref{eqn:regularityenergy} and \eqref{eqn:integrability2}, there exists $\lambda^* \leq -l^*$, defined in Section~\ref{subsec:cmd} such that for any $\lambda \in (0,\lambda^*)$, \[
\lim_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\left( \Lambda_n \leq (\lambda^*-\lambda) n^{1/3}\right) = - \theta_0 \lambda, \] and for any $\epsilon>0$,
\[ \limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}(|\Lambda_n - \lambda^* n^{1/3}| \geq \epsilon n^{1/3}) < 0.\] \end{theorem}
\begin{remark} We observe that if $u \in \mathbf{T}^{(n)}$ verifies $V(u) = M_n$, then $\Lambda^*(u) \leq nv^* - M_n$. As a consequence, the inequality $L_n \leq nv^* - M_n$ holds almost surely, which proves that $\lambda^* \leq -l^*$, as soon as these quantities exist. \end{remark}
In Theorem \ref{thm:cmd}, we give the asymptotic of the consistent maximal displacement with respect to the optimal path. However, this is not the only path one may choose to consider. For example, one can choose the ``natural speed path'', in which the speed profile is a function $v \in \mathcal{C}$ defined by $v_t = \inf_{\theta > 0} \frac{\kappa_t(\theta)}{\theta}$. Note that $v_t$ is the speed of a time-homogeneous branching random walk with reproduction law $\mathcal{L}_t$. As for all $t \in [0,1]$, $K^*(v)_t=0$, for any $\lambda > 0$, the number of individuals that stayed above $\bar{v}^{(n)} - \lambda n^{1/3}$ at all time $k \leq n$ is of order $e^{O(n^{1/3})}$.
In Section \ref{sec:path}, we provide a time-inhomogeneous version of the Many-to-one lemma, that links additive moments of the branching random walk with time-inhomogeneous random walk estimates. To prove Theorems \ref{thm:main} and \ref{thm:cmd}, we use random walk estimates that are proved in Section \ref{sec:rw}.
\subsection{Airy functions and random walk estimates} \label{subsec:introrw}
We introduce a few basic property on Airy functions, that can be found in \cite{AbS64}. The \textit{Airy function of first kind} $\mathrm{Ai}$ can be defined, for $x \in \mathbb{R}$, by the improper integral \begin{equation}
\label{eqn:airy}
\mathrm{Ai}(x) = \frac{1}{\pi}\lim_{t \to + \infty} \int_0^t \cos\left( \tfrac{s^3}{3} + x s \right)ds, \end{equation} and the \textit{Airy function of second kind} $\mathrm{Bi}$ by \begin{equation}
\label{eqn:airy2}
\mathrm{Bi}(x) = \frac{1}{\pi} \lim_{t \to + \infty} \int_0^t \exp\left( -\tfrac{s^3}{3} + x s \right) + \sin\left(\tfrac{s^3}{3} + x s \right)ds. \end{equation} These two functions form a basis of the space of functions solutions to \[
\forall x \in \mathbb{R}, y''(x) - x y(x) = 0, \] and verify $\lim_{x \to +\infty} \mathrm{Ai}(x) = 0$ and $\lim_{x \to +\infty} \mathrm{Bi}(x) = +\infty$. The equation $\mathrm{Ai}(x) = 0$ has an infinitely countable number of solutions, all negative with no accumulation points, which are listed in the decreasing order in the following manner: $0 > \alpha_1 > \alpha_2 > \cdots$.
The Laplace transform of the area below a random walk, or a Brownian motion, conditioned to stay positive admits an asymptotic behaviour linked to the largest zero of $\mathrm{Ai}$, as proved by Darling \cite{Dar83}, Louchard \cite{Lou84} and Tak\'acs \cite{Tak92}. This result still holds in time-inhomogeneous settings. Let $(X_{n,k}, n \geq 1, k \leq n)$ be a triangular array of independent centred random variables. We assume that \begin{equation}
\label{eqn:variance_rw}
\exists \sigma \in \mathcal{C}([0,1],(0,+\infty)) : \forall n \in \mathbb{N}, k \leq n, \E(X_{n,k}^2) = \sigma_{k/n}^2, \end{equation} \begin{equation}
\label{eqn:integrability_rw}
\exists \mu > 0 : \E\left[ e^{\mu |X_{n,k}|} \right] < +\infty. \end{equation} We write $S^{(n)}_k = \sum_{j=1}^k X_{n,j}$ for the time-inhomogeneous random walk. \begin{theorem}[Time-inhomogeneous Tak\'acs estimate] \label{thm:taktie} Under \eqref{eqn:variance_rw} and \eqref{eqn:integrability_rw}, for any continuous function $g$ such that $g(0)>0$ and any absolutely continuous increasing function $h$ with a Riemann-integrable derivative $\dot{h}$, we have \begin{multline*}
\lim_{n \to +\infty} \frac{1}{n^{1/3}} \log \E\left[ \exp\left(- \sum_{j=1}^{n} (h_{j/n}-h_{(j-1)/n}) S^{(n)}_{j}\right) ; S_j \leq g_{j/n} n^{1/3}, j \leq n \right] \\
= \int_0^1 \left(\dot{h}_s g_s + \frac{a_1}{2^{1/3}} (\dot{h}_s \sigma_s)^{2/3}\right) ds. \end{multline*} \end{theorem}
This result is, in some sense, similar to the Mogul'ski\u\i{} estimate \cite{Mog74}, which gives the asymptotic of the probability for a random walk to stay in an interval of length $n^{1/3}$. A time-inhomogeneous version of this result, with an additional exponential weight, holds again. To state this result, we introduce a function $\Psi$, defined in the following lemma. \begin{lemma} \label{lem:existencePsi} Let $B$ be a Brownian motion. There exists a unique convex function $\Psi : \mathbb{R} \to \mathbb{R}$ such that for all $h \in \mathbb{R}$ \begin{equation}
\label{eqn:definepsi}
\lim_{t \to +\infty} \frac{1}{t} \log \sup_{x \in [0,1]} \E_x\left[ e^{ -h \int_0^t B_s ds} ; B_s \in [0,1], s \in [0,t] \right] = \Psi(h). \end{equation} \end{lemma}
\begin{remark} We show in Appendix \ref{subsec:twosided} that $\Psi$ admits the following alternative definition: \[
\forall h > 0, \Psi(h) = \frac{h^{2/3}}{2^{1/3}} \sup\left\{\lambda \leq 0 : \mathrm{Ai}\left( \lambda \right) \mathrm{Bi} \left( \lambda + (2h)^{1/3} \right) - \mathrm{Bi}\left( \lambda \right) \mathrm{Ai}\left( \lambda + (2h)^{1/3} \right) = 0 \right\}, \] and prove that $\Psi$ verifies $\Psi(0) = -\frac{\pi^2}{2}$, $\Psi(h) \sim_{h \to +\infty} \alpha_1 \frac{h^{2/3}}{2^{1/3}}$ and $\Psi(h)-\Psi(-h)=-h$ for all $h \in \mathbb{R}$. \end{remark}
\begin{proof}[Proof of Lemma \ref{lem:existencePsi}] For $h \in \mathbb{R}$ and $t \geq 0$, we write \[
\Psi_t(h) = \frac{1}{t} \log \sup_{x \in [0,1]} \E_x \left[ e^{ h \int_0^t B_s ds} ; B_s \in [0,1], s \in [0,t] \right]. \]
As $B_s \in [0,1]$, we have trivially $|\Psi_t(h)| \leq |h| < +\infty$. Let $0 \leq t_1 \leq t_2$ and $x \in [0,1]$, by the Markov property \begin{align*}
&\E_x \left[ e^{ h \int_0^{t_1 + t_2} B_s ds} ; B_s \in [0,1], s \in [0,t_1 + t_2] \right]\\
&\qquad = \E_x\left[ e^{h \int_0^{t_1} B_s ds} \E_{B_{t_1}} \left[ e^{h\int_0^{t_2} B_s ds} ; B_s \in [0,1], s \in [0,t_2]\right] ; B_s \in [0,1], s \in [0,t_1] \right]\\
&\qquad \leq e^{t_2 \Psi_{t_2}(h)} \E_x\left[ e^{h \int_0^{t_1} B_s ds}; B_s \in [0,1], s \in [0,t_1] \right]
\leq e^{t_1 \Psi_{t_1}(h)} e^{t_2 \Psi_{t_2}(h)}. \end{align*} As a consequence, for all $h \in \mathbb{R}$, $(\Psi_t(h), t \geq 0)$ is a sub-additive function and \[
\lim_{t \to +\infty} \Psi_t(h) = \inf_{t \geq 0} \Psi_t(h) =: \Psi(h). \]
In particular, for all $h \in \mathbb{R}$, we have $|\Psi(h)| \leq |h| < +\infty$.
We now prove that $\Psi$ is a convex function on $\mathbb{R}$, thus continuous. By the Hölder inequality, for all $\lambda \in [0,1]$, $(h_1,h_2) \in \mathbb{R}^2$, $x \in [0,1]$ and $t \geq 0$, we have \begin{align*}
&\E_x \left[ e^{ (\lambda h_1 + (1-\lambda)h_2) \int_0^t B_s ds} ; B_s \in [0,1], s \in [0,t] \right]\\
\leq &\E_x \left[ \left(e^{\lambda h_1 \int_0^t B_s ds} \ind{B_s \in [0,1], s \in [0,t]} \right)^{\frac{1}{\lambda}}\right]^\lambda \E_x \left[ \left(e^{(1-\lambda) h_2 \int_0^t B_s ds} \ind{B_s \in [0,1], s \in [0,t]} \right)^{\frac{1}{1-\lambda}}\right]^{1-\lambda}\\
\leq &e^{t \lambda \Psi_t(h_1)} e^{t(1-\lambda) \Psi_t(h_2)}. \end{align*} Consequently \[
\limsup_{t \to +\infty} \Psi_t(\lambda h_1 + (1-\lambda)h_2) \leq \lambda \limsup_{t \to + \infty} \Psi_t(h_1) + (1-\lambda) \limsup_{t \to +\infty} \Psi_t(h_2), \] which proves that $\Psi$ is convex, thus continuous. \end{proof}
\begin{theorem}[A time-inhomogeneous Mogul'ski\u\i{} estimate] \label{thm:mogtie} Under \eqref{eqn:variance_rw} and \eqref{eqn:integrability_rw}, for any pair of continuous functions $f<g$ such that $f(0)<0<g(0)$ and any absolutely continuous function $h$ with a Riemann-integrable derivative $\dot{h}$, we have \begin{multline*}
\lim_{n \to +\infty} \frac{1}{n^{1/3}} \log \E\left[ \exp\left( \sum_{j=1}^{n} (h_{j/n}-h_{(j-1)/n}) S^{(n)}_{j} \right) ; \frac{S_j}{n^{1/3}} \in [f_{j/n}, g_{j/n}], j \leq n \right] \\
= \int_0^1 \left( \dot{h}_s g_s + \frac{\sigma_s^2}{(g_s-f_s)^2}\Psi\left( \tfrac{(g_s-f_s)^3}{\sigma_s^2}\dot{h}_s \right) \right) ds. \end{multline*} \end{theorem}
The rest of the paper is organised as follows. Theorems \ref{thm:taktie} and \ref{thm:mogtie} are unified and proved in Section \ref{sec:rw}. These results are used in Section \ref{sec:path} to compute some branching random walk estimates, useful to bound the probability that there exists an individual that stays in a given path until time $n$. We study \eqref{eqn:existence_max} in Section \ref{sec:optimization}, proving in particular Proposition \ref{prop:regularity}. Using the particular structure of the optimal path, we prove Theorems \ref{thm:main} and \ref{thm:cmd} in Section \ref{sec:maxdis}.
\textbf{Acknowledgments.} I would like to thank Pascal Maillard, for introducing me to the time-inhomogeneous branching random walk topic, Ofer Zeitouni for his explanations on \cite{FaZ12} and Zhan Shi for help in all the stages of the research. I also thank the referees for their careful proofreading of this article and pointing out a mistake in one of the proofs. Finally, I wish to thank David Gontier and Cécile Huneau for their help with the PDE analysis in Appendix \ref{app:bm}.
\section{Random walk estimates} \label{sec:rw}
We consider an array $(X_{n,k}, n \geq 1, k \leq n)$ of independent centred random variables, such that there exist $\sigma \in \mathcal{C}([0,1], (0,+\infty))$ and $\mu\in (0,+\infty)$ verifying \eqref{eqn:variance_rw} and \eqref{eqn:integrability_rw}. We write $S^{(n)}_k = S^{(n)}_0 + \sum_{j=1}^k X_{n,j}$ for the time-inhomogeneous random walk of length $n$, with $\mathbf{P}_x(S^{(n)}_0=x)=1$. Let $\E_x$ be the expectation corresponding to the probability $\mathbf{P}_x$. Let $h$ be a continuous function on $[0,1]$ such that \begin{equation} \label{eqn:regularityh}
h \text{ is absolutely continuous, with Riemann-integrable derivative } \dot{h}. \end{equation} The main result of this section is the computation of the asymptotic, as $n \to +\infty$, of the Laplace transform of the integral of $S^{(n)}$ with respect to $h$, on the event that $S^{(n)}$ stays in a given path, that is defined by \eqref{eqn:defIn}.
Let $f$ and $g$ be two continuous functions on $[0,1]$ such that $f<g$ and $f(0)<0<g(0)$, and $F$ and $G$ be two Riemann-integrable subsets of $[0,1]$ --i.e. verifying $\mathbf{1}_F$ and $\mathbf{1}_G$ are Riemann-integrable-- such that \begin{equation}
\label{eqn:paslineaire}
\{t \in [0,1] : \dot{h}_t < 0 \} \subset F \quad \mathrm{and} \quad \{ t \in [0,1] : \dot{h}_t > 0 \} \subset G. \end{equation} Interval $F$ (respectively $G$) represent the set of times at which the barrier $f$ (resp. $g$) is put below (resp. above) the path of the time-inhomogeneous random walk. Consequently, \eqref{eqn:paslineaire} implies that when there is no barrier below, the Laplace exponent is non-negative, so that the random walk does not ``escape'' to $-\infty$ with high probability.
For $n \geq 1$, we introduce the $\frac{1}{n}^\mathrm{th}$ approximation of $F$ and $G$, defined by \begin{equation}
\label{eqn:deffnandgn}
F_n = \left\{ 1\leq k \leq n : \left[\tfrac{k}{n}, \tfrac{k+1}{n}\right] \cap F \neq \emptyset\right\} \text{, } G_n = \left\{ 0 \leq k \leq n : \left[\tfrac{k}{n}, \tfrac{k+1}{n}\right] \cap G \neq \emptyset\right\}. \end{equation} The path followed by the random walk of length $n$ is defined, for $0 \leq j \leq n$, by \begin{equation}
\label{eqn:defIn}
I_n(j) =
\begin{cases}
\left[f_{j/n}n^{1/3}, g_{j/n}n^{1/3}\right] & \text{if } j \in F_n \cap G_n,\\
\left[f_{j/n},+\infty\right[ & \text{if } j \in F_n \cap G_n^c,\\
\left]-\infty, g_{j/n}n^{1/3}\right] & \text{if } j \in F_n^c\cap G_n,\\
\mathbb{R} & \text{otherwise.}
\end{cases} \end{equation} The random walk $S^{(n)}$ follows the path $I^{(n)}$ if $ \geq f_{k/n} n^{1/3}$ at any time $k \in F_n$, and $S^{(n)}_k \leq g_{k/n} n^{1/3}$ at any time $k \in G_n$. Choosing $F$ and $G$ in an appropriate way, we obtain Theorem \ref{thm:taktie} --where $F=\emptyset$ and $G=[0,1]$-- and Theorem \ref{thm:mogtie} --where $F=G=[0,1]$.
We introduce the quantity \begin{multline}
\label{eqn:defineH}
H^{F,G}_{f,g} = \int_0^1 \dot{h}_s g_s ds + \int_{F\cap G} \frac{\sigma_s^2}{(g_s - f_s)^2} \Psi\left( \tfrac{(g_s-f_s)^3}{\sigma_s^2} \dot{h}_s \right) ds\\
+ \int_{F^c \cap G} \frac{\alpha_1}{2^{1/3}} (\dot{h}_s \sigma_s)^{2/3} ds + \int_{G \cap F^c} \left( \dot{h}_s(f_s - g_s) + \frac{\alpha_1}{2^{1/3}} (-\dot{h}_s \sigma_s)^{2/3} \right) ds, \end{multline} where $\Psi$ is the function defined by \eqref{eqn:definepsi}. The first integral in this definition enables to ``center'' the path interval in a way that $g$ is replaced by $0$. The integral term over $F\cap G$ comes from the set of times in which the random walk is blocked in an interval of finite length as in Theorem \ref{thm:mogtie}, and the last two integral terms correspond to paths with only one bound, above or below the random walk respectively.
The rest of the section is devoted to the proof of the following result. \begin{theorem} \label{thm:general_rw} Under the assumptions \eqref{eqn:variance_rw} and \eqref{eqn:integrability_rw}, for any continuous function $h$ satisfying \eqref{eqn:regularityh}, for any pair of continuous functions $f<g$ such that $f(0)<0<g(0)$, for any Riemann-integrable $F,G \subset [0,1]$ such that \eqref{eqn:paslineaire} holds, we have \begin{equation}
\label{eqn:limsup}
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \sup_{x \in \mathbb{R}} \log \E_x\left[ e^{\sum_{j=1}^n (h_{(j+1)/n}-h_{j/n}) S^{(n)}_j}; S^{(n)}_j \in I^{(n)}_j, j \leq n \right] = H^{F,G}_{f,g}(1). \end{equation} Moreover, setting $\tilde{I}^{(n)}_j = I^{(n)}_j \cap [-n^{2/3},n^{2/3}]$, for all $f_1 < a < b < g_1$ we have \begin{equation}
\label{eqn:liminf}
\liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \E_0\left[ e^{\sum_{j=1}^n (h_{(j+1)/n} - h_{j/n}) S^{(n)}_j} \ind{S^{(n)}_n \in [an^{1/3},bn^{1/3}]} ; S^{(n)}_j \in \tilde{I}^{(n)}_j, j \leq n \right] = H^{F,G}_{f,g}. \end{equation} \end{theorem}
\begin{remark} Observe that when \eqref{eqn:paslineaire} does not hold, the correct rate of growth of the expectations in \eqref{eqn:limsup} and \eqref{eqn:liminf} is exponential, instead of the order $e^{O(n^{1/3})}$. \end{remark}
To prove this theorem, we decompose the time interval $[0,n]$ into $A$ intervals, each of length $\frac{n}{A}$. On these smaller intervals, the functions $f$, $g$ and $\dot{h}$ can be approached by constants. These intervals are divide into $\frac{n^{1/3}}{tA}$ subintervals of length $tn^{2/3}$. On these subintervals, the time-inhomogeneous random walk can be approached by a Brownian motion, on which the quantities can be explicitly computed, using the Feynman-Kac formula. Letting $n$, $t$ then $A$ grow to $+\infty$, we conclude the proof of Theorem \ref{thm:general_rw}. We give in Section \ref{subsec:bm} the asymptotic of the area under a Brownian motion constrained to stay non-negative or in an interval, and use the Sakhanenko exponential inequality in Section \ref{subsec:rw} to quantify the approximation of a random walk by a Brownian motion, before proving Theorem \ref{thm:general_rw} in Section \ref{subsec:conclusion}.
\subsection{Brownian estimates through the Feynman-Kac formula} \label{subsec:bm}
The asymptotic of the Laplace transform of the area under a Brownian motion, constrained to stay non-negative or in an interval, is proved in Appendix~\ref{app:bm}. In this section, $(B_t, t \geq 0)$ is a standard Brownian motion, which starts at position $x \in \mathbb{R}$ at time $0$ under the law $\mathbf{P}_x$. We give the main results that are used in the next section to compute similar quantities for time-inhomogeneous random walks. First, for a Brownian motion that stay non-negative: \begin{lemma} \label{lem:bmOnesided} For all $h>0$, $0<a<b$ and $0<a'<b'$, we have \begin{multline}
\label{eqn:bmOnesided}
\lim_{t \to +\infty} \frac{1}{t} \log \sup_{x \in \mathbb{R}} \E_x\left[ e^{-h \int_0^t B_s ds} ; B_s \geq 0, s \leq t \right]\\
= \lim_{t \to +\infty} \frac{1}{t} \log \inf_{x \in [a,b]} \E_x\left[ e^{-h \int_0^t B_s ds} \ind{B_t \in [a',b']} ; B_s \geq 0, s \leq t \right] = \frac{\alpha_1}{2^{1/3}}h^{2/3}. \end{multline} \end{lemma}
A similar estimate holds for a Brownian motion constrained to stay in the interval $[0,1]$: \begin{lemma} \label{lem:bmTwosided} Let $B$ be a Brownian motion. For all $h \in \mathbb{R}$, $0<a<b<1$ and $0<a'<b'<1$, we have \begin{multline}
\label{eqn:bmTwosided}
\lim_{t \to +\infty} \frac{1}{t} \sup_{x \in [0,1]} \log \E_x\left[ e^{-h \int_0^t B_s ds} ; B_s \in [0,1], s \leq t \right]\\ =
\lim_{t \to +\infty} \frac{1}{t} \inf_{x \in [a,b]} \log \E_x\left[ e^{-h \int_0^t B_s ds} \ind{B_t \in [a',b']} ; B_s \in [0,1], s \leq t \right] = \Psi(h). \end{multline} Moreover, for all $h>0$, we have \begin{equation}
\label{eqn:alternativeDefinition}
\Psi(h) = \frac{h^{2/3}}{2^{1/3}} \sup\left\{\lambda \leq 0 : \mathrm{Ai}\left( \lambda \right) \mathrm{Bi} \left( \lambda + (2h)^{1/3} \right) - \mathrm{Bi}\left( \lambda \right) \mathrm{Ai}\left( \lambda + (2h)^{1/3} \right) = 0 \right\}. \end{equation} We also have $\Psi(0)=-\frac{\pi^2}{2}$, $\lim_{h \to +\infty} \frac{\Psi(h)}{h^{2/3}} = \frac{\alpha_1}{2^{1/3}}$ and, for $h \in \mathbb{R}$, $\Psi(h) - \Psi(-h) = h$. \end{lemma}
\subsection{From a Brownian motion to a random walk} \label{subsec:rw}
We use the Sakhanenko exponential inequality to extend the Brownian estimates to similar quantities for time-inhomogeneous random walks. We obtain here the correct $n^{1/3}$ order, but non-optimal upper and lower bounds. These results are used in the next section to prove Theorem \ref{thm:general_rw}. The Sakhanenko exponential inequality links a time-inhomogeneous random walk with a Brownian motion, in a similar way that the KMT coupling links a classical random walk with a Brownian motion. \begin{theorem}[Sakhanenko exponential inequality \cite{Sak84}] \label{thm:sakhanenko} Let $X=(X_1,\ldots X_n)$ be a sequence of independent centred random variables. We suppose there exists $\lambda>0$ such that for all $j \leq n$ \begin{equation}
\label{eqn:integrabilityRwBis}
\lambda \E\left( |X_j|^3 e^{\lambda |X_j|}\right) \leq \E\left(X_j^2\right). \end{equation} We can construct a sequence $\tilde{X}=(\tilde{X}_1,\ldots \tilde{X}_n)$ with the same law as $X$; and $Y$ a sequence of centred Gaussian random variables with the same covariance as $\tilde{X}$ such that for some universal constant $C_0$ and all $n \geq 1$ \[ \E\left[ \exp(C_0 \lambda \Delta_n)\right] \leq 1 + \lambda \sqrt{\sum_{j=1}^n \Var(X_j)}, \]
where $\Delta_n = \max_{j \leq n} \left| \sum_{k=1}^j \tilde{X}_k-Y_k \right|$. \end{theorem}
Using this theorem, we couple a time-inhomogeneous random walk with a Brownian motion in such a way that they stay at distance $O(\log n)$ with high probability. Technically, to prove Theorem \ref{thm:general_rw}, we simply need a uniform control on $\mathbf{P}(\Delta_{n} \geq \epsilon n^{1/3})$. To obtain it, the polynomial Sakhanenko inequality is enough, that only impose a uniform bound on the third moment of the array of random variables instead of \eqref{eqn:integrability_rw}. However in the context of branching random walks, exponential integrability conditions are needed to guarantee the regularity of the optimal path (see Section \ref{subsec:heuristic}).
Let $(X_{n,k}, n \in \mathbb{N}, k \leq n)$ be a triangular array of independent centred random variables, such that there exists a continuous positive function $\sigma^2$ verifying \begin{equation}
\label{eqn:variance1}
\forall n \in \mathbb{N}, k \leq n, \E\left[ X_{n,k}^2 \right] = \sigma^2_{k/n}. \end{equation} We set $\underline{\sigma}= \min_{t \in [0,1]} \sigma_t > 0$ and $\bar{\sigma} = \max_{t \in [0,1]} \sigma_t < +\infty$. We also assume that \begin{equation}
\label{eqn:integrabilityRw1}
\exists \lambda > 0 : \sup_{n \in \mathbb{N}, k \leq n} \E\left( e^{\lambda |X_{n,k}|} \right) < + \infty. \end{equation} Observe that for all $\mu<\lambda$, there exists $C>0$ such that for all $x \geq 0$, $x^3 e^{\mu x} \leq C e^{\lambda x}$. Thus \eqref{eqn:integrabilityRw1} implies \begin{equation}
\label{eqn:integrabilityRwBis1}
\exists \mu > 0 : \sup_{n \geq 1, k \leq n} \mu \E\left( |X_{n,j}|^3 e^{\mu |X_{n,j}|} \right) \leq \underline{\sigma}^2. \end{equation}
In the first instance, we bound from above the asymptotic of the Laplace transform of the area under a time-inhomogeneous random walk. \begin{lemma} \label{lem:upperboundRw} We assume \eqref{eqn:variance1} and \eqref{eqn:integrabilityRw1} are verified. For all $h>0$, we have \begin{equation}
\label{eqn:upperboundRwOnesided}
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \sup_{x \in \mathbb{R}} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{n-1} S^{(n)}_j} ; S^{(n)}_j \geq 0, j \leq n \right] \leq \frac{\alpha_1}{2^{1/3}} (h \underline{\sigma})^{2/3}. \end{equation} For all $h \in \mathbb{R}$ and $r>0$, we have \begin{equation}
\label{eqn:upperboundRwTwosided}
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \sup_{x \in \mathbb{R}} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{n-1} S^{(n)}_j} ; S^{(n)}_j \in [0, rn^{1/3}] \right]\\
\leq \frac{\underline{\sigma}^2}{r^2} \Psi\left( \tfrac{r^3}{\underline{\sigma}^2} h \right). \end{equation} \end{lemma}
\begin{proof} In this proof, we assume $h \geq 0$ --and $h>0$ if $r=+\infty$. The result for $h<0$ in \eqref{eqn:upperboundRwTwosided} can be deduced by symmetry and the formula $\Psi(h) - \Psi(-h)= -h$.
For all $r \in [0, +\infty)$, we write $f(r) = \frac{\underline{\sigma}^2}{r^2} \Psi\left( \tfrac{r^3}{\underline{\sigma}^2} h \right)$ and $f(+\infty) = \frac{\alpha_1}{2^{1/3}} (h \underline{\sigma})^{2/3}$. For all $x \in \mathbb{R}$, we use the convention $+\infty + x = x + \infty = +\infty$. By Lemmas \ref{lem:bmOnesided} and \ref{lem:bmTwosided}, for all $r \in [0,+\infty]$, we have \begin{equation}
\label{eqn:generalUpperbound}
\limsup_{t \to +\infty} \frac{1}{t} \log \sup_{x \in \mathbb{R}} \E_x\left[ e^{-h \int_0^t B_{\underline{\sigma}^2s}ds} ; B_{\underline{\sigma}^2s} \in [0,r], s \leq t \right]
\leq f(r), \end{equation} using the scaling property of the Brownian motion.
Let $A \in \mathbb{N}$ and $n \in \mathbb{N}$, we write $T = \ceil{A n^{2/3}}$ and $K = \floor{n/T}$. For all $k \leq K$, we write $m_k = k T$; applying the Markov property at time $m_K,m_{K-1},\ldots m_1$, we have \begin{multline}
\label{eqn:decompositionUpper}
\sup_{x \geq 0} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{n-1} S^{(n)}_j} ; S^{(n)}_j \in [0,rn^{1/3}], j \leq n \right]\\
\leq \prod_{k=0}^{K-1} \sup_{x \in \mathbb{R}} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{n-1} S^{(n,k)}_j} ; S^{(n,k)}_j \in [0,rn^{1/3}], j \leq T \right], \end{multline} where we write $S^{(n,k)}_j = S^{(n)}_0 + S^{(n)}_{m_k+j}- S^{(n)}_{m_k}$ for the time-inhomogeneous random walk starting at time $m_k$ and at position $x$ under $\mathbf{P}_x$. We now bound, uniformly in $k<K$, the quantity \[
E^{(n)}_k(r) = \sup_{x \in \mathbb{R}} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{T-1} S^{(n,k)}_j} ; S^{(n,k)}_j \in [0,rn^{1/3}], j \leq T \right]. \]
Let $k < K$, we write $t^k_j = \sum_{i=kT+1}^{kT + j} \sigma^2_{j/n}$. We apply Theorem \ref{thm:sakhanenko}, by \eqref{eqn:variance1} and \eqref{eqn:integrabilityRw1}, there exist Brownian motions $B^{(k)}$ such that, denoting by $\tilde{S}^{(n,k)}$ a random walk with same law as $S^{(n,k)}$ and $\Delta_n^k = \max_{j \leq T} \left| B^{(k)}_{t^k_j} - \tilde{S}^{(n,k)}_j \right|$, there exists $\mu>0$ such that for all $\epsilon>0$, $n \geq 1$ and $k \leq K$, \[
\mathbf{P}\left(\Delta_n^k \geq \epsilon n^{1/3}\right) \leq e^{-C_0 \mu \epsilon n^{1/3}} \E\left( e^{C_0 \mu \Delta_n^k} \right) \leq e^{-C_0 \mu \epsilon n^{1/3}} \left(1+\mu \bar{\sigma} A^{1/2} n^{1/3}\right), \] where we used \eqref{eqn:integrabilityRw1} (thus \eqref{eqn:integrabilityRwBis1}) and the exponential Markov inequality. Note in particular that for all $\epsilon>0$, $\mathbf{P}(\Delta_n^k \geq \epsilon n^{1/3})$ converges to 0 as $n \to +\infty$, uniformly in $k \leq K$. As a consequence, for all $\epsilon>0$ \begin{align*}
E^{(n)}_k(r)
&= \sup_{x \in \mathbb{R}} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{T-1} \tilde{S}^{(n,k)}_j} ; \tilde{S}^{(n,k)}_j \in [0,rn^{1/3}], j \leq T \right]\\
&\leq \sup_{x \in \mathbb{R}} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{T-1} \tilde{S}^{(n,k)}_j} \ind{\Delta_n^k \leq \epsilon n^{1/3}} ; \tilde{S}^{(n,k)}_j \in [0,rn^{1/3}], j \leq T \right] + \mathbf{P}\left(\Delta_n^k \geq \epsilon n^{1/3}\right). \end{align*} Moreover, \begin{align*}
&\sup_{x \in \mathbb{R}} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{T-1} \tilde{S}^{(n,k)}_j} \ind{\Delta_n^k \leq \epsilon n^{1/3}} ; \tilde{S}^{(n,k)}_j \in [0,rn^{1/3}], j \leq T \right]\\
& \qquad \leq \sup_{x \in \mathbb{R}} \E_x\left[ e^{h\frac{T}{n}\Delta_n^k-\frac{h}{n} \sum_{j=0}^{T-1} B_{t^k_j}} \ind{\Delta_n^k \leq \epsilon n^{1/3}} ; B_{t^k_j} \in [-\Delta_n^k,rn^{1/3}+\Delta_n^k], j \leq T \right]\\
& \qquad \leq \sup_{x \in \mathbb{R}} \E_x\left[ e^{h \frac{T}{n} \epsilon n^{1/3}-\frac{h}{n} \sum_{j=0}^{T-1} B_{t^k_j}}; B_{t^k_j} \in [-\epsilon n^{1/3}, (r+\epsilon)n^{1/3}], j \leq T \right] \leq e^{3 h A \epsilon} \tilde{E}^{(n)}_k(r+2\epsilon), \end{align*} setting $\tilde{E}^{(n)}_k(r) = \sup_{x \in \mathbb{R}} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{T-1} B_{t^k_j}} ; B_{t^k_j} \in [0, r n^{1/3}], j \leq T \right]$ for all $r \in [0,+\infty]$. We set $\tau^k_j = n^{-2/3}t^k_j$; by the scaling property of the Brownian motion, we have \[
\tilde{E}^{(n)}_k(r) = \sup_{x \in \mathbb{R}} \E_x\left[ e^{-\frac{h}{n^{2/3}} \sum_{j=0}^{T-1} B_{\tau^k_j}} ; B_{\tau^k_j} \in [0,r], j \leq T \right]. \]
We now replace the sum in $\tilde{E}$ by an integral: we set \[
\omega_{n,A} = \sup_{|t-s| \leq 2 A n^{-1/3}} \left| \sigma^2_t - \sigma^2_s \right| \quad \mathrm{and} \quad \Omega_{n,A} = \sup_{\substack{s,t \leq 2\bar{\sigma}^2 A + \omega_{n,A}\\ |t-s| \leq 2 A \omega_{n,A} + \bar{\sigma}^2 n^{-1/3}}} \left| B_t - B_s\right|. \] For all $k < K$ and $j \leq T$, we have \[
\left| \tau^k_j - j \sigma^2_{kT/n} n^{-2/3} \right| \leq n^{-2/3} \sum_{i=m_k+1}^{m_k+j} \left| \sigma^2_{i/n} - \sigma^2_{kT/n} \right| \leq 2 A \omega_{n,A}, \]
and $\sup_{s \in [j\underline{\sigma}^2/n,(j+1)\underline{\sigma}^2/n}\left| B_s - B_{t^k_j} \right| \leq \Omega_{n,A}$. As a consequence, for all $\epsilon>0$, we obtain \begin{align*}
\tilde{E}^{(n)}_k(r)
&\leq \sup_{x \in \mathbb{R}} \E_x\left[ e^{-\frac{h}{n^{2/3}} \sum_{j=0}^{T-1} B_{\tau^k_j}} \ind{\Omega_{n,A} \leq \epsilon} ; B_{\tau^k_j} \in [0,r], j \leq T \right] + \mathbf{P}(\Omega_{n,A} \geq \epsilon)\\
&\leq e^{3hA \epsilon} \sup_{x \in \mathbb{R}} \E_x\left[ e^{-h \int_0^A B_{\underline{\sigma}^2 s} ds} ; B_{\underline{\sigma}^2 s} \in [0,(r+2\epsilon)], s \leq A \right] + \mathbf{P}(\Omega_{n,A} \geq \epsilon). \end{align*} We set $\bar{E}^{A}(r) = \sup_{x \in \mathbb{R}} \E_x\left[ e^{-h \int_0^A B_{\underline{\sigma}^2 s} ds} ; B_{\underline{\sigma}^2 s} \in [0,r], s \leq A \right]$. As $B$ is continuous, we have $\lim_{n \to +\infty} \mathbf{P}(\Omega_{n,A} \geq \epsilon) = 0$ uniformly in $k<K$. Therefore \eqref{eqn:decompositionUpper} leads to \begin{align*}
&\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \sup_{x \geq 0} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{n-1} S^{(n)}_j} ; S^{(n)}_j \in [0,rn^{1/3}], j \leq n \right]\\
&\qquad \qquad \leq \limsup_{n \to +\infty} \frac{K}{n^{1/3}} \max_{k \leq K} \log E^{(n)}_k(r)\\
&\qquad \qquad \leq \frac{1}{A} \limsup_{n \to +\infty} \left[ 3 h A \epsilon + \max_{k \leq K} \log \left( \tilde{E}^{(n)}_k(r+2\epsilon) + \mathbf{P}\left(\Delta_n^k \geq \epsilon n^{1/3}\right) \right) \right]\\
&\qquad \qquad \leq 6 h \epsilon + \limsup_{n \to +\infty} \log \left[\bar{E}^A(r+4\epsilon) + \mathbf{P}(\Omega_{n,A} \geq \epsilon) + \max_{k \leq K} \mathbf{P}\left(\Delta_n^k \geq \epsilon n^{1/3}\right) \right]\\
&\leq 6h \epsilon + \frac{1}{A} \log \bar{E}^A(r+4\epsilon). \end{align*} We now use \eqref{eqn:generalUpperbound}, letting $A \to +\infty$, and thereby letting $\epsilon \to 0$, this yields \[
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \sup_{x \geq 0} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{n-1} S^{(n)}_j} ; S^{(n)}_j \in [0,rn^{1/3}], j \leq n \right] \leq f(r), \] which ends the proof. \end{proof}
Next, we derive lower bounds with similar computations. We set $I^{(n)}_{a,b} = [an^{1/3},bn^{1/3}]$. \begin{lemma} \label{lem:lowerboundRw} We assume \eqref{eqn:variance1} and \eqref{eqn:integrabilityRw1}. For all $h>0$, $0<a<b$ and $0<a'<b'$, we have \begin{equation}
\label{eqn:lowerboundRwOnesided}
\liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \inf_{x \in I^{(n)}_{a,b}} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{n-1} S^{(n)}_j} \ind{S_n \in I^{(n)}_{a',b'}} ; S^{(n)}_j \geq 0, j \leq n \right] \geq \frac{\alpha_1}{2^{1/3}} (h \bar{\sigma})^{2/3}, \end{equation} and for all $h \in \mathbb{R}$, $r>0$, $0<a<b<r$ and $0<a'<b'<r$, we have \begin{equation}
\label{eqn:lowerboundRwTwosided}
\liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \inf_{x \in I^{(n)}_{a,b}} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{n-1} S^{(n)}_j} \ind{S_n \in I^{(n)}_{a',b'}} ; S^{(n)}_j \in I^{(n)}_{0,r}, j \leq n \right]
\geq \frac{\bar{\sigma}^2}{r^2} \Psi\left( \tfrac{r^3}{\bar{\sigma}^2} h \right). \end{equation} \end{lemma}
\begin{proof} We once again assume $h \geq 0$; as if $h<0$ we can deduce \eqref{eqn:lowerboundRwTwosided} by symmetry and the formula $\Psi(h) - \Psi(-h)= h$. We write, for all $r \in [0, +\infty)$, $f(r) = \frac{\bar{\sigma}^2}{r^2} \Psi\left( \tfrac{r^3}{\bar{\sigma}^2} h \right)$ and $f(+\infty) = \frac{\alpha_1}{2^{1/3}} (h \bar{\sigma})^{2/3}$. By Lemmas \ref{lem:asymptoticOnesided} and \ref{lem:asymptoticTwosided}, for all $r \in [0,+\infty]$, $0<a<b<r$ and $0<a'<b'<r$, we have \begin{equation}
\label{eqn:generalLowerbound}
\liminf_{t \to +\infty} \frac{1}{t} \log \inf_{x \in [a,b]} \E_x\left[ e^{-h \int_0^t B_{\bar{\sigma}^2s}ds} \ind{B_t \in [a',b']} ; B_s \in [0,r], s \leq t \right]
\geq f(r). \end{equation}
We choose $u \in (a',b')$ and $\delta > 0$ such that $(u-3\delta,u+3\delta) \subset (a',b')$, and we introduce $J^{(n)}_{\delta} = I^{(n)}(u-\delta, u+\delta)$. We decompose again $[0,n]$ into subintervals of length of order $n^{2/3}$. Let $A \in \mathbb{N}$ and $n \in \mathbb{N}$, we write $T = \floor{A n^{2/3}}$ and $K = \floor{n/T}$. For all $k \leq K$, we set again $m_k = kT$, for all $r \in [0,+\infty]$, applying the Markov property at times $m_K,m_{K-1},\ldots m_1$ leads to \begin{multline}
\label{eqn:decompositionLower}
\inf_{x \in I^{(n)}_{a,b}} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{n-1} S^{(n)}_j} \ind{S_n \in I^{(n)}_{a',b'}} ; S^{(n)}_j \in I^{(n)}_{0,r}, j \leq n \right]\\
\geq \inf_{x \in I^{(n)}_{a,b}} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{T-1} S^{(n)}_j} \ind{S^{(n)}_T \in J^{(n)}_\delta} ; S^{(n)}_j \in I^{(n)}_{0,r}, j \leq T \right] \qquad \qquad \qquad\\
\qquad \quad\times \prod_{k=1}^{K-1} \inf_{x \in J^{(n)}_\delta} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{T-1} S^{(n,k)}_j} \ind{S^{(n,k)}_T \in J^{(n)}_\delta} ; S^{(n,k)}_j \in I^{(n)}_{0,r}, j \leq T \right]\\
\times\inf_{x \in J^{(n)}_\delta} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{n-KT} S^{(n,K)}_j} ; S^{(n,k)}_j \in I^{(n)}_{a',b'}, j \leq n-KT \right], \end{multline} where $S^{(n,k)}_j = S^{(n)}_0 + S^{(n)}_{m_k+j}- S^{(n)}_{m_k}$. Let $0<a<b<r$ and $0<a'<b'<r$, we set $\epsilon>0$ such that $a>8\epsilon$, $r-b>8\epsilon$ and $b'-a'>8\epsilon$. We bound uniformly in $k$ the quantity \[
E^{(n)}_k(r) = \inf_{x \in I^{(n)}_{a,b}} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{T-1} S^{(n,k)}_j} \ind{S^{(n,k)}_T \in I^{(n)}_{a',b'}} ; S^{(n,k)}_j \in I^{(n)}_{0,r}, j \leq T \right]. \]
To do so, we set once again, for $k<K$, $t^k_j = \sum_{i=kT+1}^{kT + j} \sigma^2_{j/n}$. By Theorem \ref{thm:sakhanenko}, we introduce a Brownian motion $B$ such that, denoting by $\tilde{S}^{(n,k)}$ a random walk with the same law as $S^{(n,k)}$ and setting $\Delta_n^k = \max_{j \leq T} \left| B_{t^k_j} - \tilde{S}^{(n,k)}_j \right|$, for all $\epsilon>0$, by \eqref{eqn:integrabilityRwBis1} and the exponential Markov inequality we get \[
\sup_{k \leq K} \mathbf{P}\left(\Delta_n^k \geq \epsilon n^{1/3}\right) \leq e^{-C_0 \mu \epsilon n^{1/3}} \left(1+\mu \bar{\sigma} A^{1/2} n^{1/3}\right), \] which converges to $0$, uniformly in $k$, as $n \to +\infty$. As a consequence, for all $\epsilon>0$ and $k < K$, \begin{align*}
E^{(n)}_k(r)
&= \inf_{x \in I^{(n)}_{a,b}} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{T-1} \tilde{S}^{(n,k)}_j} \ind{\tilde{S}^{(n,k)}_T \in I^{(n)}_{a',b'}} ; \tilde{S}^{(n,k)}_j \in I^{(n)}_{0,r}, j \leq T \right]\\
&\geq \inf_{x \in I^{(n)}_{a,b}} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{T-1} \tilde{S}^{(n,k)}_j} \ind{\tilde{S}^{(n,k)}_T \in I^{(n)}_{a',b'}} \ind{\Delta_n^k \leq \epsilon n^{1/3}} ; \tilde{S}^{(n,k)}_j \in I^{(n)}_{0,r}, j \leq T \right]\\
&\geq \inf_{x \in I^{(n)}_{a-\epsilon,b+\epsilon}} e^{-3hA \epsilon}\E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{T-1} B_{t^k_j}} \ind{B_{t^k_T} \in I^{(n)}_{a'+\epsilon,b'-\epsilon}} \ind{\Delta_n^k \leq \epsilon n^{1/3}} ; B_{t^k_j} \in I^{(n)}_{\epsilon,r-\epsilon}, j \leq T \right]\\
&\geq e^{-3 h A \epsilon} \left(\tilde{E}^{(n)}_k(r-2\epsilon) - \mathbf{P}(\Delta_n^k \geq \epsilon n^{1/3}) \right), \end{align*} where we set \[
\tilde{E}^{(n)}_k(r) = \inf_{x \in [a-2\epsilon,b+2\epsilon]} \E_x\left[ e^{-\frac{h}{n^{2/3}} \sum_{j=0}^{T-1} B_{\tau^k_j}} \ind{B_{\tau^k_T} \in [a'+2\epsilon,b'-2\epsilon]} ; B_s \in [0,r], s \leq \tau^k_T \right], \] and $\tau^k_j = t^k_jn^{-2/3}$. We also set \[
\omega_{n,A} = \sup_{|t-s| \leq 2 A n^{-1/3}} \left| \sigma^2_t - \sigma^2_s \right| \quad \mathrm{and} \quad \Omega_{n,A} = \sup_{\substack{s,t \leq 2\bar{\sigma}^2 A + \omega_{n,A}\\ |t-s| \leq 2 A \omega_{n,A} + \bar{\sigma}^2 n^{-1/3}}} \left| B_t - B_s\right|, \] so that for all $k < K$ and $j \leq T$, we have \[
\left| \tau^k_j - j \sigma^2_{kT/n} n^{-2/3} \right| \leq n^{-2/3} \sum_{i=m_k+1}^{m_k+j} \left| \sigma^2_{i/n} - \sigma^2_{kT/n} \right| \leq 2 A \omega_{n,A}, \]
and $\sup_{s \in [\underline{\sigma}^2\frac{j}{n},\underline{\sigma}^2\frac{j+1}{n}]}\left| B_s - B_{t^k_j} \right| \leq \Omega_{n,A}$. As a consequence, \begin{multline*}
\tilde{E}^{(n)}_k(r) e^{3hA\epsilon} \geq \\
\inf_{x \in [a-4\epsilon, b + 4\epsilon]} \E_x\left[ e^{-h \int_0^A B_{\underline{\sigma}^2 s} ds} \ind{B_{\underline{\sigma}^2 A} \in [a'+4\epsilon, b'-4\epsilon]} ; B_{\underline{\sigma}^2 s} \in [0,r-2\epsilon], s \leq A \right]
- \mathbf{P}(\Omega_{n,A} \geq \epsilon). \end{multline*}
This last estimate gives a lower bound for $E^{(n)}_k(r)$ which is uniform in $k \leq K$. As a consequence, \eqref{eqn:decompositionLower} yields \begin{multline*}
\liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \inf_{x \in I^{(n)}_{a,b}} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{n-1} S^{(n)}_j} \ind{S_n \in I^{(n)}_{a',b'}} ; S^{(n)}_j \in I^{(n)}_{0,r}, j \leq n \right] \geq \\
-6h\epsilon + \frac{1}{A} \log \inf_{x \in [a-4\epsilon, b + 4 \epsilon]} \E_x\left[ e^{-h \int_0^A B_{\underline{\sigma}^2 s} ds} \ind{B_{\underline{\sigma}^2 A} \in [a'+4\epsilon, b'-4\epsilon]} ; B_{\underline{\sigma}^2 s} \in [0,r-4\epsilon], s \leq A \right]. \end{multline*} Letting $A \to +\infty$, then $\epsilon \to 0$ leads to \[
\liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \inf_{x \in I^{(n)}_{a,b}} \E_x\left[ e^{-\frac{h}{n} \sum_{j=0}^{n-1} S^{(n)}_j} \ind{S_n \in I^{(n)}_{a',b'}} ; S^{(n)}_j \in I^{(n)}_{0,r}, j \leq n \right] \geq f(r), \] which ends the proof. \end{proof}
\subsection{Proof of Theorem \ref{thm:general_rw}} \label{subsec:conclusion}
We prove Theorem \ref{thm:general_rw} by decomposing $[0,n]$ into $A$ intervals of length $n/A$, and apply Lemmas~\ref{lem:upperboundRw} and \ref{lem:lowerboundRw}.
\begin{proof}[Proof of Theorem \ref{thm:general_rw}] We set $n \in \mathbb{N}$ and $A \in \mathbb{N}$. For all $0 \leq a \leq A$, we write $m_a = \floor{na/A}$, and $d_a = m_{a+1}-m_a$.
\paragraph*{Upper bound in \eqref{eqn:limsup}.} We apply the Markov property at times $m_{A-1}, m_{A-2}, \ldots m_1$, to see that \begin{multline*}
\sup_{x \in I^{(n)}_0} \E_x\left[ e^{\sum_{j=0}^{n-1} (h_{(j+1)/n}-h_{j/n}) S^{(n)}_j}; S^{(n)}_j \in I^{(n)}_j, j \leq n \right]\\
\leq \prod_{a=0}^{A-1} \underbrace{\sup_{x \in I^{(n)}_{m_a}} \E_x\left[ e^{\sum_{j=0}^{d_a-1} (h_{(m_a+j+1)/n}-h_{(m_a+j)/n}) S^{(n,a)}_j}; S^{(n,a)}_j \in I^{(n)}_{m_a+j}, j \leq d_a \right]}_{R^{(n)}_{a,A}}, \end{multline*} where $S^{(n,a)}_j = S^{(n)}_0 + S^{(n)}_{m_a+j} - S^{(n)}_{m_a}$ is the time-inhomogeneous random walk starting at time $m_a$ and position $x$. Letting $n \to +\infty$, this yields \begin{multline}
\label{eqn:upperDivision}
\limsup_{n \to +\infty} \sup_{x \in \mathbb{R}} \frac{1}{n^{1/3}} \log \E_x\left[ e^{\sum_{j=1}^n (h_{(j+1)/n}-h_{j/n}) S^{(n)}_j}; S^{(n)}_j \in I^{(n)}_{j}, j \leq n \right]\\
\leq \sum_{a=0}^{A-1} \limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log R^{(n)}_{a,A}. \end{multline}
To bound $R^{(n)}_{a,A}$, we replace functions $f,g$ and $\dot{h}$ by constants. We set, for all $A \in \mathbb{N}$ and $a \leq A$, \begin{multline*}
\bar{h}_{a,A} = \sup_{t \in [\frac{a-1}{A},\frac{a+2}{A}]} \dot{h}_t, \quad \underline{h}_{a,A} = \inf_{t \in [\frac{a-1}{A}, \frac{a+2}{A}]} \dot{h}_t,\\
g_{a,A} = \sup_{t \in [\frac{a-1}{A},\frac{a+2}{A}]} g_t, \quad f_{a,A} = \inf_{t \in [\frac{a-1}{A}, \frac{a+2}{A}]} f_t \quad \mathrm{and} \quad \sigma_{a,A} = \inf_{t \in [\frac{a-1}{A},\frac{a+2}{A}]} \sigma_s. \end{multline*} For any $n \in \mathbb{N}$ and $k \leq n$, by \eqref{eqn:paslineaire}, if $h_{(k+1)/n}>h_{k/n}$, then $k \in G_n$, and if $h_{(k+1)/n}<h_{k/n}$, then $k \in F_n$. Consequently, for all $x \in I^{(n)}_k$, \begin{equation}
\label{eqn:paslineaireConsequence}
(h_{(k+1)/n} - h_{k/n}) x \leq (h_{(k+1)/n}-h_{k/n})_+ g_{k/n} n^{1/3} - (h_{k/n} - h_{(k+1)/n})_+ f_{k/n} n^{1/3}. \end{equation} We bound from above $R^{(n)}_{a,A}$ in four different cases.
First, for all $a<A$, by \eqref{eqn:paslineaireConsequence}, we have \[
R^{(n)}_{a,A} \leq \exp\left( \sum_{j=m_a}^{m_{a+1}-1} (h_{(j+1)/n}-h_{j/n})_+ g_{j/n} n^{1/3} - (h_{j/n} - h_{(j+1)/n})_+ f_{k/n} n^{1/3} \right), \] and thus, \begin{equation}
\label{eqn:free}
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log R^{(n)}_{a,A} \leq \int_{a/A}^{(a+1)/A} (\dot{h}_s)_+ g_s - (\dot{h}_s)_- f_sds = \int_{a/A}^{(a+1)/A} \dot{h}_s g_s - (\dot{h}_s)_- (f_s-g_s) ds. \end{equation}
This crude estimate can be improved as follows. If $\underline{h}_{a,A} > 0$, then $[\frac{a}{A},\frac{a+1}{A}] \subset G$ and the upper bound $g_{k/n}n^{1/3}$ of the path is present at all times $k \in [m_a, m_{a+1}]$. As a consequence, \eqref{eqn:paslineaireConsequence} becomes \begin{equation}
\label{eqn:paslineaireConsequenceUpperfrontier}
\forall k \in [m_a,m_{a+1}), \sup_{x \in I^{(n)}_k} (h_{(k+1)/n}-h_{k/n})x \leq (h_{(k+1)/n}-h_{k/n})g_{a,A}n^{1/3} + \frac{1}{n} \underline{h}_{a,A}(x - g_{a,A}n^{1/3}). \end{equation} We have \begin{align*}
R^{(n)}_{a,A} &= \sup_{x \in I^{(n)}_{m_a}} \E_x\left[ e^{\sum_{j=0}^{d_a-1} (h_{(m_a+j+1)/n}-h_{(m_a+j)/n}) S^{(n),a}_j}; S^{(n),a}_j \in I^{(n)}_{m_a+j}, j \leq d_a \right]\\
&\leq e^{\sum_{j=m_a}^{m_{a+1}-1} (h_{(j+1)/n}-h_{j/n}) g_{a,A} n^{1/3}}\\
&\qquad\qquad \times \sup_{x \in I^{(n)}_{m_a}} \E_x \left[ e^{\frac{1}{n} \sum_{j=0}^{d_a-1} \underline{h}_{a,A} (S^{(n),a}_j - g_{a,A}n^{1/3})}; S^{(n),a}_j \in I^{(n)}_{m_a+j}, j \leq d_a \right]\\
&\leq e^{(h_{m_{a+1}/n} - h_{m_a/n})g_{a,A}n^{1/3}} \sup_{x \leq 0} \E_x\left[ e^{\frac{1}{n} \sum_{j=0}^{d_a-1} \underline{h}_{a,A} S^{(n),a}_j} ; S^{(n),a}_{m_a+j} \leq 0, j \leq d_a\right]. \end{align*} Letting $n \to +\infty$, we have \begin{multline*}
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log R^{(n)}_{a,A} \leq (h_{(a+1)/A}-h_{a/A}) g_{a,A} \\
+ \limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \sup_{x \leq 0} \E_x\left[ e^{\frac{\underline{h}_{a,A}}{n} \sum_{j=0}^{d_a-1} S^{(n),a}_j} ; S^{(n),a}_j \leq 0, j \leq d_a\right]. \end{multline*} As $d_a \sim_{n \to +\infty} n/A$, by \eqref{eqn:upperboundRwOnesided}, \begin{multline*}
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \sup_{x \leq 0} \E_x\left[ e^{\frac{\underline{h}_{a,A}}{A (d_a+1)} \sum_{j=0}^{d_a-1} S^{(n),a}_j} ; S^{(n),a}_j \leq 0, j \leq d_a \right]\\
\leq \frac{1}{A^{1/3}}\frac{\alpha_1}{2^{1/3}} \left(\tfrac{1}{A}\underline{h}_{a,A} \sigma_{a,A} \right)^{2/3} = \frac{\alpha_1}{2^{1/3}A} \left(\underline{h}_{a,A} \sigma_{a,A} \right)^{2/3}. \end{multline*} We conclude that \begin{equation}
\label{eqn:premierOnesided}
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log R^{(n)}_{a,A} \leq (h_{(a+1)/A}-h_{a/A}) g_{a,A} + \frac{\alpha_1}{2^{1/3}A} \left(\underline{h}_{a,A} \sigma_{a,A} \right)^{2/3}. \end{equation} By symmetry, if $\bar{h}_{a,A}<0$, then $[\frac{a}{A},\frac{a+1}{A}] \subset F$, $h_{(k+1)/n}<h_{k/n}$ and the lower bound of the path is present at all time, which leads to \begin{equation}
\label{eqn:secondOnesided}
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log R^{(n)}_{a,A} \leq (h_{(a+1)/A}-h_{a/A}) f_{a,A} + \frac{\alpha_1}{2^{1/3}A} \left(-\bar{h}_{a,A} \sigma_{a,A} \right)^{2/3}. \end{equation}
Fourth and the smallest upper bound; if $[\frac{a}{A},\frac{a+1}{A}] \subset F\cap G$, then both bounds of the path are present at any time in $[m_a,m_{a+1}]$, and, by \eqref{eqn:paslineaireConsequenceUpperfrontier}, setting $r_{a,A} = g_{a,A}-f_{a,A}$, \begin{multline*}
R^{(n)}_{a,A} \leq e^{(h_{m_{a+1}/n} - h_{m_a/n})g_{a,A}n^{1/3}} \\
\times \sup_{x \in [r_{a,A} n^{1/3},0]} \E_x\left[ e^{\frac{1}{n} \sum_{j=0}^{d_a-1} \underline{h}_{a,A} S^{(n),a}_j} ; S^{(n),a}_j \in [-r_{a,A} n^{1/3},0], j \leq d_a\right]. \end{multline*} We conclude that \begin{multline*}
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log R^{(n)}_{a,A} \leq (h_{(a+1)/A}-h_{a/A}) g_{a,A} \\
+ \limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \sup_{x \in [-r_{a,A} n^{1/3},0]} \E_x\left[ e^{\frac{\underline{h}_{a,A}}{n} \sum_{j=0}^{d_a-1} S^{(n),a}_j} ; S^{(n),a}_j \in [-r_{a,A} n^{1/3},0], j \leq d_a\right]. \end{multline*} Applying then \eqref{eqn:upperboundRwTwosided}, this yields \begin{multline*}
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \sup_{x \in [-r_{a,A} n^{1/3},0]} \E_x\left[ e^{\frac{\underline{h}_{a,A}}{n} \sum_{j=0}^{d_a-1} S^{(n),a}_j} ; S^{(n),a}_j \in [-r_{a,A} n^{1/3},0], j \leq d_a\right]\\
\leq \frac{\sigma_{a,A}^2}{A r_{a,A}^2} \Psi\left( \frac{r_{a,A}^3}{\sigma_{a,A}^2} \underline{h}_{a,A} \right), \end{multline*} which yields \begin{equation}
\label{eqn:twoSided}
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log R^{(n)}_{a,A} \leq (h_{(a+1)/A}-h_{a/A}) g_{a,A} + \frac{\sigma_{a,A}^2}{A (g_{a,A} - f_{a,A})^2} \Psi\left( \frac{(g_{a,A}- f_{a,A})^3}{\sigma_{a,A}^2} \underline{h}_{a,A} \right). \end{equation} We now let $A$ grow to $+\infty$ in \eqref{eqn:upperDivision}. By Riemann-integrability of $F,G$ and $\dot{h}$, we have \begin{multline}
\label{eqn:twoSidedLimit}
\limsup_{A \to +\infty} \sum_{\substack{0\leq a < A \\ [\frac{a}{A},\frac{a+1}{A}] \subset F\cap G}} \left[ (h_{(a+1)/A}-h_{a/A}) g_{a,A} + \frac{\sigma_{a,A}^2}{A (g_{a,A} - f_{a,A})^2} \Psi\left( \frac{(g_{a,A}- f_{a,A})^3}{\sigma_{a,A}^2} \underline{h}_{a,A} \right)\right]\\ \leq \int_{F\cap G} \dot{h}_s g_s + \frac{\sigma^2_s}{(g_s-f_s)^2} \Psi\left( \frac{(g_s-f_s)^3}{\sigma_s^2} \dot{h}_s \right) ds. \end{multline} Similarly, using the fact that $\dot{h}$ is non-negative on $F^c$, and non-positive on $G^c$, \eqref{eqn:premierOnesided} and \eqref{eqn:secondOnesided} lead respectively to \begin{multline}
\label{eqn:premierOnesidedLimit}
\limsup_{A \to +\infty} \sum_{\substack{0 \leq a < A \\ \underline{h}_{a,A}>0, [\frac{a}{A},\frac{a+1}{A}] \not\subset F\cap G}} \left[ (h_{(a+1)/A}-h_{a/A}) g_{a,A} + \frac{\alpha_1}{A 2^{1/3}} \left( \underline{h}_{a,A} \sigma_{a,A} \right)^{2/3}\right]\\
\leq \int_{F^c \cap G} \dot{h}_s g_s + \frac{\alpha_1}{2^{1/3}} \left( \dot{h}_s \sigma_s\right)^{2/3} ds , \end{multline} and to \begin{multline}
\label{eqn:secondOnesidedLimit}
\limsup_{A \to +\infty} \sum_{\substack{0 \leq a < A \\ \bar{h}_{a,A}<0, [\frac{a}{A},\frac{a+1}{A}] \not\subset F\cap G}} \left[(h_{(a+1)/A}-h_{a/A}) f_{a,A} + \frac{\alpha_1}{A 2^{1/3}} \left( - \bar{h}_{a,A} \sigma_{a,A} \right)^{2/3}\right]\\
\leq \int_{F^c \cap G} \dot{h}_s g_s + \dot{h}_s (f_s - g_s) + \frac{\alpha_1}{2^{1/3}} \left( -\dot{h}_s \sigma_s\right)^{2/3} ds. \end{multline} Finally, by \eqref{eqn:free}, \eqref{eqn:twoSidedLimit}, \eqref{eqn:premierOnesidedLimit} and \eqref{eqn:secondOnesidedLimit}, letting $A \to +\infty$, \eqref{eqn:upperDivision} yields \[
\limsup_{n \to +\infty} \sup_{x \in \mathbb{R}} \frac{1}{n^{1/3}} \log \E_x\left[ e^{\sum_{j=1}^n (h_{(j+1)/n}-h_{j/n}) S^{(n)}_j}; S^{(n)}_j \in I^{(n)}_{j}, j \leq n \right] \leq H^{F,G}_{f,g}. \]
\paragraph*{Lower bound in \eqref{eqn:liminf}} We now take care of the lower bound. We start by fixing $H>0$, and we write \[
I^{(n,H)}_{j} = I^{(n)}_j \cap [-Hn^{1/3}, Hn^{1/3}], \] letting $H$ grow to $+\infty$ at the end of the proof. We only need \eqref{eqn:lowerboundRwTwosided} here.
We choose $k \in \mathcal{C}([0,1])$ a continuous function such that $k_0=0$ and $k_1 \in (a,b)$ and $\epsilon>0$ such that for all $t \in [0,1]$, $k_t \in [f_t + 4\epsilon, g_t-4\epsilon]$ and $k_1 \in [a+4\epsilon, b-4\epsilon]$. We set \[
J^{(n)}_a = \left[ (k_{a/A} - \epsilon)n^{1/3}, (k_{a/A} + \epsilon)n^{1/3}\right]. \] We apply the Markov property at times $m_{A-1},\ldots m_1$, only considering random walk paths that are in interval $J^{(n)}_a$ at any time $m_a$. For all $n \geq 1$ large enough, we have \begin{align*}
\E & \left[ e^{\sum_{j=0}^{n-1} (h_{(j+1)/n}-h_{j/n}) S^{(n)}_j} \ind{\frac{S^{(n)}_n}{n^{1/3}} \in [a',b']} ; S^{(n)}_j \in \tilde{I}^{(n)}_j, j \leq n \right]\\
&\geq \prod_{a=0}^{A-1} \inf_{x \in I^{(n)}_{m_a}} \E_x\left[ e^{\sum_{j=0}^{d_a-1} (h_{(m_a+j+1)/n}-h_{(m_a+j)/n}) S^{(n,a)}_j} \ind{S^{(n,a)}_{d_a} \in J^{(n)}_{a+1}}; S^{(n,a)}_j \in I^{(n,H)}_{m_a+j}, j \leq d_a \right]\\
&=: \prod_{a=0}^{A-1} \tilde{R}^{(n)}_{a,A}, \end{align*} with the same random walk notation as in the previous paragraph. Therefore, \begin{equation}
\label{eqn:lowerDivision}
\liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \E\left[ e^{\sum_{j=1}^n (h_{(j+1)/n}-h_{j/n}) S^{(n)}_j}; S^{(n)}_j \in I^{(n)}_{j}, j \leq n \right]
\geq \sum_{a=0}^{A-1} \liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \tilde{R}^{(n)}_{a,A}. \end{equation} We now bound from below $\tilde{R}^{(n)}_{a,A}$, replacing functions $f,g$ and $\dot{h}$ by constants. We write here \[
f_{a,A} = \sup_{t \in [\frac{a-1}{A},\frac{a+2}{A}]} f_t, \quad g_{a,A} = \inf_{t \in [\frac{a-1}{A}, \frac{a+2}{A}]} g_t \quad \mathrm{and} \quad \sigma_{a,A} = \inf_{t \in [\frac{a-1}{A}, \frac{a+2}{A}]} \sigma_t, \] keeping notations $\bar{h}_{a,A}$ and $\underline{h}_{a,A}$ as above. We assume $A > 0$ is chosen large enough such that \[
\sup_{|t-s| \leq \frac{2}{A}} |f_t - f_s| + |g_t-g_s| + |k_t-k_s| \leq \epsilon. \]
We first observe that $[f_{a,A}n^{1/3}, g_{a,A} n^{1/3}] \subset I^{(n,H)}_j$ for all $j \in [m_a,m_{a+1}]$, therefore, writing $r_{a,A} = g_{a,A} - f_{a,A}$, \begin{multline*}
\tilde{R}^{(n)}_{a,A} \geq e^{(h_{m_{a+1}/n} - h_{m_a/n}) g_{a,A}n^{1/3}} \\
\times \inf_{x \in J^{(n)}_a} \E_x\left[ e^{\frac{\bar{h}_{a,A}}{n} \sum_{j=0}^{d_a-1} S^{(n,a)}_j} \ind{S^{(n,a)}_{d_a} \in J^{(n)}_{a+1}}; S^{(n,a)}_j \in [-r_{a,A}n^{1/3}, 0], j \leq d_a \right]. \end{multline*} Thus, by \eqref{eqn:lowerboundRwTwosided}, we have \begin{equation}
\label{eqn:twosidedLower}
\liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \tilde{R}^{(n)}_{a,A} \geq (h_{(a+1)/A} - h_{a/A}) g_{a,A} + \frac{\sigma_{a,A}^2}{A(g_{a,A}-f_{a,A})^2} \Psi\left( \tfrac{(g_{a,A}-f_{a,A})^3}{\sigma_{a,A}^2} \bar{h}_{a,A} \right). \end{equation}
This lower bound can be improved, if $[\frac{a}{A},\frac{a+1}{A}] \subset F^c$. We have $[-Hn^{1/3}, g_{a,A}n^{1/3}] \subset I^{(n,H)}_j$ for all $j \in [m_a,m_{a+1}]$, thus \begin{multline*}
\tilde{R}^{(n)}_{a,A} \geq e^{(h_{m_{a+1}/n} - h_{m_a/n}) g_{a,A}n^{1/3}} \\ \times \inf_{x \in J^{(n)}_a} \E_x\left[ e^{\frac{\bar{h}_{a,A}}{n} \sum_{j=0}^{d_a-1} S^{(n,a)}_j} \ind{S^{(n,a)}_{d_a} \in J^{(n)}_{a+1}}; S^{(n,a)}_j \in [-(H-g_{a,A})n^{1/3}, 0], j \leq d_a \right], \end{multline*} which leads to \begin{equation}
\label{eqn:premierOnesidedLower}
\liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \tilde{R}^{(n)}_{a,A} \geq (h_{(a+1)/A} - h_{a/A}) g_{a,A} + \frac{\sigma_{a,A}^2}{A(g_{a,A}+H)^2} \Psi\left( \tfrac{(g_{a,A}+H)^3}{\sigma_{a,A}^2} \bar{h}_{a,A} \right). \end{equation} By symmetry, if $[\frac{a}{A},\frac{a+1}{A}] \subset G^c$, we have \begin{equation}
\label{eqn:secondOnesidedLower}
\liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \tilde{R}^{(n)}_{a,A} \geq (h_{(a+1)/A} - h_{a/A}) f_{a,A} + \frac{\sigma_{a,A}^2}{A(H-f_{a,A})^2} \Psi\left(- \tfrac{(H-f_{a,A})^3}{\sigma_{a,A}^2} \underline{h}_{a,A} \right). \end{equation} As a consequence, letting $A \to +\infty$, by Riemann-integrability of $F$, $G$ and $\dot{h}$, \eqref{eqn:twosidedLower} leads to \begin{equation}
\label{eqn:twosidedLowerLimit}
\liminf_{A \to +\infty} \sum_{\substack{0 \leq a \leq A\\ [\frac{a}{A},\frac{a+1}{A}] \cap F\cap G \neq \emptyset}} \liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \tilde{R}^{(n)}_{a,A}\\
\geq \int_{F\cap G} \dot{h}_t g_t + \frac{\sigma^2_t}{(g_t-f_t)^2} \Psi\left( \tfrac{(g_t-f_t)^3}{\sigma_t^2} \dot{h}_t \right) dt. \end{equation} Similarly, \eqref{eqn:premierOnesidedLower} gives \begin{equation}
\label{eqn:premierOnesidedLowerLimit}
\liminf_{A \to +\infty} \sum_{\substack{0 \leq a \leq A\\ [\frac{a}{A},\frac{a+1}{A}] \subset F^c}} \liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \tilde{R}^{(n)}_{a,A}\\
\geq \int_{F^c} \dot{h}_t g_t + \frac{\sigma^2_t}{(g_t+H)^2} \Psi\left( \tfrac{(g_t+H)^3}{\sigma_t^2} \dot{h}_t \right) dt, \end{equation} and \eqref{eqn:secondOnesidedLower} gives \begin{multline}
\label{eqn:secondOnesidedLowerLimit}
\liminf_{A \to +\infty} \sum_{\substack{0 \leq a \leq A\\ [\frac{a}{A},\frac{a+1}{A}] \subset F \cap G^c}} \liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \tilde{R}^{(n)}_{a,A}\\
\geq \int_{F \cap G^c} \dot{h}_t g_t + \dot{h}_t(f_t - g_t) + \frac{\sigma^2_t}{(H-f_t)^2} \Psi\left( -\tfrac{(H-f_t)^3}{\sigma_t^2} \dot{h}_t \right) dt. \end{multline}
Finally, we recall that \[
\lim_{H \to +\infty} \frac{1}{H^{2/3}} \Psi(H) = \frac{\alpha_1}{2^{1/3}}. \] As $\dot{h}$ is non-negative on $G^c$ and null on $F^c \cap G^c$, by dominated convergence, we have \[
\lim_{H \to +\infty} \int_{F^c} \frac{\sigma^2_s}{(g_s+H)^2} \Psi\left( \tfrac{(g_s+H)^3}{\sigma_s^2} \dot{h}_s \right) ds = \int_{F^c} \frac{\alpha_1}{2^{1/3}} (\dot{h}_s \sigma_s)^{2/3} ds, \] and as $\dot{h}$ is non-positive on $F^c$, we have similarly, \[
\lim_{H \to +\infty} \int_{F \cap G^c} \frac{\sigma^2_s}{(H-f_s)^2} \Psi\left( - \tfrac{(H-f_s)^3}{\sigma^2_s} \dot{h}_s \right) ds = \int_{F\cap G^c} \frac{\alpha_1}{2^{1/3}} (-\dot{h}_s \sigma_s)^{2/3}ds. \] Consequently, letting $n$, then $A$, then $H$ grow to $+\infty$ --observe that $\epsilon$, given it is small enough, does not have any impact on the asymptotic-- we have \[
\liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \E_0\left[ e^{\sum_{j=0}^{n-1} (h_{(j+1)/n} - h_{j/n}) S^{(n)}_j} \ind{S^{(n)}_n \in [an^{1/3},bn^{1/3}]} ; S^{(n)}_j \in \tilde{I}^{(n)}_j, j \leq n \right]
\geq H_{f,g}^{F,G}. \]
\paragraph*{Conclusion} Using the fact that \begin{multline*}
\sup_{x \in \mathbb{R}} \E_x\left[ e^{\sum_{j=0}^{n-1} (h_{(j+1)/n} - h_{j/n})S^{(n)}_j} ; S^{(n)}_j \in I^{(n)}_j, j \leq n \right]\\
\geq \E_0\left[ e^{\sum_{j=1}^n (h_{(j+1)/n} - h_{j/n}) S^{(n)}_j} \ind{S^{(n)}_n \in [an^{1/3},bn^{1/3}]} ; S^{(n)}_j \in \tilde{I}^{(n)}_j, j \leq n \right], \end{multline*} the two inequalities we obtained above allow to conclude the proof. \end{proof}
\section{The many-to-one lemma and branching random walk estimates} \label{sec:path} In this section, we introduce a time-inhomogeneous version of the many-to-one lemma, that links some additive moments of the branching random walk with the random walk estimates obtained in the previous section. Using the well-established method in the branching random walk theory (see e.g. \cite{Aid13,AiJ11,AiS10,FaZ11,FaZ12,GHS11,HaR11,HuS09,MaZ13,Mal14} and a lot of others) that consists in proving the existence of a frontier via a first moment method, then bounding the tail distribution of maximal displacement below this frontier by estimation of first and second moments of the number of individuals below this frontier, and the Cauchy-Schwarz inequality. The frontier will be determined by a differential equation, which is solved in Section 5.
\subsection{Branching random walk notations and the many-to-one lemma} \label{subsec:manytoone}
The many-to-one lemma can be traced back at least to the early works of Peyrière \cite{Pey74} and Kahane and Peyrière \cite{KaP76}. This result has been used under many forms in the past years, extended to branching Markov processes in \cite{BiK04}. This is a very powerful tool that has been used to obtain different branching random walk estimates, see e.g. \cite{Aid13,AiJ11,AiS10,FaZ11,FaZ12,HaR11}. We introduce some additional branching random walk notation in a first time.
Let $(\mathbf{T},V)$ be a BRWtie of length $n$ with environment $(\mathcal{L}_t, t \in [0,1])$. We recall that $\mathbf{T}$ is a tree of height $n$ and that for any $u \in \mathbf{T}$, $|u|$ is the generation to which $u$ belongs, $u_k$ the ancestor of $u$ at generation $k$ and $V(u)$ the position of $u$. We introduce, for $k \leq n$, $\mathcal{F}_k = \sigma \left( (u,V(u)), |u| \leq k \right)$ the $\sigma$-field generated by the branching random walk up to generation $k$.
For $y \in \mathbb{R}$ and $k \leq n$, we denote by $\mathbf{P}_{k,y}$ the law of the time-inhomogeneous branching random walk $(\mathbf{T}^k, V^k)$ such that $\mathbf{T}^k$ is a tree of length $n-k$, and that $\{L^{u'}, u' \in \mathbf{T}^k, |u'| \leq n-k-1\}$ is a family of independent point processes, with $L^{u'}$ of law $\mathcal{L}_{(|u'|+k+1)/n}$. With this definition, we observe that conditionally on $\mathcal{F}_k$, for every individual $u \in \mathbf{T}$ alive at generation $k$, the subtree $\mathbf{T}^u$ of $\mathbf{T}$ rooted at $u$, with marks $V_{|\mathbf{T}^u}$ is a time-inhomogeneous branching random walk with law $\mathbf{P}_{|u|,V(u)}$, independent of the rest of the branching random walk $(\mathbf{T} \backslash \mathbf{T}^u, V)$.
We introduce $\varphi$ a continuous positive function on $[0,1]$ such that \begin{equation}
\label{eqn:phibiendef}
\forall t \in [0,1], \kappa_t(\varphi_t) < +\infty, \end{equation} and set, for $t \in [0,1]$ \begin{equation}
\label{eqn:meanandvariance}
b_t = \partial_\theta \kappa_t(\varphi_t) \quad \mathrm{and} \quad \sigma^2_t = \partial^2_\theta \kappa_t(\varphi_t). \end{equation} Let $(X_{n,k}, n \geq 1, k \leq n)$ be a triangular array of independent random variables such that for all $n \geq 1$, $k\leq n$ and $x \in \mathbb{R}$, we have \[
\mathbf{P}(X_{n,k} \leq x) = \E\left[ \sum_{\ell \in L_{k/n}} \ind{\ell \leq x} e^{\varphi_{k/n} \ell - \kappa_{k/n}(\varphi_{k/n})} \right], \] where $L_{k/n}$ is a point process of law $\mathcal{L}_{k/n}$. By \eqref{eqn:regularity} and \eqref{eqn:meanandvariance}, we have \[
\E(X_{n,k}) = b_{k/n} \quad \mathrm{and} \quad \E\left( (X_{n,k}-b_{k/n})^2 \right) = \sigma^2_{k/n}. \] For $k \leq n$, we denote by $S_k = \sum_{j=1}^k X_{n,j}$ the time-inhomogeneous random walk associated to $\varphi$, by $\bar{b}^{(n)}_k = \sum_{j=1}^k b_{j/n}$, by $\tilde{S}_k = S_k - \bar{b}^{(n)}_k$ the centred version of this random walk and by \begin{equation}
\label{eqn:energyDef}
E_k := \sum_{j=1}^k \varphi_{j/n} b_{j/n} - \kappa_{j/n}(\varphi_{j/n}) = \sum_{j=1}^k \kappa^*_{j/n}(b_{j/n}), \end{equation} by \eqref{eqn:legendreestimate}. Under the law $\mathbf{P}_{k,y}$, $(S_j, j \leq n-k)$ has the same law as $\left(y + \sum_{i=k+1}^{k+j+1} X_{n,i},j \leq n-k\right)$.
\begin{lemma}[Many-to-one lemma] \label{lem:manytoone} Let $n \geq 1$ and $k \leq n$. Under assumption \eqref{eqn:phibiendef}, for any measurable non-negative function $f$, we have \[
\E\left( \sum_{|u|=k} f(V(u_j),j \leq k) \right) = e^{-E_k}\E\left[ e^{-\varphi_{k/n} \tilde{S}_k + \sum_{j=0}^{k-1} (\varphi_{(j+1)/n}-\varphi_{j/n}) \tilde{S}_j} f(S_j, j \leq k) \right]. \] \end{lemma}
\begin{remark} As an immediate corollary of the many-to-one lemma, we have, for $p \leq n$, $y \in \mathbb{R}$ and $k \leq n-p$, \begin{multline*}
\E_{p,y} \left( \sum_{|u|=p} f(V(u_j), j \leq k) \right)\\
= e^{E_p-E_{k+p}} e^{\varphi_{p/n} y} \E_{p,y} \left[ e^{-\varphi_{(k+p)/n} \tilde{S}_k + \sum_{j=p}^{p+k-1}(\varphi_{(j+1)/n}-\varphi_{j/n}) \tilde{S}_{j-p}} f(S_j, j \leq k) \right]. \end{multline*} \end{remark}
\begin{proof} Let $n \geq 1$, $k \leq n$ and $f$ non-negative and measurable, we prove by induction on $k \leq n$ that \[
\E\left( \sum_{|u|=k} f(V(u_j),j \leq k) \right) = \E\left[ e^{-\sum_{j=1}^k \varphi_{j/n}X_{n,j} - \kappa_{j/n}(\varphi_{j/n})} f(S_j, j \leq k) \right]. \] We first observe that if $k=1$, by definition of $X_{n,1}$, we have \[
\E\left( \sum_{|u|=1} f(V(u)) \right) = \E\left[ e^{-\varphi_{1/n}X_{n,1} + \kappa_{1/n}(\varphi_{1/n})} f(X_{n,1}) \right]. \] Let $k \geq 2$. By conditioning on $\mathcal{F}_{k-1}$, we have \begin{align*}
\E\left( \sum_{|u|=k} f(V(u_j), j \leq k) \right) &= \E\left[ \sum_{|u|=k-1} \sum_{u' \in \Omega(u)} f(V(u'_j), j \leq k) \right]\\
&= \E\left( \sum_{|u|=k-1} g(V(u_j), j \leq k-1) \right), \end{align*} where, for $(x_j, j \leq k-1) \in \mathbb{R}^{k-1}$, \begin{align*}
g(x_j, j \leq k-1) &= \E\left[ \sum_{\ell \in L_{k/n}} f(x_1,\ldots x_{k-1}, x_{k-1} + \ell) \right]\\
&= \E\left[ e^{-\varphi_{k/n} X_{n,k} + \kappa_{k/n}(\varphi_{k/n})} f(x_1,\ldots x_{k-1}, x_{k-1} + X_{n,k}) \right]. \end{align*} Using the induction hypothesis, we conclude that \begin{align*}
\E\left( \sum_{|u|=k} f(V(u_j), j \leq k) \right) &= \E\left[ e^{-\sum_{j=1}^k \varphi_{j/n} X_{n,j} - \kappa_{j/n}(\varphi_{j/n})} f(S_j, j \leq k) \right]\\
&= e^{-E_k} \E\left[ e^{-\sum_{j=1}^k \varphi_{j/n} (X_{n,j}-b_{j/n})} f(S_j, j \leq k) \right]. \end{align*} Finally, we modify the exponential weight by the Abel transform, \begin{align*}
\sum_{j=1}^k \varphi_{j/n} (X_{n,j} - b_j)
&= \sum_{j=1}^k \varphi_{j/n} (\tilde{S}_j - \tilde{S}_{j-1})
= \sum_{j=1}^k \varphi_{j/n} \tilde{S}_j - \sum_{j=1}^k \varphi_{j/n} \tilde{S}_{j-1}\\
&= \sum_{j=1}^k \varphi_{j/n} \tilde{S}_j - \sum_{j=1}^{k-1} \varphi_{(j+1)/n} \tilde{S}_j
= \varphi_{k/n} \tilde{S}_k - \sum_{j=1}^{k-1} (\varphi_{(j+1)/n}-\varphi_{j/n}) \tilde{S}_j, \end{align*} which ends the proof. \end{proof}
\subsection{Number of individuals staying along a path} \label{subsec:moments}
In this section, we bound some quantities related to the number of individuals that stay along a path. We start with an upper bound of the expected number of individuals that stay in the path until some time $k \leq n$, and then exit the path by the upper frontier. Subsequently, we bound the probability that there exists an individual that stays in the path until time $n$. We then compute the first two moments of the number of such individuals, and apply the Cauchy-Schwarz inequality to conclude. We assume in this section that \begin{equation}
\label{eqn:regularphi}
\varphi \text{ is absolutely continuous, with a Riemann-integrable derivative } \dot{\varphi}, \end{equation} as we plan to apply Theorem \ref{thm:general_rw} with function $h = \varphi$. Under this assumption, $\varphi$ is Lipschitz, thus so is $b$. As a consequence, we have \begin{equation}
\label{eqn:energy}
\sup_{\substack{n \in \mathbb{N}\\ k \leq n}} \sup_{t \in [\frac{k-1}{n},\frac{k+2}{n}]} \left| E_k - n K^*(b)_t \right| < +\infty \quad \mathrm{and} \quad \sup_{\substack{n \in \mathbb{N}\\ k \leq n}} \sup_{t \in [\frac{k-1}{n},\frac{k+2}{n}]} \left| \bar{b}^{(n)}_k - n \int_0^t b_s ds \right| < +\infty. \end{equation}
Let $f<g$ be two continuous functions such that $f(0)<0<g(0)$, and $F$ and $G$ two Riemann-integrable subsets of $[0,1]$ such that \begin{equation}
\label{eqn:FetG}
\{ t \in [0,1] : \dot{\varphi}_t < 0 \} \subset F \quad \mathrm{and} \quad \{ t \in [0,1] : \dot{\varphi}_t > 0 \} \subset G. \end{equation} We write, for $t \in [0,1]$ \begin{multline}
\label{eqn:defineK}
H_t^{F,G}(f,g,\varphi) = \int_0^t \dot{\varphi}_s g_s ds + \int_0^t \mathbf{1}_{F \cap G}(s) \frac{\sigma_s^2}{(g_s - f_s)^2} \Psi\left( \tfrac{(g_s-f_s)^3}{\sigma_s^2} \dot{\varphi}_s \right) ds\\
+ \int_0^t \mathbf{1}_{F^c \cap G}(s) \frac{a_1}{2^{1/3}} (\dot{\varphi}_s \sigma_s)^{2/3} + \mathbf{1}_{F \cap G^c} \left( \dot{\varphi}_s(f_s - g_s) + \frac{a_1}{2^{1/3}} (-\dot{\varphi}_s \sigma_s)^{2/3} \right) ds. \end{multline} We keep notation of Section \ref{sec:rw}: $F_n$ and $G_n$ are the subsets of $\{0,\ldots n-1\}$ defined in \eqref{eqn:deffnandgn}, and the path $I^{(n)}_k$ as defined in \eqref{eqn:defIn}. We are interested in the individuals $u$ alive at generation $n$ such that for all $k \leq n$, $V(u_k)-\bar{b}^{(n)}_k \in I^{(n)}_k$.
\subsubsection{A frontier estimate} \label{subsubsec:frontier}
We compute the number of individuals that stayed in $\bar{b}^{(n)} + I^{(n)}$ until some time $k-1$ and then crossed the upper boundary $\bar{b}^{(n)}_k + g_{k/n} n^{1/3}$ of the path at time $k \in G_n$. We denote by \[
\mathcal{A}_n^{F,G}(f,g) = \left\{ u \in \mathbf{T}, |u| \in G_n : V(u) - \bar{b}^{(n)}_{|u|} > g_{|u|/n}n^{1/3} , V(u_j) - \bar{b}^{(n)}_j \in I^{(n)}_j , j < |u|\right\}, \] the set of such individuals, and by $A_n^{F,G}(f,g) = \# \mathcal{A}_n^{F,G}(f,g)$.
\begin{lemma} \label{lem:estimate_upperfrontier} Under the assumptions \eqref{eqn:regularity}, \eqref{eqn:phibiendef}, \eqref{eqn:regularphi} and \eqref{eqn:FetG}, if $G \subset \{ t \in [0,1] : K^*(b)_t = 0\}$, \begin{align*}
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \E(A_n^{F,G}(f,g)) \leq \sup_{t \in [0,1]} \left[K^{F,G}_t(f,g,\varphi) - \varphi_t g_t \right]. \end{align*} \end{lemma}
\begin{remark} Observe that in order to use this lemma, we need to assume that \[ \{ t \in [0,1] : \dot{\varphi}_t > 0 \} \subset G \subset \{ t \in [0,1] : K^*(b)_t = 0\}, \] we cannot consider paths of speed profile $b$ such that the associated parameter $\varphi$ increases at a time when there is an exponentially large number of individuals following the path. For such paths, the mean of $A_n$ grows exponentially fast. \end{remark}
\begin{proof} By \eqref{eqn:phibiendef} and Lemma \ref{lem:manytoone}, we have \begin{align*}
\E(A_n^{F,G}(f,g)) &= \sum_{k\in G_n} \E\left[ \sum_{|u|=k} \ind{V(u) - \bar{b}^{(n)}_{k} > g_{k/n}n^{1/3}} \ind{V(u_j) - \bar{b}^{(n)}_j \in I^{(n)}_j , j < k} \right]\\
&= \sum_{k \in G_n} e^{-E_k} \E\left[ e^{-\varphi_{k/n} \tilde{S}_k + \sum_{j=0}^{k-1} (\varphi_{(j+1)/n} - \varphi_{j/n})\tilde{S}_j} \ind{\tilde{S}_k > g_{k/n} n^{1/3}} \ind{\tilde{S}_j \in I^{(n)}_j, j < k} \right]. \end{align*} For all $k \in G_n$, there exists $t \in [k/n,(k+1)/n]$ such that $t \in G$, thus $K^*(b)_t=0$. By \eqref{eqn:energy}, this implies that $\sup_{n \in \mathbb{N}, k \in G_n} E_k< +\infty$, hence \[
\E(A_n^{F,G}(f,g))
\leq C \sum_{k \in G_n} e^{-\varphi_{k/n} g_{k/n} n^{1/3}} \E\left[ e^{\sum_{j=0}^{k-1} (\varphi_{(j+1)/n}-\varphi_{j/n})\tilde{S}_j} \ind{\tilde{S}_k > g_{k/n} n^{1/3}} \ind{\tilde{S}_j \in I^{(n)}_j, j < k} \right]. \] As \eqref{eqn:FetG} is verified, similarly to \eqref{eqn:paslineaireConsequence}, for all $k \leq n$ and $x \in I^{(n)}_k$, we have \begin{equation}
\label{eqn:FetGconsequence}
(\varphi_{(k+1)/n} - \varphi_{k/n}) x \leq (\varphi_{(k+1)/n}-\varphi_{k/n})_+ g_{k/n} n^{1/3} - (\varphi_{k/n} - \varphi_{(k+1)/n})_+ f_{k/n} n^{1/3}. \end{equation}
In particular, $(\varphi_{(k+1)/n}-\varphi_{k/n}) x \leq \left| \varphi_{(k+1)/n} - \varphi_{k/n} \right| \left( \norme{f}_\infty + \norme{g}_\infty \right)$. Let $A>0$ be a large integer. For $a < A$, we set $m_a = \floor{an/A}$ and \begin{multline*}
\underline{g}_{a,A} = \inf\left\{g_t,t \in \left[\tfrac{a-1}{A},\tfrac{a+2}{A}\right]\right\}, \quad \underline{\varphi}_{a,A} = \inf\left\{ \varphi_t, t \in \left[\tfrac{a-1}{A},\tfrac{a+2}{A}\right] \right\}\\ \mathrm{and} \quad d_{a,A} = (\norm{f}_\infty + \norm{g}_\infty)\int_{(a-1)/A}^{(a+2)/A} |\dot{\varphi}_s| ds. \end{multline*} For $k \in (m_a, m_{a+1}]$, applying the Markov property at time $m_a$, we have \[
\E\left[ e^{\sum_{j=0}^{k-1} (\varphi_{(j+1)/n}-\varphi_{j/n}) \tilde{S}_j} \ind{\tilde{S}_k > g_{k/n} n^{1/3}} \ind{\tilde{S}_j \in I^{(n)}_j, j < k} \right] \leq \exp\left(d_{a,A} n^{1/3}\right) \Phi_{a,A}^{(n)}, \] where $\Phi_{a,A}^{(n)} = \E\left[ e^{\sum_{j=1}^{m_a}(\varphi_{(j+1)/n}-\varphi_{j/n}) \tilde{S}_j} \ind{\tilde{S}_j \in I^{(n)}_j, j \leq m_a} \right]$. We observe that $\tilde{S}$ is a centred random walk which, by \eqref{eqn:meanandvariance}, verifies \eqref{eqn:variance_rw} with variance function $\sigma^2$. Moreover, as \begin{multline*}
\E\left[ e^{\mu|X_{n,k}|} \right] \leq \E\left[ e^{\mu X_{n,k}} + e^{-\mu X_{n,k}} \right]
\leq \E\left[ \sum_{\ell \in L_{k/n}} e^{(\varphi_t+\mu) \ell - \kappa_t(\varphi_t)} + e^{(\varphi_t - \mu) \ell - \kappa_t(\varphi_t)} \right]\\ \leq e^{\kappa_t(\varphi_t+\mu) - \kappa_t(\varphi_t)} + e^{\kappa_t(\varphi_t-\mu) - \kappa_t(\varphi_t)}, \end{multline*}
by \eqref{eqn:regularity}, there exists $\mu>0$ such that $\sup_{n \in \mathbb{N}, k \leq n} \E\left[ e^{\mu|X_{n,k}|} \right] < +\infty$ and \eqref{eqn:integrability_rw} is verified. For all $a \leq A$, we apply Theorem \ref{thm:general_rw}, to $h_t = \varphi_{t\wedge a/A}$, functions $f$ and $g$ and intervals $F$ and $G$ stopped at time $a/A$. We have $ \displaystyle \limsup_{n \to +\infty} \frac{\log \Phi_{a,A}^{(n)}}{n^{1/3}} = K_{a/A}^{F,G}(f,g,\varphi)$.
We observe that \[
\E(A^{F,G}_n(f,g)) \leq C \sum_{a=0}^{A-1} \frac{n}{A} \exp\left( \left(d_{a,A} - \underline{\varphi}_{a,A} \underline{g}_{a,A}\right) n^{1/3}\right) \Phi_{a,A}^{(n)}. \] Letting $n \to +\infty$, we have \[
\limsup_{n \to +\infty} \frac{\log E(A_n(f,g))}{n^{1/3}} \leq \max_{a < A} K_{a/A}^{F,G}(f,g,\varphi) - \underline{\varphi}_{a,A} \underline{g}_{a,A} + d_{a,A}. \] By uniform continuity of $K,g,\varphi$, and as $\lim_{A \to +\infty} d_{a,A} = 0$, letting $A \to +\infty$, we have \[
\limsup_{n \to +\infty} \frac{\log E(A_n^{F,G}(f,g))}{n^{1/3}} \leq \sup_{t \in [0,1]} \left[ H_t^{F,G}(f,g,\varphi) - \varphi_t g_t \right]. \] \end{proof}
Lemma \ref{lem:estimate_upperfrontier} is used to obtain an upper bound for the maximal displacement among individuals that stay above $\bar{b}^{(n)}_k + n^{1/3}f_{k/n}$ at any time $k \in F_n$. If the quantity $\sup_{t \in [0,1]} \left[H_t^{F,G}(f,g,\varphi) - \varphi_t g_t \right]$ is negative, then with high probability, no individual crosses the frontier $\bar{b}^{(n)}_k + n^{1/3} g_{k/n}$ at time $k \in (G \cup \{1\})_n$. In particular, there is at time $n$ no individual above $\bar{b}^{(n)}_n + g_1 n^{1/3}$. If we choose $g$ and $G$ in a proper manner, the upper bound obtained here is tight.
\subsubsection{Concentration estimate by a second moment method} \label{subsubsec:secondmoment}
We take interest in the number of individuals which stay at any time $k \leq n$ in $\bar{b}^{(n)}_k + I^{(n)}_k$. For all $0<x < g_1 - f_1$, we set \[
\mathcal{B}_n^{F,G} (f,g,x) = \left\{ |u|=n : V(u_j) - \bar{b}^{(n)}_j \in \tilde{I}^{(n)}_j, j \leq n, V(u)-\bar{b}^{(n)}_n\geq (g_1-x)n^{1/3}\right\}, \] where $\tilde{I}^{(n)}_j = I^{(n)}_j \cap [-n^{2/3},n^{2/3}]$. We denote by $B_n^{F,G}(f,g,x) = \# \mathcal{B}_n^{F,G}(f,g,x)$. In order to bound from above the probability that $\mathcal{B}_n \neq \emptyset$, we compute the mean of $B_n$.
\begin{lemma} \label{lem:firstorder} We assume \eqref{eqn:regularity}, \eqref{eqn:phibiendef}, \eqref{eqn:regularphi} and \eqref{eqn:FetG}. If $K^*(b)_1=0$ then \begin{equation*}
\lim_{n \to +\infty} \frac{1}{n^{1/3}} \log \E(B^{F,G}_n(f,g,x)) = K_1^{F,G}(f,g,\varphi) - \varphi_1 (g_1-x). \end{equation*} \end{lemma}
\begin{proof}
Observe that, as $K^*(b)_1 = 0$, by \eqref{eqn:energy} $|E_n|$ is bounded by a constant uniformly in $n \in \mathbb{N}$. Using the many-to-one lemma, we have \begin{align*}
\E(B_n^{F,G}(f,g,x)) &= e^{-E_n}\E\left[ e^{-\varphi_1 \tilde{S}_n + \sum_{j=1}^n (\varphi_{(j+1)/n}-\varphi_{j/n}) \tilde{S}_j} \ind{\tilde{S}_j \in \tilde{I}^{(n)}_j, j \leq n} \ind{\tilde{S}_n \geq (g_1 - x)n^{1/3}} \right]\\
&\leq C e^{-\varphi_1 (g_1 - x)n^{1/3}} \E\left[ e^{\sum_{j=1}^n (\varphi_{(j+1)/n}-\varphi_{j/n}) \tilde{S}_j} \ind{\tilde{S}_j \in I^{(n)}_j, j \leq n} \right]. \end{align*} Therefore applying Theorem \ref{thm:general_rw}, we have \[
\limsup_{n \to +\infty} \frac{\log \E(B^{F,G}_n(f,g,x))}{n^{1/3}} = K_1^{F,G}(f,g,\varphi) - \varphi_1 (g_1-x). \]
We compute a lower bound for $\E(B_n)$. Applying Lemma \ref{lem:manytoone}, for any $\epsilon>0$ we have \begin{multline*}
\E(B_n^{F,G}(f,g,x))\\ \geq e^{-E_n}\E\left[ e^{-\varphi_1 \tilde{S}_n + \sum_{j=1}^n (\varphi_{(j+1)/n}-\varphi_{j/n}) \tilde{S}_j} \ind{\tilde{S}_j \in \tilde{I}^{(n)}_j, j \leq n} \ind{\tilde{S}_n - (g_1 - x)n^{1/3} \in [0,\epsilon n^{1/3}]} \right]\\
\geq c e^{-\varphi_1 (g_1 - x +\epsilon)n^{1/3}} \E\left[ e^{\sum_{j=1}^n (\varphi_{(j+1)/n}-\varphi_{j/n}) \tilde{S}_j} \ind{\tilde{S}_j \in \tilde{I}^{(n)}_j, j \leq n} \ind{\tilde{S}_n - (g_1 - x)n^{1/3} \in [0,\epsilon n^{1/3}]}\right]. \end{multline*} Applying Theorem \ref{thm:general_rw} again, we have \[
\liminf_{n \to +\infty} \frac{\log \E\left(B^{F,G}_n(f,g,x)\right)}{n^{1/3}} \geq K_1^{F,G}(f,g,\varphi) - \varphi_1 (g_1 - x + \epsilon). \] Letting $\epsilon \to 0$ concludes the proof. \end{proof}
To obtain a lower bound for $\mathbf{P}(\mathcal{B}_n \neq \emptyset)$, we compute an upper bound for the second moment of $B_n$. We assume \begin{equation}
\label{eqn:integrability2phi}
\sup_{t \in [0,1]} \E\left[ \left(\sum_{\ell \in L_t} e^{\varphi_t \ell} \right)^2 \right] < +\infty \end{equation} which enables to bound the second moment of $B_n$.
\begin{lemma} \label{lem:secondmoment_estimate} Under the assumptions \eqref{eqn:regularity}, \eqref{eqn:phibiendef}, \eqref{eqn:regularphi}, \eqref{eqn:FetG} and \eqref{eqn:integrability2phi}, if $G = [0,1]$, $K^*(b)_1=0$ and for all $t \in [0,1]$, $K^*(b)_t \leq 0$, then \begin{multline*}
\limsup_{n \to + \infty} \frac{1}{n^{1/3}} \log \E\left({B^{F,G}_n(f,g,x)}^2\right)\\
\leq 2 \left[ K^{F,G}_1(f,g,\varphi) - \varphi_1 (g_1 - x) \right] - \inf_{t \in [0,1]}\left[ H_t^{F,G}(f,g,\varphi)- \varphi_t g_t\right]. \end{multline*} \end{lemma}
\begin{proof} In order to estimate the second moment of $B_n$, we decompose the pairs of individuals $(u,u') \in \mathbf{T}^2$ according to their most recent common ancestor $u \wedge u'$ as follows: \begin{multline*}
\E\left[ B^{F,G}_n(f,g,x)^2 \right]
= \sum_{k = 0}^n \E\left[ \sum_{\substack{|u|=|v|=n\\ |u \wedge u'|=k}} \ind{u \in \mathcal{B}^{F,G}_n(f,g,x)} \ind{u' \in \mathcal{B}^{F,G}_n(f,g,x)} \right]\\
= \E\left[ B^{F,G}_n(f,g,x) \right] + \sum_{k=0}^{n-1} \E\left[ \sum_{|u|=k} \ind{V(u_j) - \bar{b}^{(n)}_j \in \tilde{I}^{(n)}_j, j \leq k} \sum_{u_1 \neq u_2 \in \Omega(u)} \Lambda(u_1)\Lambda(u_2) \right], \end{multline*}
where, for $u' \in \mathbf{T}$, we denote by $\Lambda(u') = \sum_{|u|=n, u > u'} \ind{u \in \mathcal{B}^{F,G}_n(f,g,x)}$ the number of descendants of $u'$ which are in $\mathcal{B}_n$. We observe that for any two distinct individuals $|u_1|=|u_2|=k$, conditionally to $\mathcal{F}_{k}$, the quantities $\Lambda(u_1)$ and $\Lambda(u_2)$ are independent.
By the Markov property applied at time $k$, for all $u' \in \mathbf{T}$ with $|u'|=k$, we have \begin{align*}
\E\left[ \left.\Lambda(u') \right| \mathcal{F}_k \right]
&= \E_{k,V(u')}\left[ \left. \sum_{|u|=n-k} \ind{V(u) - \bar{b}^{(n)}_n \geq (g_1 - x)n^{1/3}} \ind{V(u_j) - \bar{b}^{(n)}_{j+k} \in \tilde{I}^{(n)}_j, j \leq n-k} \right| \mathcal{F}_k \right]\\
&= \exp\left( - E_n + E_k + \varphi_{k/n} (V(u') - \bar{b}^{(n)}_k)\right)\\
&\qquad \times \E_{k,V(u')} \left[ e^{-\varphi_1 \tilde{S}_{n-k} + \sum_{j=0}^{n-k-1} \Delta \varphi_{n,k+j} \tilde{S}_j} \ind{\tilde{S}_j \in \tilde{I}^{(n)}_{j+k}, j \leq n-k} \ind{\tilde{S}_{n-k} \geq (g_1 - x)n^{1/3}} \right], \end{align*} using the many-to-one lemma. Therefore, \begin{multline*}
\E\left[ \left. \Lambda(u') \right| \mathcal{F}_k \right] \leq C \exp\left(E_k + \varphi_{k/n} (V(u') - \bar{b}^{(n)}_k) -\varphi_1 (g_1 - x)n^{1/3}\right) \\
\times \E_{k,V(u')}\left[ e^{\sum_{j=0}^{n-k-1}\Delta \varphi_{n,j+k} \tilde{S}_j} \ind{\tilde{S}_j \in I^{(n)}_{j+k}, j \leq n-k} \right]. \end{multline*}
Let $A > 0$ be a large integer, and for $a \leq A$, let $m_a = \floor{ an/A}$. We introduce \begin{multline*}
\Phi_{a,A}^\mathrm{start} = \E\left[ \exp\left(\sum_{j=0}^{m_a-1} (\varphi_{(j+1)/n}-\varphi_{j/n}) \tilde{S}_j\right) \ind{\tilde{S}_j \in \tilde{I}^{(n)}_j, j \leq m_a} \right] \quad \mathrm{and}\\
\Phi^\mathrm{end}_{a,A} = \sup_{y \in \mathbb{R}} \E_{m_a,y} \left[ \exp\left(\sum_{j=0}^{n-m_a-1} \Delta \varphi_{n,m_a+j} \tilde{S}_j\right) \ind{\tilde{S}_j \in I^{(n)}_{m_a+j}, j \leq n-m_a} \right]. \end{multline*} By Theorem \ref{thm:general_rw}, we have \[
\limsup_{n \to +\infty} \frac{\log \Phi^\mathrm{start}_{a,A}}{n^{1/3}} = K_{a/A}^{F,G}(f,g,\varphi)\quad \mathrm{and} \quad \limsup_{n \to +\infty} \frac{\log \Phi^\mathrm{end}_{a,A}}{n^{1/3}} = K_1^{F,G}(f,g,\varphi) - K_{a/A}^{F,G}(f,g,\varphi). \] Moreover, using the same estimates as in Lemma \ref{lem:estimate_upperfrontier}, and setting \begin{multline*}
\bar{g}_{a,A} = \sup\left\{g_t,t \in \left[\tfrac{a-1}{A},\tfrac{a+1}{A}\right]\right\}, \quad \bar{\varphi}_{a,A} = \sup\left\{ \varphi_t, t \in \left[\tfrac{a-1}{A},\tfrac{a+1}{A}\right] \right\}\\ \mathrm{and} \quad d_{a,A} = \int_{(a-1)/A}^{(a+1)/A} |\dot{\varphi}_s| ds (\norm{f}_\infty + \norm{g}_\infty), \end{multline*} for all $k \in [m_a,m_{a+1})$, applying the Markov property at time $m_{a+1}$, we have \begin{equation}
\label{eqn:endpart}
\E\left[ \Lambda(u') | \mathcal{F}_k \right] \leq C e^{E_k + \varphi_{k/n} (V(u') - \bar{b}^{(n)}_k)} \exp\left( \left(d_{a,A}-\varphi_1 (g_1 - x)\right)n^{1/3}\right) \Phi^\mathrm{end}_{a+1,A}. \end{equation}
We observe that for all $u \in \mathbf{T}$ with $|u|=k$ and $V(u) \in \tilde{I}^{(n)}_k$ we have \begin{align}
\E\left[ \left. \sum_{u_1 \neq u_2 \in \Omega(u)} e^{\varphi_{(k+1)/n} (V(u_1)+V(u_2))} \right| \mathcal{F}_k \right] &\leq e^{2 \varphi_{(k+1)/n} V(u)} \E\left[ \left(\sum_{\ell \in L_{(k+1)/n}} e^{\varphi_{(k+1)/n} \ell} \right)^2 \right] \nonumber\\
&\leq C e^{2 \varphi_{k/n} V(u)} e^{n^{2/3}|\varphi_{(k+1)/n} - \varphi_{k/n}|} \leq Ce^{2 \varphi_{k/n} V(u)},\label{eqn:middlepart} \end{align} using \eqref{eqn:integrability2phi} and the fact that $\varphi$ is Lipschitz. We now bound, for $k \in [m_a, m_{a+1})$ \begin{multline}
\E\left[ \sum_{|u|=k} e^{2 \varphi_{k/n}(V(u) - \bar{b}^{(n)}_k)} \ind{V(u_j) - \bar{b}^{(n)}_j \in \tilde{I}^{(n)}_j, j \leq k} \right]\\
= \E\left[ e^{\varphi_{k/n} \tilde{S}_k + \sum_{j=0}^{k-1} (\varphi_{(j+1)/n}-\varphi_{j/n}) \tilde{S}_j} \ind{\tilde{S}_j \in \tilde{I}^{(n)}_j, j \leq k} \right], \end{multline} using Lemma \ref{lem:manytoone}. As $\sup_{t \in [0,1]} K^*(b)_t \leq 0$ and by \eqref{eqn:energy}, $E_k$ is bounded from above uniformly in $n \in \mathbb{N}$ and $k \leq n$. As $G_n = \{0,\ldots, n\}$, for all $n \in \mathbb{N}$ large enough and $k \in [m_a, m_{a+1})$, applying the Markov property at time $m_an$, it yields \begin{equation}
\label{eqn:startpart}
\E\left[ \sum_{|u|=k} e^{2 \varphi_{k/n}(V(u) - \bar{b}^{(n)}_k)} \ind{V(u_j) \in \tilde{I}^{(n)}_j, j \leq k} \right]
\leq \exp\left( \left(\bar{\varphi}_{a,A} \bar{g}_{a,A} +d_{a,A}\right)n^{1/3}\right) \Phi^\mathrm{start}_{a,A}. \end{equation} Finally, combining \eqref{eqn:endpart} with \eqref{eqn:middlepart} and \eqref{eqn:startpart}, for all $n \geq 1$ large enough and $k \in [m_a, m_{a+1})$, \begin{multline*}
\E\left[ \sum_{|u|=k} \ind{V(u_j) - \bar{b}^{(n)}_j \in \tilde{I}^{(n)}_j, j \leq k} \sum_{u_1 \neq u_2 \in \Omega(u)} \Lambda(u_1)\Lambda(u_2) \right]\\
\leq C \exp\left[ n^{1/3}\left(-2\varphi_1 (g_1 - x) + \bar{\varphi}_{a,A} \bar{g}_{a,A} + 3 d_{a,A}\right) \right] \Phi^\mathrm{start}_{a,A} \left(\Phi^\mathrm{end}_{a+1,A}\right)^2, \end{multline*} thus \begin{multline*}
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \sum_{k=0}^{n-1} \E\left[ \sum_{|u|=k} \ind{V(u_j) - \bar{b}^{(n)}_j \in \tilde{I}^{(n)}_j, j \leq k} \sum_{u_1 \neq u_2 \in \Omega(u)} \Lambda(u_1)\Lambda(u_2) \right]\\
\leq 2 \left( K_1^{F,G}(f,g,\varphi) - (g_1-x)\right) - \min_{a < A} 2 K_{\frac{a+1}{A}}^{F,G}(f,g,\varphi)-K_{\frac{a}{A}}^{F,G}(f,g,\varphi) - \bar{\varphi}_{a,A} \bar{g}_{a,A} - 3 d_{a,A}. \end{multline*} Letting $A \to +\infty$, and using Lemma \ref{lem:firstorder}, we obtain \[
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \E(B_n(f,g)^2) \leq 2 \left( K_1^{F,G}(f,g,\varphi) - (g_1-x)\right) - \inf_{t \in [0,1]} \left(H_t^{F,G}(f,g,\varphi) - \varphi_t g_t\right). \] \end{proof}
Using the previous two lemmas, we can bound from below the probability that there exists an individual that follows the path $\bar{b}^{(n)} + I^{(n)}$. \begin{lemma} \label{lem:lowerbound} Assuming \eqref{eqn:regularity}, \eqref{eqn:phibiendef}, \eqref{eqn:regularphi}, \eqref{eqn:FetG} and \eqref{eqn:integrability2phi}, if $K^*(b)_1 = \sup_{t \in [0,1]} K^*(b)_t = 0$, then for any $x < g_1$ \begin{equation}
\label{eqn:probaLower}
\liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}(\mathcal{B}^{F,G}_n(f,g,x) \neq \emptyset)
\geq \inf_{t \in [0,1]} \left( K^{F,G}_t(f,g,\varphi) - \varphi_t g_t\right). \end{equation} \end{lemma}
\begin{proof} We first assume that $G = [0,1]$. Since $B_n \in \mathbb{Z}_+$ a.s, we have \[
\mathbf{P}(\mathcal{B}^{F,G}_n(f,g,x) \neq \emptyset) = \mathbf{P}(B^{F,G}_n(f,g,x)>0) \geq \frac{\E(B^{F,G}_n(f,g,x))^2}{\E(B^{F,G}_n(f,g,x)^2)}, \] using the Cauchy-Schwarz inequality. As a consequence, \begin{align*}
\liminf_{n \to + \infty} \frac{1}{n^{1/3}} &\log \mathbf{P}(\mathcal{B}^{F,G}_n(f,g,x) \neq \emptyset)\\
&\geq 2 \liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \E(B^{F,G}_n(f,g,x)) - \limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \E\left(B^{F,G}_n(f,g,x)^2\right)\\
&\geq \inf_{t \in [0,1]} \left( H_t^{F,G}(f,g,\varphi) - \varphi_t g_t \right). \end{align*}
We then extend this estimate for $G$ a Riemann-integrable subset of $[0,1]$, that we can, without loss of generality, choose closed --as the Lebesgue measure of the boundary of a Riemann-integrable set is null. According to \eqref{eqn:FetG}, $\{ \dot{\varphi}>0\} \subset G$. We set, for $H > 0$ \[
g^H_t = \max\left\{ g_t, -\norme{g}_\infty + Hd(t,G) \right\}. \] Observe that $g^H$ is an increasing sequence of functions, that are equal to $g$ on $G$ and increase to $+\infty$ on $G^c$. For all $n \in \mathbb{N}$, $x \in [f_1,g_1]$ and $H>0$, we have $\mathcal{B}^{F,[0,1]}_n(f,g^H,x) \subset \mathcal{B}^{F,G}_n(f,g,x)$. As a consequence, \begin{align*}
\liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}(\mathcal{B}^{F,G}_n(f,g,x) \neq \emptyset)
&\geq \lim_{H \to +\infty} \liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}(\mathcal{B}^{F,[0,1]}_n(f,g^H,x) \neq \emptyset)\\
&\geq \lim_{H \to +\infty} \inf_{t \in [0,1]} \left( H_t^{F,[0,1]}(f,g^H, \varphi) - \varphi_t g^H_t\right). \end{align*} By Lemma \ref{lem:bmTwosided}, we have $\Psi(h) \sim_{h \to +\infty} \frac{\alpha_1}{2^{1/3}} h^{2/3}$. Thus, using \eqref{eqn:FetG}, this yields \[
\liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}(\mathcal{B}^{F,G}_n(f,g,x) \neq \emptyset) \geq \inf_{t \in [0,1]} \left( H_t^{F,G}(f,g, \varphi) - \varphi_t g_t\right). \] \end{proof}
\begin{remark} Observe that the inequality in Lemma \ref{lem:lowerbound} is sharp when \[
\inf_{t \in [0,1]} \left(H_t^{F,G}(f,g,\varphi) - \varphi_t g_t\right) = K_1^{F,G}(f,g,\varphi) - \varphi_1 g_1. \] \end{remark}
\section{Identification of the optimal path} \label{sec:optimization}
We denote by $\mathcal{R} = \left\{ b \in \mathcal{D} : \forall t \in [0,1], K^*(b)_t \leq 0 \right\}$. In this section, we take interest in functions $a \in \mathcal{R}$ that verify \begin{equation}
\int_0^1 a_s ds = \sup\left\{ \int_0^1 b_s ds, b \in \mathcal{R} \right\}, \end{equation} i.e. which are solution of \eqref{eqn:existence_max}. This equation is an optimisation problem under constraints. Information on its solution can be obtained using a theorem of existence of Lagrange multipliers in Banach spaces.
Let $E,F$ be two Banach spaces, a function $f : E \to F$ is said to be differentiable at $u \in E$ if there exists a linear continuous mapping $D_u f : E \to F$ called its \textit{Fréchet derivative at $u$}, verifying \[
f(u+h) = f(u) + D_u f(h) + o(\norm{h}), \quad \norm{h} \to 0, \quad h \in E. \] A set $R$ is a \textit{closed convex cone} of $F$ if it is a closed subset of $F$ such that \[
\forall x, y \in R, \forall \lambda, \mu \in [0,+\infty)^2 , \lambda x + \mu y \in K. \] Finally, we set $F^*$ the set of linear continuous mappings from $F$ to $\mathbb{R}$. We now introduce a result on the existence of Lagrange multipliers in Banach spaces obtained in \cite{Kur76}, Theorem 4.5. \begin{theorem}[Kurcyusz \cite{Kur76}] \label{thm:lagrange} Let $E,F$ be two Banach spaces, $J : E \to \mathbb{R}$, $g : E \to F$ and $R$ be a closed convex cone of $F$. If $\hat{u}$ verifies \[J(\hat{u}) = \max\{ J(u), u \in E : g(u) \in R\} \quad \mathrm{and} \quad g(\hat{u}) \in R,\] and if $J$ and $g$ are both differentiable at $\hat{u}$, and $D_{\hat{u}} g$ is a bijection, then there exists $\lambda \in F^*$ such that \begin{align}
&\forall h \in E, D_{\hat{u}} J (h) = \lambda^*\left[ D_{\hat{u}} g(h)\right] \label{eqn:lagrange1}\\
&\forall h \in R, \lambda^*(h) \leq 0 \label{eqn:lagrange2} \\
&\lambda^*(g(\hat{u})) = 0 \label{eqn:lagrange3}. \end{align} \end{theorem}
We first introduce the \textit{natural speed path} of the branching random walk, which is the path driven by $(v_t, t \in [0,1])$. \begin{lemma} Under the assumptions \eqref{eqn:breeding} and \eqref{eqn:regularity}, there exists a unique $v\in \mathcal{R}$ such that for all $t \in [0,1]$, $\kappa^*_t(v_t)=0$. Moreover, for all $t \in [0,1]$, $\bar{\theta}_t := \partial_a \kappa^*_t(v_t)>0$, and $v$ and $\bar{\theta}$ are $\mathcal{C}_1$ function. \end{lemma}
\begin{proof} For any $t \in [0,1]$ we have $\inf_{a \in \mathbb{R}} \kappa^*_t(a) = -\kappa_t(0)< 0$, as $\kappa^*_t$ is the Fenchel-Legendre transform of $\kappa_t$. Moreover, $a \mapsto \kappa^*_t(a)$ is convex, continuous on the interior of its definition set and increasing. By \eqref{eqn:regularity}, we have $\kappa^*(a) \to +\infty$ when $a$ increases to $\sup\{ b \in \mathbb{R} : \kappa^*_t(b) < +\infty\}$. As a consequence, by continuity, there exists $x \in \mathbb{R}$ such that $\kappa^*_t(x)=0$. As $\inf_{a \in \mathbb{R}} \kappa^*_t(a)<0$, $\kappa^*_t$ is strictly increasing at point $x$. Therefore the point $v_t=x$ is uniquely determined, and $\bar{\theta}_t = \partial_a\kappa^*_t(x)$ at point $x$ is positive. Finally, $v \in \mathcal{C}_1$ by the implicit function theorem; thus so is $\bar{\theta}$, by composition with $\partial_a \kappa^*$. \end{proof}
We now observe that if $a$ is a solution of \eqref{eqn:existence_max}, then $a$ is a regular point of $\mathcal{R}$ --i.e. we can apply Theorem \ref{thm:lagrange}. \begin{lemma} \label{lem:regularity_existence_max} Under the assumptions \eqref{eqn:breeding} and \eqref{eqn:regularity}, if $a$ is a solution of \eqref{eqn:existence_max}, then for all $t \in [0,1]$, $\partial_a \kappa^*_t(a_t)>0$. \end{lemma}
\begin{proof} Let $a \in \mathcal{R}$ be a solution of \eqref{eqn:existence_max}. For $t \in [0,1]$, we set $\theta_t = \partial_a \kappa^*_t(a_t)$. We observe that $\theta \in \mathcal{D}$ is non-negative.
We first assume that for all $t \in [0,1]$, $\theta_t=0$, in which case $\kappa^*_t(a_t)$ is the minimal value of $\kappa^*_t$. By \eqref{eqn:breeding}, we have $\inf_{t \in [0,1]} \kappa_t(0)>0$, thus $\sup_{t \in [0,1]} \inf_{a \in \mathbb{R}} \kappa^*_t(a) < 0$. As a consequence, by continuity, there exists $x \in \mathcal{D}$ such that for all $t \in [0,1]$, $\kappa^*_t(a_t + x_t) \leq 0$. We have $a+x \in \mathcal{R}$ and $\int_0^1 a_s + x_s ds > \int_0^1 a_s ds$, which contradicts $a$ is a solution of \eqref{eqn:existence_max}.
We now assume that $\theta$ is non-identically null, but there exists $t \in [0,1]$ such that $\theta_t = 0$. We start with the case $\theta_0>\epsilon>0$. As $\theta \in \mathcal{D}$, there exists $t>0$ and $\delta>0$ such that $\inf_{s \in [0,\delta]} \theta_s > \epsilon$ and $\sup_{s \in [t,t+\delta]} \theta_s < \epsilon/3$. For $x>0$, we set $a^x = a- x \mathbf{1}_{[0,\delta]} + 2x \mathbf{1}_{[t,t+\delta]}$. We observe that uniformly for $s \in [0,1]$, as $x \to 0$ \[
K^*(a^x)_s \leq K^*(a)_s - x \epsilon s \wedge \delta + \tfrac{2}{3} x \epsilon (s-t)_+ \wedge \delta + O(x^2). \] Thus there exists $x>0$ small enough such that $a^x \in \mathcal{R}$ and $\int a^x > \int a$, which contradicts again the fact that $a$ is a solution of \eqref{eqn:existence_max}.
Finally, we assume that $\theta_0=0$. In this case, there exists $\delta > 0$ such that $K^*(a)_t < - \delta t$ for all $t \leq \delta$. Therefore, there exists $t>0$ such that for all $0<s \leq t$, $K^*(a)_s < 0$, and $\theta_t>0$. For all $\theta_t>\epsilon>0$, there exists $\delta'>0$ such that for all for all $s< \delta'$, we have $\theta_s < \epsilon/3$ and for all $s \in [t,t+\delta']$, $\theta_s > 2 \epsilon$. Therefore, setting $a^x = a + 2x \mathbf{1}_{[0,\delta)} - x \mathbf{1}_{[t,t+\delta)}$, as $x \to 0$, uniformly in $s \in [0,1]$, we have \[
K^*(a^x) \leq K^*(a)_s + \frac{2}{3}x \epsilon (s \wedge \delta') - x \epsilon ((s-t)_+\wedge \delta) + O(x^2), \] so for $x>0$ small enough we have $a^x \in \mathcal{R}$. Moreover $\int_0^1 a^x > \int_0^1 a$ which, once again, contradicts the fact that $a$ is a solution of \eqref{eqn:existence_max}. \end{proof}
Applying Theorem \ref{thm:lagrange}, and using the previous lemma, we prove Proposition \ref{prop:regularity}. \begin{proof}[Proof of Proposition \ref{prop:regularity}] We first consider a function $a \in \mathcal{R}$ that verifies \[
\int_0^1 a_s ds = \sup\left\{ \int_0^1 b_s ds , b \in \mathcal{R} \right\}, \] i.e. such that $a$ is a solution of \eqref{eqn:existence_max}. We set $\theta_t = \partial_a \kappa^*_t(a_t)$, and observe that $\theta \in \mathcal{D}$.
We introduce $J : a \mapsto \int_0^1 a_s ds$ and $g : a \mapsto \left(\kappa^*_s(a_s),s \in [0,1]\right)$. These functions are differentiable at $a$, and for $h \in \mathcal{D}$, we have $D_a J(h) = \int_0^1 h_s ds$ and $D_a g(h)_t = \theta_t h_t$. We denote by \[R = \left\{ h \in \mathcal{D} : \forall t \in [0,1], \int_0^t h_s ds \leq 0 \right\}, \] which is a closed convex cone of $\mathcal{D}$. Using Lemma \ref{lem:regularity_existence_max}, we have $\theta_t > 0$ for all $t \in [0,1]$, thus $D_a g$ is a bijection.
By Theorem \ref{thm:lagrange}, there exists $\lambda^* \in \mathcal{D}^*$ --which is a measure by the Riesz representation theorem-- such that \begin{align}
& \forall h \in \mathcal{D} , \int_0^1 h_s ds = \int_0^1 D_a g(h)_s \lambda^*(ds)\label{eqn:z1}\\
& \forall h \in R, \int_0^1 h_s \lambda^*(ds) \leq 0 \label{eqn:z2}\\
& \int_0^1 g(a)_s \lambda^*(ds) = 0 \label{eqn:z3}. \end{align}
We observe easily that \eqref{eqn:z1} implies that $\lambda^*$ admits a Radon-Nikod\'ym derivative with respect to the Lebesgue measure, and that $\frac{\lambda^*(ds)}{ds} = \frac{1}{\theta_s}$. As a consequence, we can rewrite \eqref{eqn:z2} as \[
\forall h \in R, \int_0^1 h_s \frac{ds}{\theta_s} \leq 0. \] We set $f_t = \int_0^t \frac{ds}{\theta_s}$, for all $s, t \in [0,1]$, and $\mu \in (0,1)$, by \eqref{eqn:z2}, we have \[
\mu f_t + (1 - \mu) f_s - f_{\mu t + (1 - \mu) s} = \int_0^1 \left(\mu \ind{u < t} + (1 - \mu) \ind{u<s} - \ind{u<\mu t + (1 - \mu)s}\right) \frac{du}{\theta_u}
\leq 0. \] As a consequence, $f$ is concave. In particular, its right derivative function $\frac{1}{\theta}$ is non-increasing. Consequently $\theta$ is non-decreasing.
The last equation \eqref{eqn:z3} gives \[
0 = \int_0^1 \kappa^*_s(a_s) \lambda^*(ds)= \int_0^1 \kappa^*_s(a_s) \theta_s^{-1} ds = K^*(a)_1 \frac{1}{\theta_1} - \int_0^1 K^*(a)_s d\theta_s^{-1}, \] by Stieltjès integration by part. But for all $t \in [0,1]$, $K^*(a)_t \leq 0$, and $\frac{1}{\theta}$ is non-increasing. This yields \[
K^*(a)_1 = 0 \quad \mathrm{and} \quad \int_0^1 K^*(a)_s d\theta^{-1}_s = 0. \] In particular, as $a \in \mathcal{R}$, $\theta$ increases on $\{t \in [0,1] : K^*(a)_t = 0\}$.
Conversely, we consider a function $a \in \mathcal{R}$ such that the function $\theta : t \mapsto \partial_a \kappa^*_t(a_t)$ is non-decreasing, $K^*(a)_1=0$ and $\int_0^1 K^*(a)_s d\theta^{-1}_s=0$. Our aim is to prove that $\int_0^1 a_s ds = v^*$, by observing that $\int_0^1 a_s ds \geq \int_0^1 b_s ds$ for all $b \in \mathcal{R}$. By convexity of $\kappa^*_t$, we have, for all $t \in [0,1]$ \[
\kappa^*_t(b_t) \geq \kappa^*_t(a_t) + \theta_t(b_t - a_t), \] and integrating with respect to $t$, we obtain \begin{align*}
\int_0^1 a_t - b_t dt &\geq \int_0^1 \frac{\kappa^*_t(a_t) - \kappa^*_t(b_t)}{\theta_t} dt\\
&\leq K^*(a)_1 - K^*(b)_1 - \int_0^1 \left(K^*(a)_t - K^*(b)_t\right) d\theta^{-1}_t, \end{align*} by Stieltljès integration by parts. Using the specific properties of $a$, we get \[
\int_0^1 a_t - b_t dt \leq -K^*(b)_1 + \int_0^1 K^*(b)_t d\theta^{-1}_t. \] As $K^*(b)$ is non-positive, and $\theta^{-1}$ is non-increasing, we conclude that the left-hand side is non-positive, which leads to $\int_0^1 a_s ds \geq \int_0^1 b_s ds$. Optimizing this inequality over $b \in \mathcal{R}$ proves that $a$ is a solution of \eqref{eqn:existence_max}.
We now prove that if $a$ is a solution of \eqref{eqn:existence_max}, then $a$ is continuous, by proving that this function has no jump. In order to do so, we assume that there exists $t \in (0,1)$ such that $a_t \neq a_{t-}$, i.e. such that $a$ jumps at time $t$. Then, $\theta_t \neq \theta_{t-}$ by continuity of $\partial_a \kappa^*$ on $D^*$. As $\int_0^1 K^*(a)_s d\theta^{-1}_s = 0$ and $d\theta^{-1}$ has an atom at point $t$, thus $K^*(a)_t=0$.
Therefore, if $a$ jumps at time $t$, then the continuous function $s \mapsto K^*(a)_s$ with right and left derivatives at each point, bounded from above by $0$, hits a local maximum at time $t$. Its left derivative $\kappa^*_t (a_{t-})$ is then non-negative and its right derivative $\kappa^*_t(a_t)$ non-positive. As $\kappa^*t$ is a non-decreasing function, we obtain $a_{t-} \geq a_t$.
Moreover, by convexity of $\kappa^*_t$, $x \mapsto \partial_a \kappa^*_t(x)$ is also non-decreasing, and as a consequence $\theta_{t-} \geq \theta_t$, which is a contradiction with the hypothesis $\theta_{t-} \neq \theta_t$ and $\theta$ non-decreasing. We conclude that $a$ (and $\theta$) is continuous as a càdlàg function with no jump.
We now assume there exists another solution $b \in \mathcal{R}$ to \eqref{eqn:existence_max}. Using the previous computations, we have $\int_0^1 K^*(b)_s d\theta^{-1}_s = 0$, and $b$ is continuous. As a consequence, denoting by $T$ the support of $d\theta^{-1}$, for all $t \in T$, $K^*(b)_t=0$. Moreover, $K^*(b)$ is a $\mathcal{C}^1$ function, with a local maximum at time $t$, thus $\kappa^*_t(b_t)=0$, or in other words, $b_t = v_t$, by Lemma \ref{lem:regularity_existence_max}.
Consequently, if we write $\varphi_t = \partial_a \kappa^*_t(b_t)$, we know from previous results that $\varphi$ is continuous and increasing. Furthermore, $\varphi$ increases only on $T$, and $\varphi_t = \partial_a \kappa^*_t(v_t) = \bar{\theta}_t$. For all $t \in [0,1]$, we set $\sigma_t=\sup\{s \leq t : s \in T\}$ and $\tau_t = \inf\{s \geq t : s \in T\}$. If $\sigma$ and $\tau$ are finite then \[
\bar{\theta}_{\sigma_t} = \varphi_{\sigma_t} = \varphi_t = \varphi_{\tau_t} = \bar{\theta}_{\tau_t}. \] As $a$ is also a solution of \eqref{eqn:existence_max}, we have $\bar{\theta}_{\sigma_t} = \theta_t = \bar{\theta}_{\tau_t}$, therefore $\theta = \varphi$. As a consequence, we have \[
a_t = \partial_\theta \kappa_t(\theta_t) = \partial_\theta \kappa_t(\varphi_t) = b_t, \] which proves the uniqueness of the solution.
We now prove that $a$ and $\theta$ are Lipschitz functions. For all $t \in [0,1]$, $\int_0^t \kappa^*_s(a_s)ds \leq 0$, and $\int_0^t \kappa^*_s(a_s)d\theta^{-1}_s = 0$. In particular, this means that $\kappa^*_t(a_t)$ vanishes $d\theta^{-1}_t$ almost everywhere, thus $\theta_t = \bar{\theta}_t$ $d\theta^{-1}_t$ almost everywhere. By continuity of $\theta$ and $\bar{\theta}$, these functions are identical on $T$. In addition, for all $s<t$ such that $(s,t) \subset [0,1]\backslash T$, we have $\int_s^t d\theta^{-1}_u = 0$, hence $\theta_t=\theta_s$, which proves that $\theta$ is constant on $[0,1]\backslash T$. As a result, for all $s<t \in [0,1]$, we have $\theta_t=\theta_s$ if $(s,t) \subset [0,1]\backslash T$, otherwise \[
\theta_s = \inf_{u \geq s, u \in T} \bar{\theta}_u \quad \mathrm{and} \quad \theta_t = \inf_{u \leq t, u \in T} \bar{\theta}_u, \]
In consequence $ |\theta_t - \theta_s| \leq \sup_{r,r' \in [s,t]} |\bar{\theta}_r - \bar{\theta}_{r'}|$. As $\bar{\theta}$ is $\mathcal{C}^1$ on $[0,1]$, $\bar{\theta}$ and $\theta$ are Lipschitz functions. As $a_t = \partial_\theta \kappa_t(\theta_t)$, the function $a$ is also Lipschitz.
Finally, we prove the existence of a solution to \eqref{eqn:existence_max}. To do so, we reformulate this optimization problem in terms of an optimization problem for $\theta$. The aim is to find a positive function $\theta \in \mathcal{C}$ such that \begin{equation}
\int_0^1 \partial_\theta \kappa_t(\theta_t) dt = \max\left\{ \int_0^1 \partial_\theta \kappa_t(\varphi_t) dt : \varphi \in \mathcal{C}, \forall t \in [0,1], E(\varphi)_t < +\infty \right\}, \end{equation} where $E(\varphi)_t = \int_0^t \varphi_s \partial_\theta \kappa_t(\varphi_t) - \kappa_t(\varphi_t)$. By Theorem \ref{thm:lagrange}, if $\theta$ exists, then it is a non-decreasing function. Moreover $E(\theta)_1=0$ and for any $t \in [0,1]$, $\int_0^t E(\theta)_sd\theta^{-1}_s = 0$.
Using these three properties, we have $\theta=\bar{\theta}$ on the support of the measure $d\theta^{-1}$. Moreover, as $E_0(\theta)=E_1(\theta)=0$ and $E_t$ is non-positive, we observe that $E_t(\theta)$ is locally non-increasing in the neighbourhood of 0 and locally non-decreasing in the neighbourhood of 1, in particular \[
\theta_0 \partial_\theta \kappa_0(\theta_0) - \kappa_0(\theta_0) \leq 0 \quad \mathrm{and} \quad \theta_1 \partial_\theta \kappa_1(\theta_1) - \kappa_1(\theta_1) \geq 0. \] As for all $t \in [0,1]$, the function $\varphi \mapsto \varphi \partial_\theta \kappa_t(\varphi) - \kappa_t(\varphi)$ is increasing, we conclude that $\theta_0 \leq \bar{\theta}_0$ and $\theta_1 \geq \bar{\theta}_1$. As a consequence, $T = \{t \in [0,1] : \theta_t = \bar{\theta}_t\}$ is non-empty, and, setting $\sigma_t = \sup\{s \leq t : s \in T \}$ and $\tau_t = \inf\{s \geq t : s \in T\}$ we have $\theta_t = \bar{\theta}_{\sigma_t}$ if $\sigma_t > -\infty$ and $\theta_t = \bar{\theta}_{\tau_t}$ if $\tau_t < +\infty$.
We write \[\Theta = \left\{ \theta \in \mathcal{C} : \theta \text{ non-decreasing}, \theta_0 \geq 0, \forall t \in [0,1], \int_0^t E_s(\theta)d\theta^{-1}_s = 0 \text{ and } E_t(\theta) \leq 0\right\}.\] This set is uniformly equicontinuous and bounded, thus by Arzelà-Ascoli theorem, it is compact. It is non-empty as for all $\epsilon>0$ small enough, the function $t \mapsto \epsilon$ belongs to $\Theta$. We write $\theta$ a maximizer of $\int_0^1 \partial_\theta\kappa_s(\theta_s)ds$ on $\Theta$.
By continuity, if $E(\theta)_1<0$, then we can increase a little $\theta$ in the neighbourhood of $1$, thus $\theta$ is non-optimal. As a result, $\theta$ is non-decreasing, verifies $E(\theta)_1=0$ and $\int E(\theta)_s d\theta^{-1}_s=0$, which proves that $a = \partial_\theta \kappa(\theta)$ is a solution of \eqref{eqn:existence_max}. \end{proof}
The previous proof gives some characteristics of the unique solution $a$ of \eqref{eqn:existence_max}. In particular, if we set $\theta_t = \partial_a \kappa^*_t(a_t)$, we know that $\theta$ is positive, non-decreasing, and that on the support of the measure $d\theta^{-1}$, $\theta$ and $\bar{\theta}$ are identical. Consequently, the optimal speed path of the branching random walk verifies the following property: while in the bulk of the branching random walk, it follows an equipotential line, and when close to the boundary it follows the natural speed path.
In certain specific cases, \eqref{eqn:existence_max} can be explicitly solved. For example, when $\bar{\theta}$ increases --which corresponds to the ``decreasing variance'' case in Gaussian settings-- we have $a=v$. \begin{lemma} \label{lem:specialcase} We assume \eqref{eqn:breeding} and \eqref{eqn:regularity}. \begin{description}
\item[Non-decreasing case] If $\bar{\theta}$ is non-decreasing, then $a=v$ (and $\bar{\theta} = \theta$).
\item[Non-increasing case] If $\bar{\theta}$ is non-increasing, then there exists $\theta \in [0,+\infty)$ such that $a_t = \partial_\theta \kappa_t(\theta)$.
\item[Mixed case] If $\bar{\theta}$ is non-increasing on $[0,1/2]$ and increasing on $[1/2,1]$, then there exists $t \in [1/2,1]$ such that
\[
\forall s \in [0,1], \partial_a \kappa^*_s(a_s) = \bar{\theta}_{s \vee t}.
\] \end{description} \end{lemma}
\begin{proof} We first assume that $\bar{\theta}$ is a non-decreasing function. As $K^*(v)_t=0$ for all $t \in [0,1]$, we have $K^*(v)_1 = \int_0^t K^*(v)_t d \bar{\theta}^{-1}_t = 0$ which, by Proposition \ref{prop:regularity} implies that $v$ is the solution of \eqref{eqn:existence_max}.
We now denote by $a$ the solution of \eqref{eqn:existence_max}, and by $\theta_t = \partial_a \kappa^*_t(a_t)$. Let $T$ be the support of the measure of $d\theta^{-1}_t$, we know by Proposition \ref{prop:regularity} that $\theta$ is non-decreasing and equal to $\bar{\theta}_t$ when it is increasing. In particular, we have \[
\frac{1}{\theta_t} = \frac{1}{\theta_0} + \int_0^t d\theta^{-1}_s = \frac{1}{\theta_0} + \int_0^t \ind{s \in T} d\theta^{-1}_s = \frac{1}{\theta_0} + \int_0^t \ind{s \in T} d\bar{\theta}^{-1}_s. \]
As a consequence, if $\bar{\theta}$ is non-increasing on $[0,t]$, then $\int_0^u \ind{s \in T} d\bar{\theta}^{-1}_s \geq 0$ for all $u \leq t$. As $\theta^{-1}$ is non-increasing, we conclude that $\int_0^u \ind{s \in T} d\bar{\theta}^{-1}_u = 0$, and $\theta_u = \theta_0$. In particular, in the non-increasing case, we conclude that $\theta$ is a constant.
In the mixed case, we have just shown that $\theta$ is constant up to time $1/2$. We set $u=\inf\{t > 1/2 : \bar{\theta}_t = \theta_0\}$. Since $\theta=\bar{\theta}$ on $T$, we know that $T \cap [1/2,u) = \emptyset$. Hence $\theta$ is constant up to point $u$. For $t>u$, as $\bar{\theta}$ increases, we have \[
\frac{1}{\theta_t} = \frac{1}{\theta_0} + \int_0^t \ind{s \in T} d\bar{\theta}^{-1}_s = \frac{1}{\bar{\theta}_u} + \int_{u}^t \ind{s \in T} d\bar{\theta}^{-1}_s \geq \frac{1}{\bar{\theta}_t}, \] which yields $\bar{\theta}_t \geq \theta_t$. We now observe that $K^*(a)_1=0$, thus $K^*(a)$ attains a local maximum at time 1, and its left derivative $\kappa^*_1(a_1)$ is non-negative. This implies that $\theta_1 \geq \bar{\theta}_1$. If there exists $s > u$ such that $\theta_s <\bar{\theta}_s$, then $T \cap [s,1] = \emptyset$, and $\theta_1 = \theta_s < \bar{\theta}_s \leq \bar{\theta}_1$, which contradicts the previous statement. In consequence, for $t \geq u$, we have $\theta_t = \bar{\theta}_t$, which ends the proof of the mixed case. \end{proof}
\section{Maximal and consistent maximal displacements} \label{sec:maxdis}
We apply the estimates obtained in the previous section to compute the asymptotic of some quantities of interest for the BRWtie. In Section \ref{subsec:maxdisplacement}, we take interest in the maximal displacement in a BRWtie with selection. In Section \ref{subsec:cmd}, we obtain a formula for the consistent maximal displacement with respect to a given path. If we apply these estimates in a particular case, we prove Theorems \ref{thm:main} and \ref{thm:cmd}.
\subsection{Maximal displacement in a branching random walk with selection} \label{subsec:maxdisplacement}
We first define the \textit{maximal displacement in a branching random walk with selection}, which is the position of the rightmost individual among those alive at generation $n$ that stayed above a prescribed curve. We consider a positive function $\varphi$ that satisfies \eqref{eqn:phibiendef} and \eqref{eqn:regularphi}. We introduce functions $b$ and $\sigma$ according to \eqref{eqn:meanandvariance}. Let $f$ be a continuous function on $[0,1]$ with $f(0)<0$, and $F$ be a Riemann-integrable subset of $[0,1]$. The set of individuals we consider is \[
\mathcal{W}_n^\varphi(f,F) = \left\{ |u|=n : \forall j \in F_n, V(u_j) \geq \bar{b}^{(n)}_j + f_{j/n} n^{1/3} \right\}. \]
This set is the tree of the BRWtie with selection $(\mathcal{W}_n^\varphi(f,F), V_{|\mathcal{W}_n^\varphi (f,F)})$, in which every individual $u$ alive at time $k \in F_n$ with position below $\bar{b}^{(n)}_k + f_{k/n} n^{1/3}$ is immediately killed, as well as all its descendants. Its maximal displacement at time $n$ is denoted by \[
M^\varphi_n(f,F) = \max\left\{ V(u), u \in \mathcal{W}^\varphi_n(f,F) \right\}. \]
To apply the results of the previous section, we assume here that $b$ satisfies \begin{equation}
\label{eqn:brealpath}
\sup_{t \in [0,1]} K^*(b)_t = 0 = K^*(b)_1; \end{equation} in other words, there exists individuals that follow the path with speed profile $b$ with positive probability, and at time 1, there are $e^{o(n)}$ of those individuals. We set $G$ the set of zeros of $K^*(b)$, and we assume that \begin{equation}
\label{eqn:pourG}
G = \{ t \in [0,1] : K^*(b)_t = 0 \} \text{ is Riemann-integrable}. \end{equation} For $\lambda \in \mathbb{R}$, we set $g^\lambda \in \mathcal{C}([0,t_\lambda))$ the function solution of \begin{equation}
\label{eqn:pourg}
\forall t \in [0,t_\lambda), \varphi_t g^\lambda_t - H_t^{F,G}(f,g^\lambda,\varphi) = \varphi_0 \lambda. \end{equation} To study $M^\varphi_n(f,F)$, we first prove the existence of a unique maximal $t_\lambda \in (0,1]$, and a function $g^\lambda$ solution of \eqref{eqn:pourg}. We recall the following theorem of Carathéodory, that can be found in \cite{Fil88}. \begin{theorem}[Existence and uniqueness of solutions of Carathéodory's ordinary differential equation] \label{thm:existenceuniqueness} Let $0 \leq t_1 < t_2 \leq 1$, $x_1 < x_2$, $M>0$ and $f : [t_1,t_2] \times [x_1,x_2] \to [-M,M]$ a bounded function. Let $t_0 \in [t_1,t_2]$ and $x_0 \in [x_1,x_2]$, we consider the differential equation consisting in finding $t > 0$ and a continuous function $\gamma : [t_0,t_0 + t] \to \mathbb{R}$ such that \begin{equation}
\label{eqn:differential}
\forall s \in [t_0,t_0+t], \gamma(s) = x_0 + \int_{t_0}^s f(u,\gamma(u)) du. \end{equation}
If for all $x \in [x_1,x_2]$, $t \mapsto f(t,x)$ is measurable and for all $t \in [t_1,t_2]$, $x \mapsto f(t,x)$ is continuous, then for all $(t_0,x_0)$ there exists $t\geq \min(t_2 - t_0, \frac{x_2-x_0}{M}, \frac{x_0-x_1}{M})$ and $\gamma$ that satisfy \eqref{eqn:differential}.
If additionally, there exists $L>0$ such that for all $x,y \in [x_1,x_2]$ and $t \in [t_1,t_2]$, $|f(t,x)-f(t,y)| \leq L |x-y|$, then for every pair of solutions $(t,\gamma)$ and $(\tilde{t},\tilde{\gamma})$ of \eqref{eqn:differential}, we have \[
\forall s \leq \min(t,\tilde{t}), \gamma(s) = \tilde{\gamma}(s). \] Consequently, there exists a unique solution defined on a maximal interval $[t_0,t_0+t_{\mathrm{max}}]$. \end{theorem}
We use this theorem to prove there exists a unique solution $g$ to \eqref{eqn:pourg}. \begin{lemma} Let $f$ be a continuous function, $\varphi$ that verifies \eqref{eqn:regularphi}, and $F,G$ two Riemann-integrable subsets of $[0,1]$. For all $\lambda > f_0$, there exists a unique $t_\lambda \in [0,1]$ and a unique continuous function defined on $[0,t_\lambda]$ such that for all $t < t_\lambda$, we have \[
g^\lambda_t > f_t \quad \mathrm{and} \quad \varphi_t g^\lambda_t = \varphi_0\lambda + K_s^{F,G}(f,g^\lambda,\varphi). \] Moreover, there exists $\lambda_c$ such that for all $\lambda > \lambda_c$, $t_\lambda = 1$ and $\lambda \mapsto g^\lambda$ is continuous with respect to the uniform norm and strictly increasing. \end{lemma}
\begin{proof} Let $f$ be a continuous function, and $F$ be a Riemann-integrable subset of $[0,1]$, we set \[D = \{ (t,x) \in [0,1] \times \mathbb{R} : \text{if } t \in F,\text{ then } x > \varphi_t f_t\},\] and, for $(t,x) \in D$, \begin{multline*}
\Phi(t,x) = \frac{\dot{\varphi}_t}{\varphi_t} x + \mathbf{1}_{F \cap G}(t) \frac{\sigma_t^2}{(\frac{x}{\varphi_t} - f_t)^2} \Psi\left( \tfrac{(\frac{x}{\varphi_t}-f_t)^3}{\sigma_t^2} \dot{\varphi}_t \right)\\
+\mathbf{1}_{F^c \cap G}(t) \frac{a_1}{2^{1/3}} (\dot{\varphi}_t \sigma_t)^{2/3} + \mathbf{1}_{F \cap G^c} \left( \dot{\varphi}_t(f_t - \tfrac{x}{\varphi_t}) + \frac{a_1}{2^{1/3}} (-\dot{\varphi}_t \sigma_t)^{2/3} \right). \end{multline*} For all $\lambda > f_0$, we introduce \[
\Gamma^\lambda = \left\{ (t,h), t \in [0,1], h \in \mathcal{C}([0,t]) : \forall s \leq t, h_s = \varphi_0 \lambda + \int_0^s \Phi(u,h_u)du \right\}, \] the set of functions such that $g=\frac{h}{\varphi}$ is a solution of \eqref{eqn:pourg}.
We observe that for all $[t_1,t_2] \times [x_1,x_2] \subset D$, $\Phi_{|[t_1,t_2] \times [x_1,x_2]}$ is measurable with respect to $t$, and uniformly Lipschitz with respect to $x$. As a consequence, by Theorem \ref{thm:existenceuniqueness}, for all $(t_0,x_0) \in D$, there exists $t>0$ such that there exists a unique function $h \in \mathcal{C}([0,t])$ satisfying \begin{equation}
\label{eqn:differentialEquation}
\forall s \leq t, h_s = x_0 + \int_{t_0}^s \Phi(u,h_u)du. \end{equation}
Using this result, we first prove that $\Gamma^\lambda$ is a set of consistent functions. Indeed, let $(t_1,h^1)$ and $(t_2,h^2)$ be two elements of $\Gamma^\lambda$, and let $\tau = \inf\{ s \leq \min(t_1,t_2) : h^1_s \neq h^2_s \}$. We observe that if $\tau < \min(t_1,t_2)$, then by continuity of $h^1$ and $h^2$, we have $h^1_\tau = h^2_\tau$. Furthermore, $s \mapsto h^1_{\tau +s}$ and $s \mapsto h^2_{\tau+s}$ are two different functions satisfying \eqref{eqn:differentialEquation} with $t_0=\tau$ and $x_0 = h^1_\tau = h^2_\tau$, which contradicts the uniqueness of the solution. We conclude that $\tau \geq \min(t_1,t_2)$, every pair of functions in $\Gamma^\lambda$ are consistent up to the first terminal point.
We now define $t_\lambda = \max\left\{ t \in [0,1] : (t,h) \in \Gamma^\lambda\right\}$. We observe easily that $t_\lambda \in (0,1]$ by noting the existence of a local solution starting at time $0$ and position $\varphi_0 \lambda$. For $s < t_\lambda$, we write $h^\lambda_s = h_s$, where $(s,h_s) \in \Gamma^\lambda$. By definition, for any $s < t_\lambda$, $h^\lambda_s = \varphi_0 \lambda + \int_0^s \Phi(u,h^\lambda_u)du$. By local uniqueness of the solution, if there exists $t \in (0,1)$ such that $h^\lambda_t = h^{\lambda'}_t$, then for all $s \leq t$, $h^\lambda_s = h^{\lambda'}_s$, and in particular $\lambda = \lambda'$. We deduce that for all $\lambda < \lambda'$, if $s < \min(t_\lambda, t_\lambda')$ then $h^{\lambda}_s < h^{\lambda'}_s$.
Moreover, as there exist $C_1$ and $C_2>0$ such that for all $t \in [0,1]$ and $x > C_1$, $\Phi(t,x)< C_2$, we have $\limsup_{t \to t_\lambda} h^\lambda_t < +\infty$. Hence, if $\lambda < \lambda'$ and $t_\lambda > t_\lambda'$, if $x_0 \in \left[ \liminf_{t \to t_\lambda} h^\lambda_t, \limsup_{t \to t_\lambda} h^\lambda_t\right]$, then as $x_0 > h^\lambda_{t_{\lambda'}}$, we can extend $h^{\lambda'}$ on $[t_{\lambda'},t_{\lambda'}+\delta]$, which contradicts the fact that $t_{\lambda'}$ is maximal. We conclude that $t_{\lambda'} \geq t_\lambda$.
If $\lambda'>\lambda > \lambda_c$, the functions $h^{\lambda}$ and $h^{\lambda'}$ are defined on $[0,1]$. Moreover, the set \[
H^{\lambda,\lambda'} = \left\{ (t,x) \in [0,1] \times \mathbb{R} : x \in [h^\lambda_t, h^{\lambda'}_t] \right\}, \] is a compact subset of $D$, that can be paved by a finite number of rectangles in $D$. As a consequence, there exists $L>0$ such that \[
\forall t \in [0,1], \forall x,x' : (t,x) \in H^{\lambda,\lambda'}, (t,x') \in H^{\lambda,\lambda'}, \left| \Phi(t,x) - \Phi(t,x') \right| \leq L |x-x'|. \] As for all $\mu \in [\lambda,\lambda']$, $(t,h^\mu_t) \in H^{\lambda, \lambda'}$, we observe that \begin{align*}
\left| h^{\mu}_t - h^{\mu'}_t \right|&\leq \left| \mu - \mu' \right| + \int_0^t \left| \Phi(s,h^{\mu}_s) - \Phi(s,h^{\mu'}_s) \right| ds \\
&\leq \left| \mu - \mu' \right| + L \int_0^t \left| h^{\mu}_s - h^{\mu'}a_s \right|ds. \end{align*} Applying the Gronwall inequality, for all $\mu,\mu' \in [\lambda, \lambda']$, we have \[
\norme{h^{\mu} - h^{\mu'}}_\infty \leq \left| \mu - {\mu'} \right| e^{L} \] which proves that $\lambda \mapsto h^\lambda$ is continuous with respect to the uniform norm.
Finally, there exist $C_0$ and $C_1>0$ such that for all $t \in [0,1]$ and $x \geq C_0$, we have $\Phi(t,x) \geq -C_1$. Therefore, for all $\lambda \geq C_0 + C_1 + \norme{\varphi f}_\infty$, for all $t \in [0,1]$, $h^\lambda_t \geq \norme{\varphi f}_\infty$, and $t_\lambda = 1$. We set $\lambda_c = \inf\{ \lambda \in \mathbb{R} : t_\lambda = 1\}$, and we conclude the proof by observing that $g^\lambda = \frac{h^\lambda}{\varphi^\lambda}$ is the solution of \eqref{eqn:pourg}. \end{proof}
\begin{lemma} \label{lem:asymptoticGeneral} Assuming \eqref{eqn:regularity}, \eqref{eqn:phibiendef}, \eqref{eqn:regularphi}, \eqref{eqn:FetG}, \eqref{eqn:integrability2phi} and \eqref{eqn:brealpath}, for any $\lambda > \max(0,\lambda_c)$, we have \[
\lim_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\left( M_n^\varphi(f,F) \geq \bar{b}^{(n)}_n + g^\lambda_1 n^{1/3} \right)= - \varphi_0 \lambda. \] \end{lemma}
\begin{proof} To obtain an upper bound, we recall that $1 \in G$, as $K^*(b)_1=0$ by \eqref{eqn:brealpath}. Let $\lambda > \max(0,\lambda_c)$, we set $g = g^\lambda$ the unique solution of \eqref{eqn:pourg}. We observe that \[
\mathbf{P}\left(M_n^\varphi(f,F) \geq \bar{b}^{(n)}_n + g^\lambda_1 n^{1/3}\right) \leq \mathbf{P}\left(\mathcal{A}_n^{F,G}(f,g) \neq \emptyset\right) \leq \E\left(\mathcal{A}_n^{F,G}(f,g)\right). \] Therefore, by Lemma \ref{lem:estimate_upperfrontier}, we have \[
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}(M_n^\varphi(f,F) \geq \bar{b}^{(n)}_n + g^\lambda_1 n^{1/3}) \leq \sup_{t \in [0,1]} K^{F,G}_t(f,g,\varphi) - \varphi_t g_t = -\varphi_0\lambda. \]
When $\lambda'>\lambda$, we have $g^{\lambda'}_1 > g_1$, and \[
\mathbf{P}\left(M_n^\varphi(f,F) \geq \bar{b}^{(n)}_n + g^\lambda_1 n^{1/3}\right) \geq \mathbf{P}\left(\mathcal{B}_n^{F,G}(f,g^{\lambda'},g^{\lambda'}_1 - g_1) \neq \emptyset \right). \] Consequently, using Lemma \ref{lem:lowerbound}, we have \[
\liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\left(M_n^\varphi(f,F) \geq \bar{b}^{(n)}_n + g^\lambda_1 n^{1/3}\right) \geq \sup_{t \in [0,1]} K^{F,G}_t(f,g,\varphi) - \varphi_t g_t = -\varphi_0\lambda'. \] As $\lambda'$ decreases to $\lambda$, we have $\displaystyle \lim_{n \to + \infty} \frac{1}{n^{1/3}} \log \mathbf{P}\left(M_n^\varphi(f,F) \geq \bar{b}^{(n)}_n + g^\lambda_1 n^{1/3}\right) = -\lambda$. \end{proof}
The previous lemma gives an estimate of the right tail of $M_n^\varphi(f,F)$ for all $f \in \mathcal{C}$ and Riemann-integrable set $F \subset [0,1]$. Note that to obtain this estimate, we do not need the assumption \eqref{eqn:breeding} of supercritical reproduction, however \eqref{eqn:brealpath} implies that \[
\inf_{t \in [0,1]} \liminf_{n \to +\infty} \frac{1}{n} \log \E\left[ \# \{ u \in \mathbf{T}^{(n)}, |u|= \floor{tn} \} \right] \geq 0, \] which is a weaker supercriticality condition. Assuming \eqref{eqn:breeding}, we can strengthen Lemma \ref{lem:asymptoticGeneral} to prove a concentration estimate for $M_n^\varphi(f,F)$ around $\bar{b}^{(n)}_n + g^0_1 n^{1/3}$. \begin{lemma}[Concentration inequality] \label{lem:concentration} Under the assumptions \eqref{eqn:breeding}, \eqref{eqn:regularity}, \eqref{eqn:phibiendef}, \eqref{eqn:regularphi}, \eqref{eqn:FetG}, \eqref{eqn:integrability2phi} and \eqref{eqn:brealpath}, if $\lambda_c>0$, then for all $\epsilon>0$, we have \[
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\left( \left|M_n^\varphi(f,F) - \bar{b}^{(n)}_n - g^0_1 n^{1/3}\right| \geq \epsilon n^{1/3}\right) <0. \] \end{lemma}
\begin{proof} We set $g = g^0$ the solution of \eqref{eqn:pourg} for $\lambda = 0$. We observe that for all $\epsilon>0$ and $t \in [0,1]$, we have $H_t^{F,G}(f,g+\epsilon,\varphi) - \varphi_t (g_t + \epsilon) < 0$. Consequently, for any $\epsilon>0$, we have \begin{align*}
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\!\left( M_n^\varphi(f,F) \geq \bar{b}^{(n)}_n + (g_1+\epsilon)n^{1/3} \right)
&\leq \limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\!\left( \mathcal{A}^{F,G\cup\{1\}}_n(f,g+\epsilon) \neq \emptyset \right)\\
&\leq \sup_{t \in [0,1]} H_t^{F,G}(f,g+\epsilon,\varphi) - \varphi_t (g_t+\epsilon)<0, \end{align*} by Lemma \ref{lem:estimate_upperfrontier}.
To obtain a lower bound, we need to strengthen the tail estimate of $M_n^\varphi(f,F)$. Using \eqref{eqn:breeding}, the size of the population in the branching random walk increases at exponential rate. We set $p \in \mathbb{R}$ and $\rho>0$ such that $\rho = \inf_{t \in [0,1]} \mathbf{P}(\# \{ \ell \in L_t : \ell \geq p \} \geq 2)$. We can assume, without loss of generality, that $p < b_0$. We couple the BRWtie with a Galton-Watson process $(Z_n, n \geq 0)$ with $Z_0=1$, and reproduction law defined by $\mathbf{P}(Z_1 = 2) = 1 - \mathbf{P}(Z_1=1) = \rho$; in a way that \[
\forall n \in \mathbb{N}, \forall k \leq n, \# \{ u \in \mathbf{T}^{(n)} : |u|=k, V(u) \geq kp \} \geq Z_k. \] There exists $\alpha>0$ such that $\limsup_{n \to +\infty} \frac{1}{n} \log \mathbf{P}\left( Z_n \leq e^{\alpha n} \right) < 0$, by standard Galton-Watson theory (see, e.g. \cite{FlW07}). Consequently, with high probability, there are at least $e^{\alpha k}$ individuals to the right of $pk$ at any time $k \leq n$.
Let $\epsilon>0$ and $\eta > 0$, we set $k =\floor{\eta n^{1/3}}$. Applying the Markov property at time $k$, we have \[
\mathbf{P}\left(M_n \leq m_n - \epsilon n^{1/3} \right) \leq \mathbf{P}\left(Z_k \leq e^{\alpha k} \right) + \left[1 - \mathbf{P}_{k, kp}\left(M_{n-k} \geq m_n - \epsilon n^{1/3})\right) \right]^{e^{\alpha k}}. \] As a consequence \begin{multline*}
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\left( M_n \leq m_n - \epsilon n^{1/3} \right) \\
\leq \max\left\{ \limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\left( Z_k \leq e^{\alpha k} \right), -\liminf_{n \to +\infty} \frac{e^{\alpha k}}{n^{1/3}} \mathbf{P}_{k,kp} \left( M_{n-k} \geq m_n - \epsilon n^{1/3} \right)\right\}. \end{multline*} We now prove that \begin{equation}
\label{eqn:final}
\liminf_{n \to +\infty} e^{\alpha \eta n^{1/3}} \mathbf{P}_{k,0} \left( M_{n-k} \geq \bar{b}^{(n)}_n + (g_1-\epsilon) n^{1/3}-kp \right)>0. \end{equation}
Let $\delta > 0$, we choose $\eta = \tfrac{\epsilon}{b_0-p} + \delta$, we have \begin{multline*}
\liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}_{k,0} \left( M_{n-k} \geq \bar{b}^{(n)}_n + (g_1-\epsilon) n^{1/3}-kp \right)\\
= \liminf_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}_{k,0} \left( M_{n-k} \geq \bar{b}^{(n)}_n - \bar{b}^{(n)}_k + g_1+\delta n^{1/3} \right) \leq -\varphi_0 \lambda_\delta, \end{multline*} by applying Lemma \ref{lem:asymptoticGeneral}, where $\lambda_\delta$ is the solution of the equation $g^{\lambda_\delta}_1=\delta$. Here, we implicitly used the fact that the estimate obtained in Lemma \ref{lem:asymptoticGeneral} is true uniformly in $k \in [0,\eta n^{1/3}]$. This is due to the fact that this is also true for Theorem \ref{thm:general_rw}. Finally, letting $\delta \to 0$, we have $\lambda_\delta \to 0$, hence \[
\liminf_{n \to +\infty} \frac{e^{\alpha k}}{n^{1/3}} \mathbf{P}_{k,kp} \left( M_{n-k} \geq m_n - \epsilon n^{1/3} \right) = +\infty, \] which concludes the proof. \end{proof}
\subsubsection*{Proof of Theorem \ref{thm:main}}
We denote by $a$ the solution of \eqref{eqn:existence_max} and by $\theta$ the function defined by $\theta_t = \partial_a \kappa^*_t(a_t)$. We assume that \eqref{eqn:regularity} is verified, i.e. $\theta$ is absolutely continuous with a Riemann-integrable derivative $\dot{h}$. For all $n \in \mathbb{N}$ and $k \leq n$, we set $\bar{a}^{(n)}_k = \sum_{j=1}^k a_{j/n}$. We recall that $l^* = \frac{\alpha_1}{2^{1/3}} \int_0^1 \frac{(\dot{\theta}_s \sigma_s)^{2/3}}{\theta_s} ds$, where $\alpha_1$ is the largest zero of the Airy function of first kind.
\begin{proof}[Proof of Theorem \ref{thm:main}] We observe that with the previous notation, we have $M_n = M^\theta_n(-l^*-1,\emptyset)$. By Proposition \ref{prop:regularity}, $a$ satisfies \eqref{eqn:brealpath}, $\theta$ is non-decreasing and increases only on $G$. As a consequence, \eqref{eqn:FetG} is verified, and \eqref{eqn:differential} can be written, for $\lambda \in \mathbb{R}$, \begin{equation}
\label{eqn:differentialPourmax}
\forall t \in [0,1], \theta_t g^\lambda_t = \theta_0 \lambda + \int_0^t \dot{\theta}_s g^\lambda_s + \frac{\alpha_1}{2^{1/3}} (\dot{\theta}_s \sigma_s)^{2/3} ds. \end{equation} By integration by parts, $g^\lambda_t = \lambda + \int_0^t \frac{\alpha_1}{2^{1/3}} (\dot{\theta}_s \sigma_s)^{2/3}ds$. In particular, $g^\lambda_1 = \lambda + l^*$. As a consequence, applying Lemma \ref{lem:asymptoticGeneral} to $\lambda = l$, we have \[
\lim_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}(M_n \geq \bar{a}^{(n)}_n + (l^*+l)n^{1/3}) = -\theta_0 l. \] Similarly, using Lemma \ref{lem:concentration}, for any $\epsilon>0$, \[
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\left( \left|M_n - \bar{a}^{(n)}_n - l^* n^{1/3} \right| \geq \epsilon n^{1/3} \right) < 0. \] As $a$ is Lipschitz, we have $\displaystyle \bar{a}^{(n)}_n = \sum_{j=1}^n a_{j/n} = n \int_0^1 a_s ds + O(1)$, concluding the proof. \end{proof}
Mixing Lemma \ref{lem:specialcase} and Theorem \ref{thm:main}, we obtain explicit asymptotic for the maximal displacement, in some particular cases. If $\bar{\theta}$ is non-decreasing, then $\theta = \bar{\theta}$. As a result, setting \[
\bar{l}^* = \frac{\alpha_1}{2^{1/3}} \int_0^1 \frac{\left( \dot{\bar{\theta}}_s \sigma_s \right)^{2/3}}{\bar{\theta}_s}ds, \] we have $M_n = n \int_0^1 v_s ds + \bar{l}^* n^{1/3} + o(n^{1/3})$ in probability.
\begin{remark} Let $\sigma \in \mathcal{C}^2$ be a positive decreasing function. For $t \in [0,1]$, we define the point process $L_t = (\ell^1_t,\ell^2_t)$ with $\ell^1_t,\ell^2_t$ two i.i.d. centred Gaussian random variables with variance $\sigma_t$. We consider the BRWtie with environment $(\mathcal{L}_t, t \in [0,1])$. We have $\bar{\theta}_t = \frac{\sqrt{2 \log 2}}{\sigma_t}$, which is increasing. Consequently, by Theorem \ref{thm:main} and Lemma \ref{lem:specialcase} \[ M_n = n \sqrt{2 \log 2} \int_0^1 \sigma_s ds + n\frac{\alpha_1}{2^{1/3}(2 \log 2)^{1/6}} \int_0^1 (-\sigma'_s)^{2/3} \sigma_s^{1/3} ds,\] which is consistent with the results obtained in \cite{MaZ13} and \cite{NRR13}. \end{remark}
If $\bar{\theta}$ is non-increasing, then $\theta$ is constant. Applying Theorem \ref{thm:main}, we have $M_n = n v^* + o(n^{1/3})$.
\subsection{Consistent maximal displacement with respect to a given path} \label{subsec:cmd}
Let $\varphi$ be a continuous positive function, we write $b_t = \partial_\theta \kappa_t(\varphi_t)$, and we assume that $b$ satisfies \eqref{eqn:brealpath}. We take interest in the consistent maximal displacement with respect to the path with speed profile $b$, defined by \begin{equation}
\label{eqn:defineCmdGeneral}
\Lambda^\varphi_n = \min_{|u|=n} \max_{k \leq n} \left( \bar{b}^{(n)}_k - V(u_k) \right). \end{equation} In other words, this is the smallest number such that, killing every individual at generation $k$ and in a position below $\bar{b}^{(n)}_k - \Lambda^\varphi_n$, an individual remains alive until time $n$.
We set, for $u \in \mathbf{T}$, $\Lambda^\varphi(u)= \max_{k \leq |u|} \left( \bar{b}^{(n)}_k - V(u_k) \right)$ the maximal delay of individual $u$. In particular, with the definition of Section \ref{sec:path}, for any $\mu \geq 0$, we have \begin{equation}
\label{eqn:important}
M^\varphi_n(-\mu, [0,1]) = \max\left\{ V(u), |u|=n, \Lambda^\varphi(u) \leq \mu n^{1/3} \right\}, \end{equation} in particular $M^\varphi_n(-\mu, [0,1]) > - \infty \iff \Lambda^\varphi_n \leq \mu n^{1/3}$.
For $\lambda,\mu > 0$, we denote by $g^{\lambda,\mu}$ the solution of \begin{equation}
\label{eqn:differentialCmd}
\varphi_t g_t = \varphi_0\lambda + \int_0^t \dot{\varphi}_s g_s + \ind{K^*(b)_s = 0} \frac{\sigma_s^2}{(g_s + \mu)^2} \Psi\left( \tfrac{(g_s + \mu)^3}{\sigma_s^2} \dot{\varphi}_s \right) ds. \end{equation} Using the structure of this differential equation, for any $\lambda, \mu > 0$, we have $g^{\lambda,\mu}= g^{\lambda + \mu,0}- \mu$. Indeed, let $\lambda > 0$ and $\mu>0$, and let $g = g^{\lambda +\mu,0}-\mu$. By definition, the differential equation satisfied by $g+\mu$ is \begin{align*}
\varphi_t (g_t + \mu) &= \varphi_0( \lambda + \mu) + \int_0^t \dot{\varphi}_s (g_s + \mu) + \ind{K^*(b)_s = 0} \frac{\sigma_s^2}{(g_s + \mu)^2} \Psi\left( \tfrac{(g_s + \mu)^3}{\sigma_s^2} \dot{\varphi}_s \right)ds\\
\varphi_t g_t &= \varphi_0\lambda + \int_0^t \dot{\varphi}_s g_s + \ind{K^*(b)_s = 0} \frac{\sigma_s^2}{(g_s + \mu)^2} \Psi\left( \tfrac{(g_s + \mu)^3}{\sigma_s^2} \dot{\varphi}_s \right) ds, \end{align*} and by uniqueness of the solution of the equation, we have $g = g^{\lambda, \mu}$.
For $\lambda > 0$, we set $g^\lambda = g^{\lambda,0}$. We observe that if $\{\dot{\varphi}_t > 0\} \subset \{ K^*(b)_t = 0\}$, then, for any $\lambda \geq 0$, $g^\lambda$ is a decreasing function. As $\lambda \mapsto g^\lambda$ is strictly increasing and continuous, there exists a unique non-negative $\lambda^*$ that verifies \begin{equation}
\label{eqn:definelambdastar}
g^{\lambda^*}_1 = 0. \end{equation} Alternatively, $\lambda^*$ can be defined as $\tilde{g}_1/\varphi_0$, where $\tilde{g}$ is the unique solution of the differential equation \[
\forall t \in [0,1), \varphi_t \tilde{g}_t = - \int_t^1 \dot{\varphi}_s g_s + \ind{K^*(b)_s = 0} \frac{\sigma_s^2}{\tilde{g}_s^2} \Psi\left( \tfrac{\tilde{g}_s^3}{\sigma_s^2} \dot{\varphi}_s \right) ds. \]
\begin{lemma}[Asymptotic of the consistent maximal displacement] \label{lem:asymptoticCmd} Under the assumptions \eqref{eqn:regularity}, \eqref{eqn:phibiendef}, \eqref{eqn:regularphi}, \eqref{eqn:FetG}, \eqref{eqn:integrability2phi} and \eqref{eqn:brealpath}, for any $\lambda < \lambda^*$, we have \[
\lim_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\left( \Lambda^\varphi_n \leq (\lambda^*-\lambda) n^{1/3} \right)= - \varphi_0\lambda. \] Moreover, for any $\epsilon>0$, \[
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\left( \left| \Lambda^\varphi_n - \lambda^* n^{1/3} \right| \geq \epsilon n^{1/3} \right) < 0. \] \end{lemma}
\begin{proof} Let $\lambda \in (0,\lambda^*)$, we set $g_t = g^{\lambda^*}_t$. Note first that $\Lambda^\varphi_n \leq \lambda n^{1/3}$ if and only if there exists an individual $u$ alive at generation $n$ such that $\Lambda^\varphi(u) \leq \lambda n^{1/3}$. To bound this quantity from above, we observe that such an individual either crosses $\bar{b}^{(n)}_{.} + n^{1/3} (g_{./n}-\lambda+\epsilon)$ at some time before $n$, or stays below this boundary until time $n$. Consequently, for any $\epsilon>0$, we have \begin{multline*}
\mathbf{P}\left( \mathcal{B}_n^{[0,1],G}(-\lambda, g-\lambda + \epsilon, -\lambda \right)\\ \leq \mathbf{P}\left( \Lambda^\varphi_n \leq \lambda n^{1/3} \right) \leq \mathbf{P}\left(\mathcal{A}_n^{[0,1],G}(-\lambda,g-\lambda+\epsilon) \neq \emptyset\right) + \mathbf{P}\left( \mathcal{B}_n^{[0,1],G}(-\lambda, g-\lambda + \epsilon, -\lambda \right). \end{multline*} Using Lemma \ref{lem:estimate_upperfrontier}, Lemma \ref{lem:firstorder} and Lemma \ref{lem:lowerbound}, and letting $\epsilon \to 0$, we conclude \begin{equation*}
\lim_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\left( \Lambda^\varphi_n \leq \lambda n^{1/3} \right) = -\varphi_0 (\lambda^* - \lambda). \end{equation*}
Finally, to bound $\mathbf{P}(\Lambda^\varphi_n \geq (\lambda^*+\epsilon)n^{1/3})$ we apply \eqref{eqn:important}, and we get \[
\mathbf{P}(\Lambda^\varphi_n \geq (\lambda^*+\epsilon)n^{1/3}) = \mathbf{P}(M_n(-\lambda^*-\epsilon,[0,1]) = -\infty). \] By Lemma \ref{lem:concentration}, we conclude that $\displaystyle \limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}(\Lambda^\varphi_n \geq (\lambda^*+\epsilon)n^{1/3}) < 0$. \end{proof}
\subsubsection*{Proof of Theorem \ref{thm:cmd}}
We now prove Theorem \ref{thm:cmd}, applying Lemma \ref{lem:asymptoticCmd} to $\Lambda_n = \Lambda_n^\theta$. \begin{proof}[Proof of Theorem \ref{thm:cmd}] We denote by $G = \{ t \in [0,1] : K^*(a)_t=0\}$. For $\lambda > 0$, we write $g^\lambda$ for the solution of \begin{equation}
\label{eqn:lastdifferentialCmd}
\theta_t g_t = \theta_0\lambda + \int_0^t \dot{\theta}_s g_s + \mathbf{1}_G(s) \frac{\sigma_s^2}{g_s^2} \Psi\left( \tfrac{g_s^3}{\sigma_s^2} \dot{\theta}_s \right) ds, \end{equation}
and $\lambda^*$ for the unique non-negative real number that verifies $g^{\lambda^*}_1 = 0$. By Proposition \ref{prop:regularity}, $a$ satisfies \eqref{eqn:brealpath} and $\{\theta_t > 0\} \subset \{ K^*(b)_t = 0 \}$. Applying Lemma \ref{lem:asymptoticCmd}, for any $\lambda \in (0,\lambda^*)$, we have $\lim_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\left[ \Lambda_n^\theta \leq (\lambda^* - \lambda)n^{1/3} \right] = - \theta_0 \lambda$. Moreover, for any $\epsilon>0$, we have $\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\left[ \left| \Lambda_n^\varphi - \lambda^* n^{1/3} \right| > \epsilon n^{1/3} \right] < 0$. \end{proof}
In a similar way, we can compute the consistent maximal displacement with respect to the path with speed profile $v$, which is $\Lambda^{\bar{\theta}}_n$. We denote by $\bar{g}^\lambda$ the solution of the equation \[
\bar{\theta}_t g_t = \bar{\theta}_0\lambda + \int_0^t \dot{\bar{\theta}}_s g_s + \frac{\sigma_s^2}{g_s^2} \Psi\left( \tfrac{g_s^3}{\sigma_s^2} \dot{\bar{\theta}}_s \right) ds, \] and by $\bar{\lambda}^*$ the solution of $g^{\bar{\lambda}^*}_1 = 0$. By Lemma \ref{lem:asymptoticCmd}, for all $0\leq l \leq \bar{\lambda}^*$, \[
\lim_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\left( \Lambda^\varphi_n \leq (\lambda^*-\lambda) n^{1/3} \right)= - \varphi_0\lambda, \] and for all $\epsilon>0$, we have \[
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\left[ \left| \Lambda_n^{\bar{\theta}} - \bar{\lambda}^* n^{1/3} \right| > \epsilon n^{1/3} \right]<0. \]
\subsubsection*{Consistent maximal displacement of the time-homogeneous branching random walk}
We consider $(\mathbf{T},V)$ a time-homogeneous branching random walk, with reproduction law $\mathcal{L}$. We denote by $\kappa$ the Laplace transform of $\mathcal{L}$. The optimal speed profile is a constant $v = \inf_{\theta > 0} \frac{\kappa(\theta)}{\theta}$, and we set $\theta^* = \kappa'(\theta^*)$ and $\sigma^2 = \kappa''(\theta^*)$. The equation \eqref{eqn:lastdifferentialCmd} can be written in the simpler form \[
\theta^* g^\lambda_t = \theta^* \lambda + \int_0^t \frac{\sigma^2}{\left(g^\lambda_s\right)^2} \Psi(0)ds. \] As $\Psi(0) = - \frac{\pi^2}{2}$, the solution of this differential equation is $g^\lambda_t = \left( \lambda^3 - t\frac{3\pi^2 \sigma^2}{2 \theta^*} \right)^{1/3}$.
For $\Lambda_n = \min_{|u| = n} \max_{k \leq n} \left( kv - V(u_k)\right)$, applying Theorem \ref{thm:cmd}, and the Borel-Cantelli lemma, we obtain \[
\lim_{n \to +\infty} \frac{\Lambda_n}{n^{1/3}} = \left( \frac{3\pi^2 \sigma^2}{2 \theta^*} \right)^{1/3} \quad \mathrm{a.s.} \] This result is similar to the one obtained in \cite{FaZ10} and \cite{FHS12}.
More generally, if $(\mathbf{T},V)$ is a BRWtie such that $\bar{\theta}$ is non-increasing, then $\theta$ is a constant, and for all $\epsilon>0$, \[
\limsup_{n \to +\infty} \frac{1}{n^{1/3}} \log \mathbf{P}\left[ \left|\Lambda_n - \left( \frac{3\pi^2 \sigma^2}{2 \theta} \int_0^1 \ind{K^*(a)_s = 0} ds\right)^{1/3} n^{1/3} \right| \geq \epsilon n^{1/3} \right]. \]
\appendix
\section{Airy facts and Brownian motion estimates} \label{app:bm}
In Section \ref{subsec:onesided}, using some Airy functions --introduced in Section \ref{subsec:introrw}-- facts, the Feynman-Kac formula and PDE analysis, we compute the asymptotic of the Laplace transform of the area under a Brownian motion constrained to stay positive, proving Lemma \ref{lem:bmOnesided}. Adding some Sturm-Liouville theory, we obtain by similar arguments Lemma \ref{lem:bmTwosided} in Section \ref{subsec:twosided}. In all this section, $B$ stands for a standard Brownian motion, starting from $x$ under law $\mathbf{P}_x$.
\subsection{Asymptotic of the Laplace transform of the area under a Brownian motion constrained to stay non-negative} \label{subsec:onesided}
In this section, we write $\mathrm{L}^2 = \mathrm{L}^2([0,+\infty))$ for the set of square-integrable measurable functions on $[0,+\infty)$. This space $\mathrm{L}^2$ can be seen as a Hilbert space, when equipped with the scalar product \[
\crochet{f,g} = \int_0^{+\infty} f(x) g(x) dx. \]
We denote by $\mathcal{C}^2_0=\mathcal{C}^2_0([0,+\infty))$ the set of twice differentiable functions $w$ with a continuous second derivative, such that $w(0) = \lim_{x \to +\infty} w(x) = 0$. Finally, for any continuous function $w$, $\norme{w}_\infty = \sup_{x \geq 0} |w(x)|$. The main result of the section is: \textit{for all $h>0$, $0<a<b$ and $0<a'<b'$, we have} \begin{multline}
\label{eqn:bmOnesidedRappel}
\lim_{t \to +\infty} \frac{1}{t} \log \sup_{x \in \mathbb{R}} \E_x\left[ e^{-h \int_0^t B_s ds} ; B_s \geq 0, s \leq t \right]\\
= \lim_{t \to +\infty} \frac{1}{t} \log \inf_{x \in [a,b]} \E_x\left[ e^{-h \int_0^t B_s ds} \ind{B_t \in [a',b']} ; B_s \geq 0, s \leq t \right] = \frac{\alpha_1}{2^{1/3}}h^{2/3}. \end{multline}
We recall that $(\alpha_n, n \in \mathbb{N})$ is the set of zeros of $\mathrm{Ai}$, listed in the decreasing order. We start with some results on the Airy function $\mathrm{Ai}$, defined in \eqref{eqn:airy}. \begin{lemma} \label{lem:definepsin} For $n \in \mathbb{N}$ and $x \geq 0$, we set \begin{equation}
\label{eqn:definepsin}
\psi_n(x) = \mathrm{Ai}(x + \alpha_n)\left(\int_{\alpha_n}^{+\infty} \mathrm{Ai}(y)dy \right)^{-1/2}. \end{equation}
The following properties hold: \begin{itemize}
\item $(\psi_n, n \in \mathbb{N})$ forms an orthogonal basis of $\mathrm{L}^2$;
\item $\lim_{n \to +\infty} \alpha_n n^{-2/3} = -\frac{3\pi}{2}$;
\item for all $\lambda \in \mathbb{R}$ and $\psi \in \mathcal{C}^2$, if \begin{equation}
\label{eqn:sturmliouvilleOnesided}
\begin{cases}
\forall x > 0, \psi''(x) - x \psi(x) = \lambda \psi(x)\\
\psi(0) = \lim_{x \to +\infty} \psi(x) = 0,
\end{cases} \end{equation} then either $\psi=0$, or there exist $n \in \mathbb{N}$ and $c \in \mathbb{R}$ such that $\lambda = \alpha_n$ and $\psi = c \psi_n$. \end{itemize} \end{lemma}
\begin{proof} The fact that $\lim_{n \to +\infty} \alpha_n n^{-2/3}=-\frac{3\pi}{2}$ and that $(\psi_n, n \in \mathbb{N})$ is an orthogonal basis of $\mathrm{L}^2$ can be found in \cite{VaS04}. We now consider $(\lambda,\psi)$ a solution of \eqref{eqn:sturmliouvilleOnesided}. In particular $\psi$ verifies \[
\forall x > 0, \psi''(x) - (x+\lambda) \psi(x) = 0. \] By definition of $\mathrm{Ai}$ and $\mathrm{Bi}$, there exist $c_1,c_2$ such that $\psi(x) = c_1 \mathrm{Ai}(x+\lambda) + c_2 \mathrm{Bi}(x+\lambda).$ As $\lim_{x\to +\infty} \psi(x) = 0$, we have $c_2=0$, and as $\psi(0)=0$, either $c_1=0$, or $\mathrm{Ai}(\lambda)=0$. We conclude that either $\psi=0$, or $\lambda$ is a zero of $\mathrm{Ai}$, in which case $\psi(x) = c_1 \psi_n(x)$ for some $n \in \mathbb{N}$. \end{proof} As $\alpha_1$ is the largest zero of $\mathrm{Ai}$, note that the eigenfunction $\psi_1$ corresponding to the largest eigenvector $\alpha_1$ is non-negative on $[0,+\infty)$, and is positive on $(0,+\infty)$.
For $h>0$ and $n \in \mathbb{N}$, we define $\psi_n^h : x \mapsto (2h)^{1/6} \psi_n((2h)^{1/3}x)$. By Lemma \ref{lem:definepsin}, $(\psi_n^h, n \in \mathbb{N})$ forms an orthonormal basis of $\mathrm{L}^2$. With this lemma, we can prove the following preliminary result. \begin{lemma} \label{lem:asymptoticOnesided} Let $h>0$ and $u_0 \in \mathcal{C}^2_0 \cap \mathrm{L}^2$, such that $u_0', u_0'' \in \mathrm{L}^2$ and $\norme{u_0''}_\infty < +\infty$. We define, for $t \geq 0$ and $x \geq 0$ \[
u(t,x) = \E_x\left[ u_0(B_t) e^{-h \int_0^t B_s ds} ; B_s \geq 0, s \in [0,t] \right]. \] We have \begin{equation}
\label{eqn:asymptoticOnesided}
\lim_{t \to +\infty} \sup_{x \in \mathbb{R}} \left| e^{-\frac{h^{2/3}}{2^{1/3}} \alpha_1 t}u(t,x) - \crochet{u_0,\psi^h_1} \psi^h_1(x) \right| = 0. \end{equation} \end{lemma}
\begin{proof} Let $h>0$, by the Feynman-Kac formula (see e.g. \cite{KaS91}, Theorem 5.7.6), $u$ is the unique solution of the equation \begin{equation}
\label{eqn:fenymankacOnesided}
\begin{cases}
\forall t > 0, \forall x > 0, \partial_t u(t,x) = \frac{1}{2} \partial^2_{x} u(t,x) - h x u(t,x)\\
\forall x \geq 0, u(0,x) = u_0(x)\\
\forall t \geq 0, u(t,0) = \lim_{x \to +\infty} u(t,x) = 0.
\end{cases} \end{equation}
We define the operator \[
\mathcal{G}^h : \begin{array}{rcl}
\mathcal{C}^2_0 & \longrightarrow & \mathcal{C}\\
w & \longmapsto & \left( x \mapsto \frac{1}{2} w''(x) - h x w(x), x \in [0,+\infty) \right),
\end{array} \] By definition of $\mathrm{Ai}$ and of the $\psi_n^h$, we have $\mathcal{G}^h \psi_n^h = \frac{h^{2/3}}{2^{1/3}} \alpha_n \psi_n^h$, thus $(\psi_n^h)$ forms an orthogonal basis of eigenfunctions of $\mathcal{G}^h$.
We recall there exists $C>0$ such that for all $x \geq 0$, $\mathrm{Ai}(x) + \mathrm{Ai}'(x) \leq C z^{1/4} e^{-2x^{2/3}/3}$ (see e.g. \cite{VaS04}). For all $w \in \mathcal{C}^2_0 \cap \mathrm{L}^2$ such that $w'$ and $w''$ are bounded, by integration by parts \begin{align*}
\crochet{\mathcal{G}^h w, \psi_n^h}
&= \frac{1}{2} \int_0^{+\infty} w''(x) \psi_n^h(x) dx - h \int_0^{+\infty} x w(x) \psi_n^h(x) dx\\
&= \frac{1}{2} \int_0^{+\infty} w(x) (\psi_n^h)''(x) dx - h \int_0^{+\infty} x w(x) \psi_n^h(x) dx\\
&= \int_0^{+\infty} w(x) (\mathcal{G}^h \psi_n^h)(x) dx = \frac{h^{2/3}}{2^{1/3}} \alpha_n \crochet{w,\psi_n^h}. \end{align*} Therefore, decomposing $w$ with respect to the basis $(\psi_n^h)$, we have \[
\crochet{\mathcal{G}^h w, w} = \crochet{\mathcal{G}^h w, \sum_{n = 1}^{+\infty} \crochet{\psi_n^h, w} \psi_n^h} = \sum_{n = 1}^{+\infty} \crochet{w, \psi_n^h}\crochet{\mathcal{G}^h w,\psi_n^h}
= \frac{h^{2/3}}{2^{1/3}} \sum_{n = 1}^{+\infty} \alpha_n \crochet{w, \psi_n^h}^2. \] As $(\alpha_n)$ is a decreasing sequence, we have \begin{equation}
\label{eqn:prems}
\crochet{\mathcal{G}^h w, w} \leq \frac{h^{2/3}}{2^{1/3}}\sum_{n = 1}^{+\infty} \alpha_1 \crochet{w, \psi_n^h}^2 \leq \frac{h^{2/3}}{2^{1/3}} \alpha_1 \crochet{w,w}. \end{equation} If $\crochet{w,\psi_n^h}=0$, the inequality can be improved in \begin{equation}
\label{eqn:deuz}
\crochet{\mathcal{G}^h w, w} \leq \frac{h^{2/3}}{2^{1/3}} \sum_{n=2}^{+\infty} \alpha_2 \crochet{w, \psi_n^h}^2 \leq \frac{h^{2/3}}{2^{1/3}} \alpha_2 \crochet{w,w}. \end{equation}
Using these results, we now prove \eqref{eqn:asymptoticOnesided}. For $x \geq 0$ and $t \geq 0$, we define \[
v(t,x) = e^{- \frac{h^{2/3}}{2^{1/3}} \alpha_1 t} u(t,x) - \crochet{u_0,\psi^h_1} \psi_1^h. \] We observe first that for all $t \geq 0$, $\crochet{v(t,\cdot), \psi_1^h} = 0$. Indeed, we have $\crochet{v(0,\cdot),\psi_1^h} = 0$ by definition, and deriving with respect to $t$, we have \begin{align*}
\partial_t \crochet{v(t,\cdot),\psi_1^h}
&= - \frac{h^{2/3}}{2^{1/3}} \alpha_1 e^{- \frac{h^{2/3}}{2^{1/3}} \alpha_1 t} \crochet{u(t,x), \psi^h_1} + e^{- \frac{h^{2/3}}{2^{1/3}} \alpha_1 t} \crochet{\partial_t u(t,x), \psi_1^h} \\
&= - \frac{h^{2/3}}{2^{1/3}} \alpha_1 e^{- \frac{h^{2/3}}{2^{1/3}} \alpha_1 t} \crochet{u(t,x), \psi^h_1} + e^{- \frac{h^{2/3}}{2^{1/3}} \alpha_1 t} \crochet{\mathcal{G}^h u(t,x), \psi_1^h}\\
&= - \frac{h^{2/3}}{2^{1/3}} \alpha_1 e^{- \frac{h^{2/3}}{2^{1/3}} \alpha_1 t} \crochet{u(t,x), \psi^h_1} + \frac{h^{2/3}}{2^{1/3}} \alpha_1 e^{- \frac{h^{2/3}}{2^{1/3}} \alpha_1 t} \crochet{u(t,x), \psi_1^h} = 0. \end{align*}
We now prove that the non-negative, finite functions \[
J_1(t) = \int_0^{+\infty} |v(t,x)|^2 dx \quad \mathrm{and} \quad J_2(t) = \int_0^{+\infty} |\partial_x v(t,x)|^2 dx, \] are decreasing, and converge to 0 as $t \to +\infty$. We observe first that \begin{align*}
\frac{\partial_t J_1(t)}{2} &= \int_0^{+\infty} v(t,x) \partial_t v(t,x) dx\\
&= \int_0^{+\infty} v(t,x) \left[- \frac{h^{2/3}}{2^{1/3}} \alpha_1 e^{- \frac{h^{2/3}}{2^{1/3}} \alpha_1 t} u(t,x) + e^{- \frac{h^{2/3}}{2^{1/3}} \alpha_1 t} \partial_t u(t,x) \right] dx\\
&= \int_0^{+\infty} v(t,x) \left[- \frac{h^{2/3}}{2^{1/3}} \alpha_1 v(t,x) + \mathcal{G}^h v(t,x) \right] dx\\
&= - \frac{h^{2/3}}{2^{1/3}} \alpha_1 \crochet{v(t,\cdot),v(t,\cdot)} + \crochet{v(t,\cdot),\mathcal{G}^h v(t,\cdot)}. \end{align*} As $\crochet{v(t,\cdot),\psi_1^h} = 0$, by \eqref{eqn:deuz} we have $\partial_t J_1(t) \leq (2h)^{2/3} (\alpha_2 - \alpha_1) J_1(t) \leq -c J_1(t)$. By Grönwall inequality, $J_1(t)$ decreases to $0$ as $t \to +\infty$ exponentially fast. Similarly $J_2(0)<+\infty$ and \begin{align*}
\frac{\partial_t J_2(t)}{2}
&= \int_0^{+\infty} \partial_x v(t,x) \partial_t \partial_x v(t,x) dx = \int_0^{+\infty} \partial_x v(t,x) \partial_x \partial_t v(t,x) dx\\
&= \crochet{\partial_x v(t,\cdot), \mathcal{G}^h \partial_x v(t,\cdot)} - \frac{h^{2/3}}{2^{1/3}} \alpha_1 \crochet{\partial_x v(t,\cdot), \partial_x v(t,\cdot)} - h \int_0^{+\infty} v(t,x) \partial_x v(t,x) dx \leq 0, \end{align*} by integration by parts and \eqref{eqn:prems}. As $J_2$ is non-negative and decreasing, this function converges, as $t \to +\infty$, to $J_2(+\infty) \geq 0$. Moreover, we can write the derivative of $J_1$ as follows \begin{align*}
\frac{\partial_t J_1(t)}{2}
&= -\frac{h^{2/3}}{2^{1/3}} \alpha_1 \crochet{v(t,\cdot),v(t,\cdot)} + \crochet{v(t,\cdot),\mathcal{G}^h v(t,\cdot)}\\
&= - \frac{h^{2/3}}{2^{1/3}} \alpha_1 J_1(t) + \int_0^{+\infty} v(t,x) \partial^2_{x} v(t,x) dx - \int_0^{+\infty} h x v(t,x)^2 dx\\
&\leq -\frac{J_2(t)}{2} - \frac{h^{2/3}}{2^{1/3}} \alpha_1 J_1(t). \end{align*}
As $J_1(t)$ decreases toward 0, its derivative cannot remain bounded away from $0$ for large $t$. Therefore, $\lim_{t \to + \infty} J_2(t) = 0$. We conclude that $\lim_{t \to +\infty} \int_0^{+\infty} |v(t,x)|^2 + |\partial_x v(t,x)|^2 dx = 0$, which means that $v(t,\cdot)$ converges to 0 in $H^1$ norm, as $t \to +\infty$. By Sobolev injection in dimension 1, there exists $C>0$ such that \[
\norme{v(t,\cdot)}_{\infty} \leq C \int_0^{+\infty} |v(t,x)|^2 + |\partial_x v(t,x)|^2 dx, \] which proves \eqref{eqn:asymptoticOnesided}. \end{proof}
This lemma can be easily extended to authorize any bounded starting function $u_0$. \begin{corollary} \label{cor:asymptoticOnesided} Let $h>0$ and $u_0$ be a measurable bounded function. Setting, for $x \geq 0$ and $t \geq 0$ \[
u(t,x) = \E_x\left[ u_0(B_t) e^{-h \int_0^t B_s ds} ; B_s \geq 0, s \in [0,t] \right], \] we have \begin{equation}
\label{eqn:asymptoticOnesidedprime}
\lim_{t \to +\infty} \sup_{x \in \mathbb{R}} \left| e^{-\frac{h^{2/3}}{2^{1/3}} \alpha_1 t}u(t,x) - \crochet{u_0,\psi_1^h} \psi^h_1(x) \right| = 0. \end{equation} \end{corollary}
\begin{proof} Let $u_0$ be a measurable bounded function. We introduce, for $x \geq 0$ and $\epsilon>0$ \[
u_\epsilon(x) = u(\epsilon,x) = \E_x \left[ u_0(B_\epsilon) e^{-h\int_0^{\epsilon} B_s ds} ; B_s \geq 0, s \in [0,1] \right]. \] Observe that by the Markov property, for all $t \geq \epsilon$, we have \begin{align*}
u(t,x)
&= \E_x \left[ \E_{B_{t-\epsilon}}\left[ u_0(B_\epsilon) e^{-h \int_0^\epsilon B_s ds} ; B_s \geq 0, s \in [0,\epsilon] \right] e^{-h\int_0^{t-\epsilon} B_s ds} ; B_s \geq 0, s \in [0,t-\epsilon] \right]\\
&= \E_x \left[ u_\epsilon(B_{t-\epsilon}) e^{-h\int_0^{t-\epsilon} B_s ds} ; s \in [0,t-\epsilon] \right]. \end{align*} Therefore, $u(t,x) = u_\epsilon(t-\epsilon,x)$, wher $u_\epsilon(t,x) = \E_x \left[ u_\epsilon(B_{t}) e^{-h\int_0^{t} B_s ds} ; s \in [0,t] \right]$. As $\int_0^\epsilon B_s ds$ is, under the law $\mathbf{P}_x$, a Gaussian random variable with mean $\epsilon x$ and variance $\epsilon^3/3$, we have \[
|u_\epsilon(x)| \leq \norme{u_0}_\infty \E_x \left[ e^{-h\int_0^{\epsilon} B_s ds} \right] \leq \norme{u_0}_\infty e^{-\epsilon h x} e^{\frac{h^2 \epsilon^3}{6}}. \]
Moreover, as $h>0$, by the Ballot lemma, $|u_\epsilon(x)| \leq \norme{u_0}_\infty \mathbf{P}_x \left[ B_s \geq 0, s \in [0,\epsilon] \right] \leq C\epsilon^{-1/2} x$.
For any $\epsilon>0$, there exists $C>0$ such that for all $x \geq 0$, $u_\epsilon(x) \leq C x\wedge e^{-hx\epsilon}$. Therefore, we can find sequences $(v^{(n)})$ and $(w^{(n)})$ of functions in $\mathcal{C}^2_0 \cap \mathrm{L}^2$, such that $(v^{(n)})', (v^{(n)})'',(w^{(n)})'$ and $(w^{(n)})''$ are in $\mathrm{L}^2$, with bounded second derivatives verifying \[
w^{(n)} \leq u_\epsilon \leq w^{(n)} + \frac{1}{n} \quad \mathrm{and} \quad v^{(n)}-\frac{1}{n} \leq u_\epsilon \leq v^{(n)}. \]
For $n \in \mathbb{N}$, $x \geq 0$ and $t \geq 0$, we set \begin{multline*}
v^{(n)}(t,x) = \E_x\left[ v^{(n)}(B_t)e^{-\int_0^t B_s ds} ; B_s \geq 0, s \in [0,t] \right] \quad \mathrm{and}\\
w^{(n)}(t,x) = \E_x\left[ w^{(n)}(B_t)e^{-\int_0^t B_s ds} ; B_s \geq 0, s \in [0,t] \right]. \end{multline*} Note that for all $x \geq 0$ and $t \geq 0$ we have $w^{(n)}(t,x) \leq u_\epsilon(t,x) \leq v^{(n)}(t,x)$. Moreover, by Lemma \ref{lem:asymptoticOnesided}, we have \begin{align*}
&\lim_{t \to +\infty} \sup_{x \in \mathbb{R}} \left| e^{-\frac{h^{2/3}}{2^{1/3}} \alpha_1 t}v^{(n)}(t,x) - \crochet{v^{(n)},\psi_1^h} \psi^h_1(x) \right| = 0,\\
\mathrm{and} \quad & \lim_{t \to +\infty} \sup_{x \in \mathbb{R}} \left| e^{-\frac{h^{2/3}}{2^{1/3}} \alpha_1 t}w^{(n)}(t,x) - \crochet{w^{(n)},\psi_1^h} \psi^h_1(x) \right| = 0. \end{align*}
By the dominated convergence theorem, we have \[
\lim_{n \to +\infty} \crochet{w^{(n)},\psi_1^h} = \lim_{n \to +\infty} \crochet{v^{(n)},\psi_1^h} = \crochet{u_\epsilon, \psi_1^h}. \] As a result, letting $t$, then $n \to +\infty$, this yields \[
\lim_{t \to +\infty} \sup_{x \in \mathbb{R}} \left| e^{-\frac{h^{2/3}}{2^{1/3}} \alpha_1 t}u_\epsilon(t,x) - \crochet{u_\epsilon,\psi_1^h} \psi^h_1(x) \right| = 0. \] Finally, for almost every $x \geq 0$, letting $\epsilon \to 0$, we have $u_\epsilon(x) \to u_0(x)$, and thus by dominated convergence theorem again, $\lim_{\epsilon \to 0} \crochet{u_\epsilon,\psi_1^h} = \crochet{u_0,\psi_1^h}.$ Moreover, as $u(t,x) = u_\epsilon(t-\epsilon,x)$, we conclude that \[
\lim_{t \to +\infty} \sup_{x \in \mathbb{R}} \left| e^{-\frac{h^{2/3}}{2^{1/3}} \alpha_1 t}u(t,x) - \crochet{u_0,\psi_1^h} \psi^h_1(x) \right| = 0. \] \end{proof}
This last corollary is enough to prove the exponential decay of the Laplace transform of the area under a Brownian motion constrained to stay positive. \begin{proof}[Proof of Lemma \ref{lem:bmOnesided}] Let $h>0$, for $t, x \geq 0$ we set $u(t,x) = \E_x\left[ e^{-h\int_0^t B_s ds} ; B_s \geq 0, s \in [0,t] \right]$ and $\mu_h = \int_0^{+\infty} \psi_1^h(x)dx<+\infty$. By Corollary \ref{cor:asymptoticOnesided}, we have \[
\lim_{t \to +\infty} \sup_{x \in [0,+\infty)} \left| e^{-\frac{h^{2/3}}{2^{1/3}} \alpha_1 t}u(t,x) - \mu_h \psi^h_1(x) \right| = 0. \] As $\psi^h_1$ is bounded, we have \begin{equation}
\label{eqn:no1}
\limsup_{t \to +\infty} \sup_{x \geq 0} \frac{1}{t} \log u(t,x) = \frac{h^{2/3}}{2^{1/3}} \alpha_1. \end{equation}
Similarly, for $0<a<b$ and $0<a'<b'$, we set \[
\tilde{u}(t,x) = \E_x\left[ \ind{B_t \in [a',b']} e^{-h\int_0^t B_s ds} ; B_s \geq 0, s \in [0,t] \right] \] and $\tilde{\mu}_h = \int_{a'}^{b'} \psi_1^h(x) dx > 0$. By Corollary \ref{cor:asymptoticOnesided} again, we have \[
\lim_{t \to +\infty} \sup_{x \geq 0} \left| e^{-\frac{h^{2/3}}{2^{1/3}} \alpha_1 t}\tilde{u}(t,x) - \tilde{\mu}_h \psi^h_1(x) \right| = 0. \] In particular, as $\inf_{x \in [a,b]} \psi^h_1>0$, we have \begin{equation}
\label{eqn:no2}
\liminf_{t \to +\infty} \inf_{x \in [a,b]} \frac{1}{t} \log \tilde{u}(t,x) = \frac{h^{2/3}}{2^{1/3}} \alpha_1. \end{equation} As $\tilde{u} \leq u$, mixing \eqref{eqn:no1} and \eqref{eqn:no2}, we conclude that \[
\lim_{t \to +\infty} \sup_{x \geq 0} \frac{1}{t} \log u(t,x) = \lim_{t \to +\infty} \inf_{x \in [a,b]} \frac{1}{t} \log \tilde{u}(t,x) = \frac{h^{2/3}}{2^{1/3}} \alpha_1. \] \end{proof}
\subsection{The area under a Brownian motion constrained to stay in an interval} \label{subsec:twosided}
The main result of this section is that for all $h \in \mathbb{R}$, $0<a<b<1$ and $0<a'<b'<1$, we have \begin{multline}
\label{eqn:convergence}
\lim_{t \to +\infty} \frac{1}{t} \sup_{x \in [0,1]} \log \E_x\left[ e^{-h \int_0^t B_s ds} ; B_s \in [0,1], s \leq t \right]\\ =
\lim_{t \to +\infty} \frac{1}{t} \inf_{x \in [a,b]} \log \E_x\left[ e^{-h \int_0^t B_s ds} \ind{B_t \in [a',b']} ; B_s \in [0,1], s \leq t \right] = \Psi(h), \end{multline} where $\Psi$ is the function defined in \eqref{eqn:definepsi}. The organisation, the results and techniques of the section are very similar to Section \ref{subsec:onesided}, with a few exceptions. First, to exhibit an orthonormal basis of eigenfunctions, we need some additional Sturm-Liouville theory, that can be found in \cite{Zet05}. Secondly, we work on $[0,1]$, which is a compact set. This lightens the analysis of the PDE obtained while proving Lemma \ref{lem:asymptoticTwosided}.
In this section, we write $\mathrm{L}^2 = \mathrm{L}^2([0,1])$ for the set of square-integrable measurable functions on $[0,1]$, equipped with the scalar product \[
\crochet{f,g} = \int_0^{1} f(x) g(x) dx. \]
Moreover, we write $\mathcal{C}^2_0 = \mathcal{C}^2_0([0,1])$ for the set of continuous, twice differentiable functions $w$ on $[0,1]$ such that $w(0)=w(1)=0$. Finally, for any continuous function $w$, $\norme{w}_\infty = \sup_{x \in [0,1]} |w(x)|$ and $\norme{w}_2= \crochet{w,w}^{1/2}$. We introduce in a first time a new specific orthogonal basis of $[0,1]$. \begin{lemma} \label{lem:definephin} Let $h>0$. The set of zeros of $\lambda \mapsto \mathrm{Ai}(\lambda) \mathrm{Bi}(\lambda + (2h)^{1/3}) - \mathrm{Ai}(\lambda + (2h)^{1/3})\mathrm{Bi}(\lambda)$ is countable and bounded from above by $0$, listed in the decreasing order $\lambda^h_1> \lambda^h_2> \cdots$. In particular, we have \begin{equation}
\label{eqn:defineLambda}
\lambda^h_1 = \sup\left\{ \lambda \leq 0 : \mathrm{Ai}(\lambda) \mathrm{Bi}(\lambda + (2h)^{1/3}) = \mathrm{Ai}(\lambda + (2h)^{1/3})\mathrm{Bi}(\lambda) \right\}. \end{equation} Additionally, for $n \in \mathbb{N}$ and $x \in [0,1]$, we define \begin{equation}
\label{eqn:definephin}
\varphi_n^h(x) = \frac{\mathrm{Ai}\left(\lambda^h_n\right) \mathrm{Bi}\left(\lambda^h_n + (2h)^{1/3}x\right) - \mathrm{Ai}\left(\lambda^h_n + (2h)^{1/3}x\right)\mathrm{Bi}\left(\lambda^h_n\right)}{\norme{\mathrm{Ai}\left(\lambda^h_n\right) \mathrm{Bi}\left(\lambda^h_n + \cdot\right) - \mathrm{Ai}\left(\lambda^h_n + \cdot\right)\mathrm{Bi}\left(\lambda^h_n\right)}_2}. \end{equation} The following properties are verified: \begin{itemize}
\item $(\varphi^h_n, n \in \mathbb{N})$ forms an orthogonal basis of $\mathrm{L}^2$;
\item $\lim_{n \to +\infty} \lambda_n^h n^{-2} = - \frac{\pi^2}{2}$;
\item for all $\mu \in \mathbb{R}$ and $\varphi \in \mathcal{C}^2_0$, if \begin{equation}
\label{eqn:sturmliouvilleTwosided}
\begin{cases}
\forall x \in (0,1), \frac{1}{2} \varphi''(x) - h x \varphi(x) = \mu \varphi(x)\\
\varphi(0) = \varphi(1) = 0,
\end{cases} \end{equation} then either $\varphi=0$, or there exist $n \in \mathbb{N}$ and $c \in \mathbb{R}$ such that $\mu = \frac{h^{2/3}}{2^{1/3}} \lambda^h_n$ and $\varphi = c \varphi^h_n$. \end{itemize} \end{lemma}
\begin{proof} We consider equation \eqref{eqn:sturmliouvilleTwosided}. This is a Sturm-Liouville problem with separated boundary conditions, that satisfies the hypotheses of Theorem 4.6.2 of \cite{Zet05}. Therefore, there is an infinite but countable number of real numbers $(\mu^h_n, n \in \mathbb{N})$ such that the differential equation \[
\begin{cases}
\forall x \in (0,1), \frac{1}{2} \varphi''(x) - h x \varphi(x) = \mu^h_n \varphi(x)\\
\varphi(0) = \varphi(1) = 0,
\end{cases} \] admit non-zero solutions. For all $n \in \mathbb{N}$, we write $\varphi_n^h$ for one of such solutions normalized so that $\norme{\varphi_n^h}_2=1$. For every solution $(\lambda,\varphi)$ of \eqref{eqn:sturmliouvilleTwosided}, there exist $n \in \mathbb{N}$ and $c \in \mathbb{R}$ such that $\lambda = \mu_n^h$ and $\varphi = c \varphi_n^h$. Moreover, since $(\varphi_n^h, n \in \mathbb{N})$ forms an orthonormal basis of $\mathrm{L}^2$. By Theorem 4.3.1. of \cite{Zet05}, we have $\lim_{n \to +\infty} \lambda^h_n n^{-2} = - \frac{\pi^2}{2}$.
We identify $(\mu^h_n)$ and $(\varphi^h_n)$. By the definition of Airy functions, given $\mu \in \mathbb{R}$, the solutions of \[
\begin{cases}
\frac{1}{2} \varphi'' (x) - h x \varphi(x) = \mu \varphi(x)\\
\varphi(0)=0,
\end{cases} \] are, up to a multiplicative constant \[
x \mapsto \mathrm{Bi}\left(\tfrac{2^{1/3}}{h^{2/3}} \mu\right) \mathrm{Ai}\left(\tfrac{2^{1/3}}{h^{2/3}}\mu + (2h)^{1/3}x\right) - \mathrm{Ai}\left(\tfrac{2^{1/3}}{h^{2/3}} \mu\right) \mathrm{Bi}\left(\tfrac{2^{1/3}}{h^{2/3}}\mu + (2h)^{1/3}x\right). \] This function is null at point $x=1$ if and only if \[
\mathrm{Bi}\left(\tfrac{2^{1/3}}{h^{2/3}} \mu \right) \mathrm{Ai}\left(\tfrac{2^{1/3}}{h^{2/3}}\mu + (2h)^{1/3}\right) - \mathrm{Ai}\left(\tfrac{2^{1/3}}{h^{2/3}} \mu\right) \mathrm{Bi}\left(\tfrac{2^{1/3}}{h^{2/3}}\mu + (2h)^{1/3}\right) = 0. \] Therefore, the zeros of $\lambda \mapsto \mathrm{Ai}(\lambda) \mathrm{Bi}(\lambda + (2h)^{1/3}) - \mathrm{Ai}(\lambda + (2h)^{1/3})\mathrm{Bi}(\lambda)$, can be listed in the decreasing order as follows: $\lambda_1^h> \lambda_2^h > \ldots$, and we have $\lambda_n^h = \tfrac{2^{1/3}}{h^{2/3}}\mu_n^h$. Moreover, we conclude that the eigenfunction $\varphi_n^h$ described above is proportional to \[
x \mapsto \mathrm{Ai}\left(\lambda^h_n\right) \mathrm{Bi}\left(\lambda^h_n + (2h)^{1/3}x\right) - \mathrm{Ai}\left(\lambda^h_n + (2h)^{1/3}x\right) \mathrm{Bi}\left(\lambda^h_n\right), \] and has $\mathrm{L}^2$ norm 1, which validates the formula \eqref{eqn:definephin}.
We have left to prove that for all $n \geq 1$, $\lambda^h_n < 0$. To do so, we observe that if $(\mu,\varphi)$ is a solution of \eqref{eqn:sturmliouvilleTwosided}, we have \begin{align*}
\mu \int_0^1 \varphi(x)^2 dx &= \int_0^1 \varphi(x) \frac{1}{2} \partial^2_{x} \varphi(x) - \int_0^1 x \varphi(x)^2 dx\\
&= - \frac{1}{2} \int_0^1 (\partial_x \varphi(x))^2 dx - h \int_0^1 x \varphi(x)^2 dx < 0, \end{align*} which proves that for all $n \in \mathbb{N}$, $\mu_n^h < 0$, so $\lambda^h_1 < 0$ which justifies \eqref{eqn:defineLambda}. \end{proof} We observe that once again, the eigenfunction $\varphi^h_1$ corresponding to the largest eigenvector $\frac{h^{2/3}}{2^{1/3}} \lambda^h_n$ is a non-negative function on $[0,1]$, and positive on $(0,1)$.
Using this lemma, we can obtain a precise asymptotic of the Laplace transform of the area under a Brownian motion. \begin{lemma} \label{lem:asymptoticTwosided} Let $h>0$ and $u_0 \in \mathcal{C}^2([0,1])$ such that $u_0(0) = u_0(1) = 0$. We define, for $t,x \geq 0$ \[
u(t,x) = \E_x\left[ u_0(B_t) e^{-h \int_0^t B_s ds} ; B_s \in [0,1], s \in [0,t] \right]. \] We have \begin{equation}
\label{eqn:asymptoticTwosided}
\lim_{t \to +\infty} \sup_{x \in \mathbb{R}} \left| e^{-\frac{h^{2/3}}{2^{1/3}} \lambda^h_1 t}u(t,x) - \crochet{u_0,\varphi^h_1} \varphi^h_1(x) \right| = 0. \end{equation} \end{lemma}
\begin{proof} This proof is very similar to the proof of Lemma \ref{lem:asymptoticOnesided}. For $h>0$, by the Feynman-Kac formula, $u$ is the unique solution of the equation \begin{equation}
\label{eqn:fenymankacTwosided}
\begin{cases}
\forall t > 0, \forall x \in (0,1), \partial_t u(t,x) = \frac{1}{2} \partial^2_{x} u(t,x) - h x u(t,x)\\
\forall x \in [0,1], u(0,x) = u_0(x)\\
\forall t \geq 0, u(t,0) = \lim_{x \to +\infty} u(t,x) = 0.
\end{cases} \end{equation} We define the operator \[
\mathcal{G}^h : \begin{array}{rcl}
\mathcal{C}^2_0 & \to & \mathcal{C}\\
w & \mapsto & \left( x \mapsto \frac{1}{2} w''(x) - h x w(x), x \in [0,1] \right),
\end{array} \] By Lemma \ref{lem:definephin}, we know that $(\varphi_n^h)$ forms an orthogonal basis of $\mathrm{L}^2$ consisting of eigenvectors of $\mathcal{G}^h$. In particular, for all $n \in \mathbb{N}$, $\mathcal{G}^h \varphi_n^h = \frac{h^{2/3}}{2^{1/3}} \lambda^h_n \varphi_n^h$.
For any $w \in \mathcal{C}^2_0$, by integration by parts, we have $\crochet{\mathcal{G}^h w, \varphi_n^h} = \frac{h^{2/3}}{2^{1/3}} \lambda^h_n \crochet{w,\varphi_n^h}$. Decomposing $w$ with respect to the basis $(\varphi_n^h)$, we obtain \[
\crochet{\mathcal{G}^h w, w} = \crochet{\mathcal{G}^h w, \sum_{n = 1}^{+\infty} \crochet{\varphi_n^h, w} \varphi_n^h} = \sum_{n = 1}^{+\infty} \crochet{w, \varphi_n^h}\crochet{\mathcal{G}^h w,\varphi_n^h}
= \sum_{n = 1}^{+\infty}\frac{h^{2/3}}{2^{1/3}} \lambda^h_n \crochet{w, \varphi_n^h}^2. \] As $(\lambda^h_n)$ is a decreasing sequence, we get \begin{equation}
\label{eqn:prems2}
\crochet{\mathcal{G}^h w, w} \leq \sum_{n = 1}^{+\infty}\frac{h^{2/3}}{2^{1/3}} \lambda^h_1 \crochet{w, \varphi_n^h}^2 \leq \frac{h^{2/3}}{2^{1/3}} \lambda^h_1 \crochet{w,w}. \end{equation} In addition, if $\crochet{w,\varphi_n^h}=0$, the inequality can be strengthened in \begin{equation}
\label{eqn:deuz2}
\crochet{\mathcal{G}^h w, w} \leq \sum_{n=2}^{+\infty} \frac{h^{2/3}}{2^{1/3}} \lambda^h_2 \crochet{w, \varphi_n^h}^2 \leq\frac{h^{2/3}}{2^{1/3}} \lambda^h_2 \crochet{w,w}. \end{equation}
Using these results, we prove \eqref{eqn:asymptoticTwosided}. For $x \in [0,1]$ and $t \geq 0$, we set \[
v(t,x) = e^{- \frac{h^{2/3}}{2^{1/3}} \lambda^h_1 t} u(t,x) - \crochet{u_0,\varphi^h_1} \varphi_1^h. \] By definition, $\crochet{v(0,\cdot),\varphi_1^h} = 0$, and for any $t \geq 0$, \[
\partial_t \crochet{v(t,\cdot),\varphi_1^h} = - \frac{h^{2/3}}{2^{1/3}} \lambda^h_1 e^{- \frac{h^{2/3}}{2^{1/3}} \lambda^h_1 t} \crochet{u(t,x), \varphi^h_1} + e^{- \frac{h^{2/3}}{2^{1/3}} \lambda^h_1 t} \crochet{\partial_t u(t,x), \varphi_1^h} = 0, \] which proves that $\crochet{v(t,\cdot), \varphi_1^h} = 0$.
We now prove that the functions $J_1(t) = \int_0^1 |v(t,x)|^2 dx$ and $J_2(t) = \int_0^1 |\partial_x v(t,x)|^2 dx$ are non-negative, decreasing, and converge to 0 as $t \to +\infty$. Note that, similarly to the previous section, \[
\partial_t J_1(t) = \int_0^1 2 v(t,x) \partial_t v(t,x) dx = 2 \left[- \frac{h^{2/3}}{2^{1/3}} \lambda^h_1 \crochet{v(t,\cdot),v(t,\cdot)} + \crochet{v(t,\cdot),\mathcal{G}^h v(t,\cdot)} \right]. \] As $\crochet{v(t,\cdot),\varphi_1^h} = 0$, we have $\partial_t J_1(t) \leq (2h)^{2/3} (\lambda^h_2 - \lambda^h_1) J_1(t)$. Therefore, $J_1(t)$ decreases to $0$ as $t \to +\infty$. With same computations \[
\partial_t J_2(t) = - \frac{h^{2/3}}{2^{1/3}} \lambda^h_1 \crochet{\partial_x v(t,\cdot), \partial_x v(t,\cdot)} + 2 \crochet{\partial_x v(t,\cdot), \mathcal{G}^h \partial_x v(t,\cdot)} - 2 h \int_0^1 v(t,x) \partial_x v(t,x) dx\leq 0. \] Thus $J_2$ is non-increasing and non-negative, therefore convergent. Observing that \[
\partial_t J_1(t) \leq -J_2(t) - (2h)^{2/3} \lambda^h_1 J_1(t), \] we conclude, with the same reasoning as in the previous section that $J_2$ decreases toward $0$.
Finally, by Cauchy-Schwarz inequality, for any $x \in [0,1]$, we have \[
|v(t,x)| \leq \int_0^x |\partial_x v(t,x)| dx \leq x^{1/2} \left(\int_0^x |\partial_x v(t,x)|^2 dx \right)^{1/2} \leq J_2(t), \] so $\lim_{t \to +\infty} \norme{v(t,\cdot)}_\infty = 0$, which proves \eqref{eqn:asymptoticOnesided}. \end{proof}
This lemma can be easily extended to authorize more general starting function $u_0$. \begin{corollary} \label{cor:asymptoticTwosided} Let $h>0$ and $u_0$ be a measurable bounded function. Setting, for $x \geq 0$ and $t \geq 0$ \[
u(t,x) = \E_x\left[ u_0(B_t) e^{-h \int_0^t B_s ds} ; B_s \in [0,1], s \in [0,t] \right], \] we have \begin{equation}
\label{eqn:asymptoticTwosidedprime}
\lim_{t \to +\infty} \sup_{x \in \mathbb{R}} \left| e^{-\frac{h^{2/3}}{2^{1/3}} \lambda^h_1 t}u(t,x) - \crochet{u_0,\varphi_1^h} \varphi^h_1(x) \right| = 0. \end{equation} \end{corollary}
\begin{proof} Let $u_0$ be a measurable bounded function. Using the Ballot theorem, for any $\epsilon>0$, there exists $C>0$ such that for any $x \in [0,1]$, $u(\epsilon,x) \leq C \min(x,1-x)$. Consequently, $u_\epsilon=u(\epsilon,.)$ can be uniformly approached from above and from below by functions $v^{(n)},w^{(n)}$ in $\mathcal{C}^2_0$. Writing \begin{align*}
&v^{(n)}(t,x) = \E_x\left[ v^{(n)}(x)e^{-\int_0^t B_s ds} ; B_s \in [0,1], s \in [0,t] \right]\\
\text{and } &w^{(n)}(t,x) = \E_x\left[ w^{(n)}(x)e^{-\int_0^t B_s ds} ; B_s \in [0,1], s \in [0,t] \right], \end{align*} we have, for any $t \geq 0$ and $x \in [0,1]$, $v^{(n)}(t,x) \geq u(t+\epsilon,x) \geq w^{(n)}(t,x)$. Applying Lemma \ref{lem:asymptoticTwosided} and the dominated convergence theorem, we obtain \[
\lim_{t \to +\infty} \sup_{x \in \mathbb{R}} \left| e^{-\frac{h^{2/3}}{2^{1/3}} \lambda^h_1 t}u_\epsilon(t,x) - \crochet{u_\epsilon,\varphi_1^h} \varphi^h_1(x) \right| = 0. \] As for almost every $x \in [0,1]$, $\lim_{\epsilon \to 0} u_\epsilon(x) = u_0(x)$, by dominated convergence theorem again we conclude that \[
\lim_{t \to +\infty} \sup_{x \in \mathbb{R}} \left| e^{-\frac{h^{2/3}}{2^{1/3}}\lambda^h_1 t}u(t,x) - \crochet{u_0,\varphi_1^h} \varphi^h_1(x) \right| = 0. \] \end{proof}
\begin{proof}[Proof of Lemma \ref{lem:bmTwosided}] Let $h>0$, we set $u(t,x) = \E_x\left[ e^{-h\int_0^t B_s ds} ; B_s \in [0,1], s \in [0,t] \right]$ and write $\mu_h = \int_0^{1} \varphi_1^h(x)dx<+\infty$. By Corollary \ref{cor:asymptoticOnesided}, we have \[
\lim_{t \to +\infty} \sup_{x \geq 0} \left| e^{-\frac{h^{2/3}}{2^{1/3}} \lambda^h_1 t}u(t,x) - \mu_h \varphi^h_1(x) \right| = 0. \] As $\varphi^h_1$ is bounded, \begin{equation}
\label{eqn:majoBmTwosided}
\limsup_{t \to +\infty} \sup_{x \geq 0} \frac{1}{t} \log u(t,x) = 2^{-1/3} h^{2/3} \lambda^h_1. \end{equation}
Similarly, for $0<a<b<1$ and $0<a'<b'<1$, we set \[
\tilde{u}(t,x) = \E_x\left[ \ind{B_t \in [a',b']} e^{-h\int_0^t B_s ds} ; B_s \in [0,1], s \in [0,t] \right], \] and $\tilde{\mu}_h = \int_{a'}^{b'} \varphi_1^h(x) dx > 0$. Corollary \ref{cor:asymptoticOnesided} implies that \[
\lim_{t \to +\infty} \sup_{x \geq 0} \left| e^{-\frac{h^{2/3}}{2^{1/3}} \lambda^h_1 t}\tilde{u}(t,x) - \tilde{\mu}_h \varphi^h_1(x) \right| = 0. \] In particular, as $\inf_{x \in [a,b]} \varphi^h_1>0$, we have \begin{equation}
\label{eqn:minoBmTwosided}
\liminf_{t \to +\infty} \inf_{x \in [a,b]} \frac{1}{t} \log \tilde{u}(t,x) = \frac{h^{2/3}}{2^{1/3}} \lambda^h_1. \end{equation} Using the fact that $\tilde{u} \leq u$, \eqref{eqn:majoBmTwosided} and \eqref{eqn:minoBmTwosided} lead to \[
\lim_{t \to +\infty} \sup_{x \geq 0} \frac{1}{t} \log u(t,x) = \lim_{t \to +\infty} \inf_{x \in [a,b]} \frac{1}{t} \log \tilde{u}(t,x) = \frac{h^{2/3}}{2^{1/3}} \lambda^h_1. \]
Moreover, by definition of $\Psi$, for all $h>0$ we have $\Psi(h) = \frac{h^{2/3}}{2^{1/3}} \lambda^h_1$, and \eqref{eqn:alternativeDefinition} is a consequence of the definition of $\lambda^h_1$. By the implicit function theorem, we observe immediately that $\Psi$ is differentiable on $(0,+\infty)$. Moreover, \[
\frac{\Psi(h)}{h^{2/3}} = 2^{-1/3} \lambda^h_1 = 2^{-1/3}\sup\left\{ x \in \mathbb{R} : \mathrm{Bi}\left(\lambda\right)\mathrm{Ai}\left(\lambda + (2h)^{1/3}\right) = \mathrm{Ai}\left( \lambda \right) \mathrm{Bi}\left( \lambda + (2h)^{1/3} \right) \right\}. \] Observe that $\lim_{h \to +\infty} \sup_{\lambda \geq \alpha_2} \frac{\mathrm{Bi}\left(\lambda\right)\mathrm{Ai}\left(\lambda + (2h)^{1/3}\right)}{\mathrm{Bi}\left( \lambda + (2h)^{1/3} \right)} = 0$. As $\mathrm{Ai}(\lambda^h_1) = \frac{\mathrm{Bi}(\lambda^h_1) \mathrm{Ai}(\lambda^h_1 + (2h)^{1/3})}{\mathrm{Bi}(\lambda^h_1 + (2h)^{1/3})}$, we have $\lim_{h \to +\infty} \frac{\Psi(h)}{h^{2/3}} = 2^{-1/3} \alpha_1$.
We now observe that if $h<0$, then \begin{multline*}
\E_x\left[ u(B_t) e^{-h\int_0^t B_s ds} ; B_s \in [0,1], s \in [0,t] \right]\\
= e^{-ht}\E_{1-x} \left[ u(1-B_t) e^{h \int_0^t B_s ds} ; B_s \in [0,1], s \in [0,t] \right], \end{multline*} yielding, for any $0<a<b<1$ and $0<a'<b'<1$, \begin{multline*}
\lim_{t \to +\infty} \sup_{x \in [0,1]} \frac{1}{t} \log \E_x\left[ e^{-h\int_0^t B_s ds} ; B_s \in [0,1], s \in [0,t] \right]\\
= \lim_{t \to +\infty} \inf_{x \in [a,b]} \frac{1}{t} \log \E_x\left[ \ind{B_t \in [a',b']} e^{-h\int_0^t B_s ds} ; B_s \in [0,1], s \in [0,t] \right] = -h + \Psi(-h). \end{multline*} Moreover, for $h<0$, $\Psi(h) = \Psi(-h)-h$.
Finally, we take care of the case $h=0$. By \cite{ItK74}, \[
\mathbf{P}_x\left[ B_t \in [a,b], B_s \in [0,1], s \in [0,t] \right]
= \int_a^b 2 \sum_{n=1}^{+\infty} e^{-n^2\frac{\pi^2}{2}t}\sin(n \pi x) \sin(n\pi z)dz \] As a consequence, \begin{multline*}
\lim_{t \to +\infty} \sup_{x \in [0,1]} \frac{1}{t} \log \mathbf{P}_x\left[B_s \in [0,1], s \in [0,t] \right]\\
= \lim_{t \to +\infty} \inf_{x \in [a,b]} \frac{1}{t} \log \mathbf{P}_x\left[B_t \in [a',b'], B_s \in [0,1], s \in [0,t] \right] = \Psi(0) = -\frac{\pi^2}{2}. \end{multline*} \end{proof}
\section{Notation} \label{app:notation}
\begin{itemize}
\item {\em Point processes}
\begin{itemize}
\item $\mathcal{L}_t$: law of a point process;
\item $L_t$: point process with law $\mathcal{L}_t$;
\item $\kappa_t$: log-Laplace transform of $\mathcal{L}_t$;
\item $\kappa^*_t$: Fenchel-Legendre transform of $\mathcal{L}_t$.
\end{itemize}
\item {\em Paths}
\begin{itemize}
\item $\mathcal{C}$: set of continuous functions on $[0,1]$;
\item $\mathcal{D}$: set of càdlàg functions on $[0,1]$, continuous at point 1;
\item $\bar{b}^{(n)}_k=\sum_{j=1}^k b_{j/n}$: path of speed profile $b \in \mathcal{D}$;
\item $K^*(b)_t = \int_0^t \kappa^*_s(b_s) ds$: energy associated to the path of speed profile $b$;
\item $\varphi_t = \partial_a \kappa^*_t (b_t)$: parameter associated to the path of speed profile $b$;
\item $E(\varphi)_t = \int_0^t \varphi_s \partial_\theta \kappa_s(\varphi_s) - \kappa_s(\varphi_s)ds$: quantity equal to $K^*(b)_t$, energy associated to the path of parameter function $\varphi$;
\item $\mathcal{R} = \left\{ b \in \mathcal{D} : \forall t \in [0,1], K^*(b)_t \leq 0 \right\}$: set of speed profiles $b$ such that $\bar{b}^{(n)}$ is followed until time $n$ by at least one individual with positive probability.
\end{itemize}
\item {\em Branching random walk}
\begin{itemize}
\item $\mathbf{T}$: genealogical tree of the process;
\item $u \in \mathbf{T}$: individual in the process;
\item $V(u)$: position of the individual $u$;
\item $|u|$: generation at which $u$ belongs;
\item $u_k$: ancestor of $u$ at generation $k$;
\item $\emptyset$: initial ancestor of the process;
\item if $u \neq \emptyset$, $\pi u$: parent of $u$;
\item $\Omega(u)$: set of children of $u$;
\item $L^u = (V(v)-V(u), v \in \Omega(u))$: point process of the displacement of the children;
\item $M_n = \max_{|u|=n} V(u)$ maximal displacement at the $n^\text{th}$ generation in $(\mathbf{T},V)$;
\item $\Lambda_n = \min_{|u|=n} \max_{k \leq n} \bar{b}^{(n)}_k - V(u_k)$: consistent maximal displacement with respect to the path $\bar{b}^{(n)}$;
\item $\mathcal{W}^\varphi_n = \{ u \in \mathbf{T} : \forall k \in F_n, V(u_k) \geq \bar{b}^{(n)}_k + f_{k/n} n^{1/3} \}$: tree of a BRWtie with selection above the curve $\bar{b}^{(n)}_. + n^{1/3}f_{./n}$ at times in $F_n$.
\end{itemize}
\item {\em The optimal path}
\begin{itemize}
\item $v^* = \sup_{b \in \mathcal{R}} \int_0^1 b_s ds$: speed of the BRWtie;
\item $a \in \mathcal{R}$ such that $\int_0^1 a_s ds = v^*$: optimal speed profile;
\item $\theta_t = \partial_a \kappa^*_s(a_s)$: parameter of the optimal path;
\item $\sigma^2_t = \partial^2_\theta \kappa_t(\theta_t)$: variance of individuals following the optimal path;
\item $\dot{\theta}$: Radon-Nikod\'ym derivative of $d\theta$ with respect to the Lebesgue measure;
\item $l^*= \alpha_1 2^{-1/3}\int_0^1 \frac{\left(\dot{\theta}_s \sigma_s\right)^{2/3}}{\theta_s} ds$: $n^{1/3}$ correction of the maximal displacement;
\item $v_t = \inf_{\theta > 0} \frac{\kappa_t(\theta)}{\theta}$: natural speed profile.
\end{itemize}
\item {\em Airy functions}
\begin{itemize}
\item $\mathrm{Ai}(x) = \frac{1}{\pi} \int_0^{+\infty} \cos\left( \tfrac{s^3}{3} + x s \right) ds$: Airy function of the first kind;
\item $\mathrm{Bi}(x) = \frac{1}{\pi} \int_0^{+\infty} \exp\left( - \tfrac{s^3}{3} + x s \right) + \sin\left( \tfrac{s^3}{3} + x s \right) ds$: Airy function of the second kind;
\item $(\alpha_n)$: zeros of $\mathrm{Ai}$, listed in the decreasing order;
\item $\Psi(h) = \lim_{t \to +\infty} \frac{1}{t} \log \sup_{x \in [0,1]} \E_x\left[ e^{-h\int_0^t B_s ds} ; B_s \in [0,1], s \in [0,t] \right]$.
\end{itemize}
\item {\em Random walk estimates}
\begin{itemize}
\item $(X_{n,k}, n \in \mathbb{N}, k \leq n)$: array of independent random variables;
\item $S^{(n)}_k = S^{(n)}_0 + \sum_{j=1}^k X_{n,j}$: time-inhomogeneous random walk, with $\mathbf{P}_x(S^{(n)}_0 = x) = 1$;
\item Given $f,g \in \mathcal{C}$, and $0 \leq j \leq n$, \[
I_n(j) =
\begin{cases}
\left[f_{j/n}n^{1/3}, g_{j/n}n^{1/3}\right] & \text{if } j \in F_n \cap G_n,\\
\left[f_{j/n},+\infty\right[ & \text{if } j \in F_n \cap G_n^c,\\
\left]-\infty, g_{j/n}n^{1/3}\right] & \text{if } j \in F_n^c\cap G_n,\\
\mathbb{R} & \text{otherwise;}
\end{cases} \]
\item $\tilde{I}^{(n)}_j = I^{(n)}_j \cap [-n^{2/3}, n^{2/3}]$.
\end{itemize}
\item {\em Many-to-one lemma}
\begin{itemize}
\item $\mathbf{P}_{k,x}$: law of the BRWtie of length $n-k$ with environment $(\mathcal{L}_{(k+j)/n},j \leq n-k)$;
\item $\mathcal{F}_k = \sigma(u,V(u), |u| \leq k)$: filtration of the branching random walk;
\item Many-to-one lemma: Lemma \ref{lem:manytoone}.
\end{itemize}
\item {\em Random walk estimates}
\begin{itemize}
\item $\mathcal{A}_n^{F,G}(f,g) = \left\{ u \in \mathbf{T}, |u| \in G_n : V(u) - \bar{b}^{(n)}_{|u|} > g_{|u|/n}n^{1/3} , V(u_j) - \bar{b}^{(n)}_j \in I^{(n)}_j , j < |u|\right\}$: individuals staying in the path $\bar{b}^{(n)} + I^{(n)}$ until some time then exiting by the upper boundary;
\item $\mathcal{B}_n^{F,G} (f,g,x) = \left\{ |u|=n : V(u_j) - \bar{b}^{(n)}_j \in \tilde{I}^{(n)}_j, j \leq n, V(u)-\bar{b}^{(n)}_n\geq (g_1-x)n^{1/3}\right\}$: individuals staying in the path $\bar{b}^{(n)} + I^{(n)}$, and being above $\bar{b}^{(n)}_n + (g_1-x)n^{1/3}$ at time $n$.
\end{itemize} \end{itemize}
\nocite{*}
\def$'${$'$}
\end{document} | arXiv |
\begin{definition}[Definition:Underlying Set/Topological Space]
Let $T = \struct {S, \tau}$ be a topological space.
Then the '''underlying set''' of $T$ is the set $S$.
\end{definition} | ProofWiki |
View source for Gumbel, Emil Julius
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{| class="wikitable" !Copyright notice <!-- don't remove! --> |- | This article ''Emil Julius Gumbel'' was adapted from an original article by Sébastien Hertz, which appeared in ''StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies''. The original article ([<nowiki>http://statprob.com/encyclopedia/EmilJuliusGUMBEL.html</nowiki> StatProb Source], Local Files: [[Media:EmilJuliusGUMBEL.pdf|pdf]] | [[Media:EmilJuliusGUMBEL.tex|tex]]) is copyrighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike License'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]]. |- |} <!-- \documentclass[12pt]{article} \begin{document} \noindent --> '''Emil Julius GUMBEL''' b. 18 July 1891 - d. 10 September 1966 <!-- \noindent --> '''Summary.''' In spite of a scientific career disrupted by exile (to France in 1932, then to the United States in 1940) the German-born pacifist E.J. Gumbel was the principal architect of the statistical theory of extreme values. Emil Julius Gumbel was born in Munich (München), Germany, into a family of Jewish origin thoroughly assimilated into the Bavarian middle class and aristocracy. He was a student at the prestigious Kaiser-Wilhelms-Gymnasium where he graduated with his Abitur in July 1910. He then pursued his studies in mathematics and political economy at the Ludwig-Maximilian University in Munich. He specialized in statistics and obtained an actuarial diploma in February 1913. For the following two semesters (summer 1913 and winter 1913/14) he worked as Georg von Mayr's (q.v.) assistant in his statistical and actuarial seminar, and he was then put in charge of tutorials in mathematical statistics. At the same time, Gumbel wrote his thesis under von Mayr, and on 24 July 1914, he received his doctorate in political economy with the highest grading. His thesis, ''Die Berechnung des Bevölkerungsstandes durch Interpolation'' appeared in 1916 in the supplements of E. Roesle's periodical ''Archiv für soziale Hygiene und Demographie''. Meanwhile, war had broken out and Gumbel, like many of his German contemporaries, threw himself passionately into it. Enrolled early as a volunteer, he rapidly became aware and disillusioned by the deceptions perpetrated in the name of German imperial power. Before he could venture onto the battlefield, Gumbel was initially exempted for reasons of health, and later, at the beginning of 1916, permanently excused from military service. In the meantime, he had installed himself in Berlin, where he worked in the Civil Service as an engineer in aeronautical workshops, and later at Telefunken, while militantly active as a pacifist, and again taking up studies (in physics) at Humboldt University. He there came in contact with Albert Einstein and Ladislaus von Bortkiewicz (q.v.) at the university and with Einstein also in the pacifist league ''Bund Neues Vaterland''. Bortkiewicz, who was to give a decisive orientation to the career and statistical work of Gumbel, had succeeded Wilhelm Lexis (q.v.) as leader of the Continental Statistical School. Bortkiewicz and Lexis had views opposed to those of Georg von Mayr , who was reluctant to use the calculus of probabilities in dealing with statistical data. By the end of the first world war, Gumbel had devoted his main efforts to the political struggle. He had placed his statistical expertise at the service of the pacifist and anti-nationalist cause, publishing pamphlets and enquiries on the assassinations carried out by the extreme right, and left largely unpunished by German justice. It was only in 1923 that Gumbel, with Bortkiewicz's help, received his Habilitation at Heidelberg, becoming a Privatdozent in mathematical statistics. The hostility of an increasing number of his colleagues and students to his ideas and his political activity was to plunge his university career into chaos. He was suspended several times, leaving Germany for Moscow in 1926 to work on the mathematical archives of Marx (MEGA project). Despite his nomination as professor extraordinarius, against the advice of his Faculty, Gumbel was eventually ousted from his position at Heidelberg on the eve of the Third Reich. He then found refuge in France, where he was invited to the Institut Henri Poincaré by Emile Borel (q.v.). In 1934 he was welcomed as a foreign assistant at the Institut de Science Financière et d'Assurances in Lyon. With the support of Maurice Fréchet (q.v.) he was appointed to the CNRS (The National Council for Scientific Research) in 1937. The possibility of a university career in France was not to be fulfilled, however, as war broke out again and forced him into a new exile in the USA, where he remained until his death. Gumbel's support of Bortkiewicz's ideas became more apparent in the 1920's and 1930's. The most notable example of this is a work published by Gumbel in 1932, ''Das Zufallsgesetz des Sterbens (The Statistical Law Governing Mortality)'', which brought to a close both his research on the subject and his activities in Germany; a few months later he was forced into exile. In fact, having spent much effort in rejecting all attempts to establish as demographic laws empirical mathematical formulae depending on interpolation, or on theoretically dubious fitted curves, Gumbel found himself defending the paradoxical idea of a mathematical law relating mortality to age. Its probabilistic nature revealed his debt to Bortkiewicz. It was essentially based on work of Lexis: who had adjusted mortality tables using a Gaussian distribution for those ages considered to be beyond normal. Gumbel extended the formula to the entire table, by considering as variables not the age at death, but the life expectancy at each age. He was then able to enunciate his probabilistic principle as "Unser Leben ist in Gottes Hand,[...] Das Schicksal zieht ein schwarzes Los aus der Urnen der Lebendigen. (Our life is in God's hand [...] Destiny draws a black ball from the ballot-boxes of those living." The statistics of mortality could then be treated as analogous to an urn model, a model which was then considered to be the basis of the calculus of probabilities (as in K. Pearson's (q.v.) curves or ``Laws"). Hence the word ``Gesetz (Law)" in the title of Gumbel's work, which is an allusion to the famous "Das Gesetz der kleinen Zahlen'' (1898) of Bortkiewicz. Gumbel's French period is arguably his most fertile one, with his scientific activities hardly affected by political harassment. During the 1930's he remained an active militant anti-Nazi, together with other exiled intellectuals. When he came to France, he brought with him a new idea, that of extreme values, which was to occupy his thoughts for the rest of his life and ensure his enduring renown. Certain formulas of fit, for example those of de Moivre (q.v.) and Wittstein, consider a survival table as stopping at the age at which no further survivors remain. Around 1930, the Danish mathematician Steffensen rekindled discussion on the topic by pointing out the difficulties of such a hypothesis, in that statistical distributions most used by actuaries, among them mainly those of Gompertz-Makeham, and of Gauss-Lexis, only tend to zero asymptotically. Gumbel's innovation consisted in the redefinition of the limiting age, on the basis of the calculus of probabilities. He introduced a new random variable, the "oldest age" of a generation, whose mode was to be called ``final age" and whose expectation only designated the ``limiting age". The new distribution differs in general from that describing the population. By approximating the size of the population, the distribution of the maximum age may be modelled by one of the asymptotic laws of extreme values. Having used his ideas on different tables, Gumbel collected his contributions in a 1937 monograph of the series in mathematical statistics edited by G. Darmois (q.v.) entitled "La durée extrême de la vie humaine''. In developing his "theory of the maximum value", Gumbel rediscovered, extended and correlated several publications on extremes carried out at the turn of the century by various authors who were apparently unaware of each other's work. Some belonged to the Continental School such as Bortkiewicz and R. von Mises (q.v.), to whom one might add the Pole Jerzy Neyman (q.v.), and the American E.L. Dodd. Another contingent consisted of British biometricians, F. Galton (q.v.), K. Pearson, L.H.C. Tippett and R.A. Fisher (q.v.). Finally, the Frenchmen J. Bertrand (q.v.), J. Haag, and especially M. Fréchet and the Italian B.de Finetti also made substantial contributions to the extreme value problem. The theoretical structure was completed in 1943 with the statement and proof of the theorem on the limit of extremes, in central limit style, by B.V. Gnedenko, There are only three kinds of limit laws possible for the maximum value, all characterized by their ``max-stability": the distribution of the maximum value, apart from a change of scale and unit, remains the same as that of the sample from which it comes. This relates the statistical theory of extremes to that of the addition of random variables, as for example in Paul Lévy's work. One of the three limiting distributions had been found by M. Fréchet in 1926. The second was named after Weibull (1887-1979), a Swedish engineer , who applied it to problems of confidence from the mid 1930's. Finally, Gumbel stressed the theoretical and practical interest of the last distribution, which was logically given his name. This, apart from a change of scale, has distribution function $G(x) = exp(exp(-x)), x \geq 0 $. Gumbel, by his interdisciplinary and transnational skills, his polyglot abilities, and his applied mathematician's flair, was able to bring to light and exploit a body of theoretical work and prove its practical use by applying his results to several areas. The first was demography, followed by cases of radioactivity in collaboration with the Curies, and finally and most importantly the fields of hydrology and meteorology. From 1937 on, first in France and after 1940 in the USA, he became an expert forecaster of river floods, and later of their drought levels, and other extreme climatic phenomena. Unable to find a university position in the USA, Gumbel acted as a consultant to different government organizations, even including NATO. It was only in 1953 that he was appointed to a chair at Columbia University. He was also elected to the membership of the ISI, with the support of M. Fréchet among others. His contributions are summarized in his book "Statistics of Extremes''. This was published in New York in 1958 and is the first treatise devoted entirely to this field. It was widely disseminated, mainly to engineers, with further editions in 1960 and up to 1979, while a Japanese translation appeared in 1963 and a Russian edition under the direction of Gnedenko in 1965. ====References==== {| |- |valign="top"|{{Ref|1}}||valign="top"| Gumbel, E. J.(1922). ''Vier Jahre politischer Mord''. Verlag der Neuen Gesellschaften, Berlin. |- |valign="top"|{{Ref|2}}||valign="top"| Gumbel, E. J.(1932). Das Zufallsgesetz des Sterbens.'' Ergänzungshefte zum Deutschen Statistischen Zentralblatt'', Heft 12, 66 pp, Leipzig & Berlin. |- |valign="top"|{{Ref|3}}||valign="top"| Gumbel, E. J.(1958). ''Statistics of Extremes''. 375 pp, Columbia University Press, New York. |- |valign="top"|{{Ref|4}}||valign="top"| Hertz, S. (1997). ''Emil Julius Gumbel (1891-1966) et la statistique des extrêmes''. Thèse de doctorat (dir. P. Crépel), Université de Lyon-1. |- |valign="top"|{{Ref|5}}||valign="top"| Jansen, C. (1991). ''Emil Julius Gumbel. Portrait eines Zivilisten.'' Wunderhorn, Heidelberg. |- |valign="top"|{{Ref|6}}||valign="top"| Vogt, Annette (1991). ''Emil Julius Gumbel. Auf der Suche nach Wahrheit.'' Dietz, Berlin. |- |} <!-- \end{document} --> <references /> Reprinted with permission from Christopher Charles Heyde and Eugene William Seneta (Editors), Statisticians of the Centuries, Springer-Verlag Inc., New York, USA. [[Category:Statprob]] [[Category:Biographical]]
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\begin{document}
\title{Complexity of the homomorphism extension problem in the random case} \author{Alexandr Kazda}
\maketitle \begin{abstract} We prove that if ${\mathbb A}$ is a large random relational structure (with at least one relation of arity at least 2) then the homomorphism extension problem $\operatorname{EXT}({\mathbb A})$ is almost surely NP-complete.
Key words: homomorphism, constraint satisfaction problem, random digraph \end{abstract}
\section{Introduction} The complexity of the constraint satisfaction problem (CSP) with a fixed target structure is a well established field of study in combinatorics and computer science (see~\cite{nesetril} for an overview). In the last decade, we have seen algebraic tools brought to bear on the question of CSP complexity, yielding major new results (see e.g.~\cite{BJK}, \cite{larossetesson-introduction}, \cite{libor}).
In the algebraic approach, it is customary to study relational structures that contain all possible constants. If ${\mathbb A}$ is such a structure and we are to decide the existence of a homomorphism $f:{\mathbb B}\to {\mathbb A}$ then the constant constraints prescribe values for $f$ at some vertices of ${\mathbb B}$. We are thus deciding if some partial homomorphism $f_c:{\mathbb B} \to {\mathbb A}$ can be extended to the whole ${\mathbb B}$. Therefore, $\operatorname{CSP}({\mathbb A})$ becomes the \emph{homomorphism extension problem} with target structure ${\mathbb A}$, denoted by $\operatorname{EXT}({\mathbb A})$. It is easy to see that $\operatorname{CSP}({\mathbb A})$ reduces to $\operatorname{EXT}({\mathbb A})$, since in CSP we extend the empty partial homomorphism.
In \cite{nesetril-random}, the authors prove that $\operatorname{CSP}({\mathbb A})$ is almost surely NP-complete for ${\mathbb A}$ large random relational structure with at least one at least binary relation and without loops. We show by a different method that the same hardness result holds for $\operatorname{EXT}({\mathbb A})$ even if we allow loops.
\section{Preliminaries} A \emph{relational structure} ${\mathbb A}$ is any set $A$ together with a family of relations $\{R_i:i\in I\}$ where $R_i\subset A^{n_i}$. We call the number $n_i$ the \emph{arity} of $R_i$. The sequence $(n_i:i\in I)$ determines the \emph{similarity type} of ${\mathbb A}$. We consider only finite structures (and finitary relations) in this paper. We use the notation $[n]=\{1,2,\dots,n\}$.
Let ${\mathbb A}=(A,\{R_i:i\in I\})$ and ${\mathbb B}=(B,\{S_i:i\in I\})$ be two relational structures of the same similarity type. A mapping $f: A\to B$ is a homomorphism if for every $i\in I$ and every $(a_1,\dots,a_{n_i})\in R_i$ we have $(f(a_1),\dots,f(a_{n_i}))\in S_i$.
Let us fix some $p\in(0,1)$ and let $A$ be a set. The relation $S\subset A^l$ is an \emph{$l$-ary random relation} on $A$ if every possible $l$-tuple belongs to $S$ with probability $p$ (independently of other $l$-tuples). We will call any relational structure with one or more random relations a \emph{random relational structure}. In particular, a random relational structure with just one binary relation is a \emph{random digraph}.
The \emph{Constraint Satisfaction Problem} with the target structure ${\mathbb A}$, denoted by $\operatorname{CSP}({\mathbb A})$, consists of deciding whether a given input relational structure ${\mathbb B}$ of the same similarity type as ${\mathbb A}$ can be homomorphically mapped to ${\mathbb A}$. It is easy to come up with examples of ${\mathbb A}$ such that $\operatorname{CSP}({\mathbb A})$ is NP-complete and this is in a sense typical behavior as proved in \cite{nesetril-random}: If $R(n,k)$ is a $k$-ary random relation on the set $[n]$ (with $p=1/2$) that does not contain any elements of the form $(a,a,\dots,a)$ for $a\in A$ then \begin{align} \forall k\geq 2,\,\lim_{n\to\infty}\operatorname{Prob}(\operatorname{CSP}([n],R(n,k))\text{ is NP-complete})=1,\label{nesetril1}\\ \forall n\geq 2, \lim_{k\to\infty}\operatorname{Prob}(\operatorname{CSP}([n],R(n,k))\text{ is NP-complete})=1.\label{nesetril2} \end{align}
There is a reason why the authors of \cite{nesetril-random} disallow loops: If ${\mathbb A}$ has only one relation $R$ and $R$ contains a loop $(a,a,\dots,a)$ then every ${\mathbb B}$ of the same similarity type as ${\mathbb A}$ can be homomorphically mapped to ${\mathbb A}$ simply by sending everything to $a$, so $\operatorname{CSP}(A)$ is very simple to solve.
Given a target structure ${\mathbb A}$, the \emph{Homomorphism Extension Problem} for ${\mathbb A}$, denoted by $\operatorname{EXT}({\mathbb A})$, consists of deciding whether a given input structure ${\mathbb B}$ and a given partial mapping $f:{\mathbb B}\to {\mathbb A}$ can be extended to a homomorphism from ${\mathbb B}$ to ${\mathbb A}$.
Let ${\mathbb A}$ be a set and $a\in A$. The \emph{constant relation} $c_a$ is the unary relation consisting only of $a$, i.e. $c_a=\{(a)\}$. When searching for a homomorphism to ${\mathbb A}$, the relation $c_a$ prescribes a set of elements of $B$ that must be mapped to $a$. A little thought gives us that if ${\mathbb A}$ contains constant relations for each of its elements (as is usual in the algebraic treatment of CSP) then $\operatorname{CSP}({\mathbb A})$ and $\operatorname{EXT}({\mathbb A})$ are essentially the same problem.
Since the homomorphism extension problem is quite important to algebraists, it makes sense to ask what is the typical complexity of $\operatorname{EXT}({\mathbb A})$. We will use the phrase ``$\operatorname{EXT}({\mathbb A})$ is almost surely NP-compete for $n$ large'' as an abbreviation for ``For each $n\in{\mathbb N}$, there exists a random relational structure ${\mathbb A}_n$ (whose precise definition is obvious from the context) such that we have \[ \lim_{n\to\infty}\operatorname{Prob}(\operatorname{EXT}({\mathbb A}_n)\text{ is NP-complete})=1.\text{''} \]
Because additional relations do not make $\operatorname{CSP}$ easier to solve, the limit (\ref{nesetril1}) gives us that that $\operatorname{EXT}({\mathbb A})$ is almost surely NP-complete if ${\mathbb A}$ is a large random relational structure with no loops and at least one relation of arity greater than one. In the remainder of the paper we show that we can allow loops without making the problem any easier.
\section{The problem $\operatorname{EXT}$ for random digraphs} We will begin by investigating random digraphs and then generalize our findings to all relational structures.
\begin{theorem}\label{thmNPgraph} Let $G$ be a random digraph on $n$ vertices. Then $\operatorname{EXT}(G)$ is almost surely NP-complete for $n$ large. \end{theorem}
\begin{proof}
Let $G=(V,E)$ be a digraph. Understand $G$ as a relational structure and add to $G$ every constant relation possible. Let $v_1,\dots,v_l\in V(G)$. Consider the set \[ F_{v_1,\dots,v_l}=\{u\in V(G): \forall i, (v_i,u)\in E(G)\} \] We will call this set a \emph{subalgebra} of $G$.
For an interested reader, we note that sets $F_{v_1,\dots,v_l}$ are subalgebras in the universal algebraic sense and our technique can be greatly generalized to all primitive positive definitions (see \cite{BJK}). For our proof, however, we need a lot less: Assume that for some choice of $v_1,\dots, v_l$ the subalgebra $F_{v_1,\dots,v_l}$ induces a loopless triangle in $G$.
We claim that we can then reduce graph 3-colorability to $\operatorname{EXT}(G)$, making $\operatorname{EXT}(G)$ NP-complete.
Let $H$ be a graph whose 3-colorability we wish to test. We then understand $H$ as a symmetric digraph and add to $H$ new vertices $w_1,\dots,w_l$ and new edges $(w_i,u)$ for each $i\in \{1,\dots,n\}$ and all $u\in V(H)$, obtaining the digraph $H'$.
Our $\operatorname{EXT}(G)$ instance will then consist of the digraph $H'$ along with the partial map $f$ which maps each $w_i$ to $v_i$. Now $f$ can be extended to a homomorphism if and only if $H$ can be homomorphically mapped into the triangle induced by $F_{v_1,\dots,v_l}$ which happens if and only if $H$ is 3-colorable.
All we need to do now is to show that $G$ almost surely contains a subalgebra that induces a triangle. Our aim, roughly speaking, is to show that $G$ almost surely contains many three element subalgebras because then there is a large chance that at least one of these subalgebras will be a triangle.
We will partition $V(G)$ into two sets $A=\{1,\dots,\lfloor n/2\rfloor\}$ and $B=\{\lceil n/2 \rceil,\dots,n\}$. We will now use points of $A$ to define subalgebras lying in $B$. Denote by $S_k$ the event ``$G$ contains at least $k$ disjoint three-element subalgebras of the form $F_{v_1,\dots,v_l}\subset B$ for some $v_1,\dots,v_l\in A$.'' We can write $$S_k=\bigcup_{\substack{C_1,\dots,C_k\subset B\\ \forall i\neq j,\, C_i\cap C_j=\emptyset\\
\forall i,\, |C_i|=3}} S_{C_1,\dots,C_k}, $$ where $S_{C_1,\dots,C_k}$ is the event ``The sets $C_1,\dots,C_k$ are subalgebras of $G$''. Finally, denote by $T_{C_1,\dots,C_k}$ the event ``There exists an $i\in\{1,2.\dots,k\}$ such that the set $C_i$ induces a triangle subgraph of $G$.''
Since a probability that a fixed $C_i$ induces a triangle is $p^6(1-p^3)$, the probability of the event $T_{C_1,\dots,C_k}$ is (for $C_1,\dots,C_k$ pairwise disjoint three element sets) $$ \operatorname{Prob}(T_{C_1,\dots,C_k})=1-(1-p^6(1-p^3))^k, $$ which tends to 1 when $k$ goes to infinity.
Observe that the event $S_{C_1,\dots,C_k}$ is independent from the event $T_{C_1,\dots,C_k}$ for each choice of $C_1,\dots,C_k\subset B$ since both events talk about disjoint sets of edges of $G$.
Assume for a moment that for all $k\in {\mathbb N}$ the value of $\operatorname{Prob}(S_k)$ tends to 1 as $n$ tends to infinity. Then, given an $\epsilon>0$, we choose $k$ so that $\operatorname{Prob}(T_{C_1,\dots,C_k})\geq 1-\epsilon$. When $n$ is large enough, the digraph $G$ contains some $k$ pairwise disjoint three element subalgebras $C_1,\dots,C_k$ with probability at least $1-\epsilon$. The probability that one of the sets $C_1,\dots,C_k$ them induces a triangle is $T_{C_1,\dots,C_k}\geq 1-\epsilon$. Thus we get an NP-complete CSP problem with probability at least $(1-\epsilon)^2> 1-2\epsilon$ and since $\epsilon$ was arbitrary, we see that for large $n$ the homomorphism extension problem is almost surely NP-complete.
It remains to show $\lim_{n\to\infty}\operatorname{Prob}(S_k)=1$ for all $k$. Fix the value of $k$. For each value of $n$, let $l$ be the integer satisfying $n p^l\geq 1>n p^{l+1}$. We will now search for the three element subalgebras of $B$ for $n$ large. We proceed in steps: Assume that after $i$ steps we have already found $m$ such subalgebras $C_1,\dots,C_m$. In the $(i+1)$-th step, we take the vertices $1+il,2+il,\dots,l+il$ of $A$ and consider the subalgebra $F_{1+il,2+il,\dots,l+il}$. If this subalgebra lies in $B$, has size three and is disjoint with all the sets $C_1,\dots,C_m$, we let $C_{m+1}=F_{1+il,2+il,\dots,l+il}$, increase $m$ by one and continue with the next step. Otherwise, $F_{1+il,2+il,\dots,l+il}$ is not a good candidate for $C_{m+1}$, so we leave $m$ unchanged and continue with the next step.
What is the probability that we find the $(m+1)$-th subalgebra in a given step? Every vertex of $G$ is in $F_{1+il,2+il,\dots,l+il}$ with the probability $p^l$. The probability that $F_{1+il,2+il,\dots,l+il}$ consists of three yet-unused vertices of $B$ is then equal to \[
q={|B|-3\cdot m \choose 3}p^{3l}(1-p^l)^{n-3}\geq \frac{(n/2-3m-3)^3}{6}p^{3l}(1-p^l)^n \] If $m\geq k$, we have already won, so assume $m<k$: \[ q\geq \frac{(n/2-3k)^3}{6}p^{3l}(1-p^l)^n=\frac{(1/2-3k/n)^3}{6} n^3p^{3l}(1-p^l)^n \] Now let $r=\frac{(1/2-3k/n)^3}6$ and observe that $r>0$ for $n$ large enough. Using the the inequalities $n p^l\geq 1>n p^{l+1}$ we have: \[ q \geq r n^3 p^{3l}(1-p^l)^n \geq r (1-p^l)^n> r \left(1-\frac{1}{pn}\right)^n. \] The lower bound on $q$ tends to $r/e^{1/p}$ as $n$ tends to infinity, so there exists a $\delta$ such that $q>\delta>0$ for all $n$ large enough.
Therefore, the probability of producing a new three-element subalgebra in a given step is at least $\delta>0$ and this bound does not depend on the number of subalgebras we have already found. Now observe that $l$ is approximately $\log_{1/p} n$ and therefore we have enough vertices in $A$ for approximately $s=\frac{n}{2\log_{1/p}n}$ steps. If we choose $n$ large enough, we can have $s$ as large as we want and so the probability of finding at least $k$ subalgebras can be arbitrarily close to 1. Therefore, $\lim_{n\to\infty} \operatorname{Prob}(S_k)=1$, concluding the proof. \end{proof}
\section{Random relational structures} It is easy to see that if ${\mathbb A}$ is a relational structure with unary relations only then $\operatorname{EXT}({\mathbb A})$ is always polynomial. We would now like to investigate the case of relations of arity greater than two. Intuition tells us that greater arity means greater complexity. The intuition is right.
\begin{lemma} Let $l\geq 2$, $n$ be large and let ${\mathbb A}=([n],S)$ be a relational structure with $S$ a random $l$-ary relation. Then the homomorphism extension problem $\operatorname{CSP}({\mathbb A})$ is almost surely NP-complete. \end{lemma} \begin{proof} We have proven the result for $l=2$. If $l>2$, consider the binary relational structure ${\mathbb B}=([n],R)$ where $R=\{(x,y)\in [n]^2: (x,y,1,1,\dots,1)\in S\}$. It is easy to see that if $S$ is a random $l$-ary relation then ${\mathbb B}$ is a random digraph where each edge exists with the probability $p$. From Theorem~\ref{thmNPgraph} we see that $\operatorname{EXT}({\mathbb B})$ is almost surely NP-complete. We will now show how to reduce $\operatorname{EXT}({\mathbb B})$ to $\operatorname{EXT}({\mathbb A})$ in polynomial time, proving that $\operatorname{EXT}({\mathbb A})$ is almost surely NP-complete.
Using algebraic tools, the reduction of $\operatorname{EXT}({\mathbb B})$ to $\operatorname{EXT}({\mathbb A})$ follows from the fact that $R$ is defined by a primitive positive formula that uses only $S$ and the constant $1$. However, we will provide an elementary reduction here: Let ${\mathbb C}=(C,T)$ be a relational structure with a single binary relation $T$ and let $f:C\to [n]$ be a partial mapping. We add to $C$ a new element $e$, construct the $l$-ary relation $U=\{(x,y,e,e,\dots,e):(x,y)\in T\}$ and the partial mapping
$g:C\cup \{e\}\to [n]$ so that $g_{|C}=f$ and $g(e)=1$. A little thought gives us that $g$ can be extended to a homomorphism $(C\cup\{e\},U)\to {\mathbb A}$ if and only if $f$ can be extended to a homomorphism $(C,T)\to {\mathbb B}$, concluding the proof. \end{proof}
Additional relations in ${\mathbb A}$ do not make $\operatorname{EXT}({\mathbb A})$ easier, so we have the most general version of our NP-completeness result:
\begin{corollary} Let ${\mathbb A}$ be the relational structure $([n],\{R_i:i\in I\})$ where at least one $R_i$ is a random relation of arity greater than one. Then $\operatorname{EXT}({\mathbb A})$ is almost surely NP-complete for $n$ large. \end{corollary}
As a final note, we will now prove the analogue of the limit~(\ref{nesetril2}) for $\operatorname{EXT}$.
\begin{corollary} Let us fix a set $A$ of at least two elements and let ${\mathbb A}=(A,R)$ be a relational structure with $R$ random $k$-ary relation. Then $\operatorname{EXT}({\mathbb A})$ is almost surely NP-complete for $k$ large. \end{corollary} \begin{proof} Assume first that $k$ is even and let $m=k/2$.
Consider the relational structure ${\mathbb B}=(A^m,S)$ with $$ S=\{((a_1,\dots,a_m),(a_{m+1},\dots,a_{2m})):(a_1,\dots,a_{2m})\in R\}. $$ It is straightforward to prove that $S$ is a binary random relation on $A^m$ and therefore $\operatorname{EXT}({\mathbb B})$ is almost surely NP-complete for large even $k$. What is more, $\operatorname{EXT}({\mathbb B})$ can be easily reduced to $\operatorname{EXT}({\mathbb A})$: If ${\mathbb C}=(C,T)$ is a relational structure with $T$ binary and $f:C\to A^m$ is a partial mapping, we construct the structure ${\mathbb C}'=(C',T')$ with \begin{align*} C'&=\{(c,i):c\in C, i\in \{1,\dots,m\}\},\\ T'&=\{((c,1),\dots,(c,m),(d,1),\dots,(d,m)):(c,d)\in T\} \end{align*} and a partial mapping $g:C'\to A$ such that $g(c,i)=a_i$ whenever $f(c)$ is defined and equal to $(a_1,\dots,a_m)$.
It is easy to see that $g$ can be extended to a homomorphism from ${\mathbb C}'$ to ${\mathbb A}$ if and only if $f$ can be extended to a homomorphism from ${\mathbb C}$ to ${\mathbb A}$.
In the case that $k=2m+1$, we fix an $e\in A$, choose ${\mathbb B}=(A^m,S)$ with $$ S=\{((a_1,\dots,a_m),(a_{m+1},\dots,a_{2m}):(a_{1},\dots,a_{2m},e)\in R\} $$ and proceed similarly to the previous case.
We see that for a large enough $k$, no matter if it is odd or even, the problem $\operatorname{EXT}({\mathbb B})$ is almost surely NP-complete. \end{proof}
\section{Conclusions} We have shown that the homomorphism extension problem is almost surely NP-complete for large relational structures (assuming we have at least one non-unary relation). In a sense, our result is not surprising since the relational structures we consider are very dense, so it stands to a reason that we can find hard instances most of the time.
It might therefore be interesting to see what is the complexity of CSP or EXT for large structures obtained by other random processes, particularly when relations are sparse. Such structures might better correspond to ``typical'' cases of CSP or EXT encountered in practice. Some such results already exist; see \cite{nesetril-luczak-projective} for a criterion on the random graph process to almost surely produce projective graphs (if $G$ is projective then $\operatorname{EXT}(G)$ is NP-complete, see \cite{BJK}). Our guess is that both CSP and EXT will remain to be almost surely NP-complete in all the nontrivial cases.
\section{Acknowledgments} The research was supported by the GA\v CR project GA\v CR 201/09/H012, by the Charles University project GA UK 67410 and by the grant SVV-2011-263317. The author would like to thank Libor Barto for suggesting this problem.
\end{document} | arXiv |
specific heat of sugar
=0.9. The specific heat capacity of crystal sugar is 1.244 kJ/kg/K at 25°C Select the number of materials in the mixture you would like to calculate. \[m_{mixture}\cdot{C}_{{p}\,mixture}\cdot\Delta{T} =\cdots\]. Step 1 – The initial setup for calculating specific heat capacity of a mixture. When did the Ancient Egyptians start believing in the afterlife? Breaking Down the Rule of Mixtures Mixture? Correspondingly, the specific heat capacities used for the calculation can be selected from Thermtest's materials database, as mentioned previously. This equation can be adjusted to approximate the specific heat capacity of a mixture with an infinite number of components when the mass and specific heat capacity of each material is known, as well as the mass of the mixture. For more information regarding the thermal conductivity of this example, the Hot Disk TPS was used to conduct measurements on asphalt, and the results can be viewed in an application page. PCMs store energy in the form of sensible heat and latent heat. Use the search box in the top-right corner to narrow your selection. Data compiled as indicated in comments: ALS - Hussein Y. Afeefy, Joel F. Liebman, and Stephen E. Stein DH- Eugene S. Domalski and Elizabeth D. Hearing The results of the calculation indicate that the specific heat capacity of the saline mixture would be approximately 3853 J/kg°C. 14-18 deg. Who is the longest reigning WWE Champion of all time? Figure 4. How long was Margaret Thatcher Prime Minister? Maintained by Still Up Marketing | Sitemap, Thermal Conductivity, Thermal Diffusivity & Specific Heat, Thermal Conductivity & Thermal Resistivity, Homogeneous, Heterogeneous, Metals & Composites, Testing the Thermal Conductivity of Polymers – Introducing the Polymer Melt Cell Sample Holder, Introducing the Thermal Resistance in Series Calculator, Differential Scanning Calorimeter, DSC-L600, Thermogravimetric Analyzer, TGA-1000 & TGA-1500, Thermal Conductivity / Thermal Diffusivity. Ibat ibang katawagan sa pilipinas ng mundo? However, as seen in Figure 2, the salt will naturally increase the boiling point of the solution, and subsequently slow down the boiling time of the salt and water solution. Once this is complete, the calculator will produce a specific heat capacity estimation. Similarly, the next example demonstrates how the Rule of Mixtures Calculator can be applied to mixtures containing more than two materials. Use Calculator. Δ f H° solid: Solid phase enthalpy of formation at standard conditions (kJ/mol). Copyright 2020. For this reason, water is able to store twice as much energy as antifreeze, allowing energy to be carried away from the engine faster than pure antifreeze. The next property to investigate is the boiling point of water; expected to decrease after the addition of salt. In comparison, literature values report a specific heat capacity of 3700 J/kg °C, for for such a mixture. The mass of each individual component is equivalent to the total mass of the mixture, which can theoretically be broken down into separate parts. Cement of high specific heat and high thermal conductivity, obtained by using silane and silica fume as admixtures. The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. If you are looking to use the calculator, click the button below. Why don't libraries smell like bookstores? The specific heat capacity of an antifreeze-water solution as the volume of antifreeze increases (Ethylene Glycol Heat-Transfer Fluid, 2017). \[ \left(C_{p}\right) = \frac{J}{{kg}\cdot^{\circ}\mathrm{C}}\]. All Rights Reserved. specific heat of sugar solution having brix 2000. Hager I. The Rule of Mixtures Calculator can be used to estimate the amount of PCMs to combine with asphalt to achieve the desired specific heat capacity. specific heat of sugar solution having brix. Specific Heat Capacity Crystal Sugar. Figure 5. C p,solid: Solid phase heat capacity (J/mol×K). In other words, the specific heat capacity represents a material's ability to store energy. Select your units from the second drop-down. Specific Heat of some Liquids and Fluids - Specific heat for some common liquids and … The following post delves into the theory behind The Rule of Mixtures Calculator, how to use it, and present real-world examples to demonstrate its usefulness. How long can you keep a fresh turkey in the fridge before it has to be cooked? Although concrete is a strong and affordable construction material, like asphalt, it is susceptible to cracking when temperature gradients form. 55-70 deg. Copyright © 2020 Multiply Media, LLC. Alternatively, you can enter your materials' name and thermal resistance manually. As previously stated, the calculator uses the rule of mixtures formula to estimate the specific heat capacity of a solution with multiple components. Step 2 – Enter the name, thermal resistance, and the mass or volume of each material. Notably, this principle can be related to the following energy balance equation, where 'Q' refers to the total energy of a mixture in units of joules. Antifreeze (ethylene glycol) is the most common commercial coolant used in vehicles. 14-18 deg. How many towns in the world are named Portland? According to literature values, the specific heat capacity of water is 4184 J/kg°C, double the specific heat capacity of antifreeze. Finally, the rule of mixtures for specific heat capacity can be derived through some rearranging to provide the final equation: \[C_{p\,mixture}=\Big(\frac{m_{1}}{m_{mixture}}\Big)C_{p\,1}+\Big(\frac{m_{2}}{m_{mixture}}\Big)C_{p\,2}\]. Additionally, the formation of temperature gradients induces stress on the location where the two temperatures meet, which also causes cracks and deformations. How can understanding energy storage help to reduce deformations in asphalt? but some where i studied specific heat of sugar solution having Behaviour of cement concrete at high temperature. When did organ music become associated with baseball? The Rule of Mixtures Calculator uses the above equation to approximate the specific heat capacity of a mixture. During this study, it was found that sensible heat energy storage increased, when PCMs were mixed with asphalt. Markedly, the first law of thermodynamics states that energy cannot be created, or destroyed. How long will the footprints on the moon last? Go To: Top, References, Notes Data compilation copyrightby the U.S. Secretary of Commerce on behalf of the U.S.A.All rights reserved. Once exposed to heat, the concrete will expand. The Rule of Mixtures Calculator can be used to demonstrate how the addition of salt would affect the specific heat capacity, and therefore boiling point of the solution. Phase Change Materials Improve the Structural Integrity of Asphalt, (Ethylene Glycol Heat-Transfer Fluid, 2017). Click on the name of the material you'd like to load. Figure 2. When the PCM/asphalt composite was combined with large amounts of energy storage from latent heat, the asphalt remained at a more constant temperature. In comparison, literature values report a specific heat capacity of 3700 J/kg °C, for for such a mixture. Xu Y, Chung DDL. Up to this point, only mixtures with two components have been discussed. Sucrose is a glycosyl glycoside formed by glucose and fructose units joined by an acetal oxygen bridge from hemiacetal of glucose to the hemiketal of the fructose.It has a role as an osmolyte, a sweetening agent, a human metabolite, an algal metabolite, a Saccharomyces cerevisiae metabolite, an Escherichia coli metabolite and a mouse metabolite. Firstly, what is specific heat capacity, and why is a specific heat capacity calculator useful for mixtures? Using the mass and specific heat capacity of each component, the Rule of Mixtures Calculator calculates the specific heat capacity of the entire sample. =0.9 Consequently, mixing the two liquids merges the low freezing point of antifreeze, with the high specific heat capacity of water. Whether it's to find the perfect ratio of antifreeze to water, to approximate a value for comparison, or just to play around with, the Rule of Mixtures Calculator quickly and efficiently estimates the specific heat capacity of mixtures. How to Use the Rule of Mixtures Calculator. Additionally, specific heat capacities can be selected from Thermtest's very own materials database, which includes the thermal properties of more than 1000 different materials. As a result, this energy is therefore influenced by specific heat capacity. In a study conducted by Xu and Chung (2000), concrete with high specific heat capacity and thermal conductivity was produced by adding silane and silica fume (Figure 5). As mentioned previously, specific heat capacity determines how quickly a material will rise in temperature. How do you put grass into a personification? 55-70 deg. In which place the raw silk factories in tajikistan? Antifreeze lowers the freezing point of fluids to prevent them from freezing in sub-zero temperatures (Figure 3). Results from the Rule of Mixtures Calculator. The Rule of Mixtures Calculator, newly released by Thermtest Inc., is an invaluable tool for estimating the specific heat capacity of mixtures containing any number of materials.
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\begin{document}
\title{Strong Stationarity Conditions for Optimal Control Problems Governed by
a Rate-Independent Evolution Variational Inequality hanks{Submitted to the editors DATE.
}
\begin{abstract} We prove strong stationarity conditions for optimal control problems that are governed by a prototypical rate-independent evolution variational inequality, i.e., first-order necessary optimality conditions in the form of a primal-dual multiplier system that are equivalent to the purely primal notion of Bouligand stationarity. Our analysis relies on recent results on the Hadamard directional differentiability of the scalar stop operator and a new concept of temporal polyhedricity that generalizes classical ideas of Mignot. The established strong stationarity system is compared with known optimality conditions for optimal control problems governed by elliptic obstacle-type variational inequalities and stationarity systems obtained by regularization. \end{abstract}
\begin{keywords} optimal control, rate independence, stop operator, variational inequality, sweeping process, strong stationarity, Bouligand stationarity, Kurzweil integral, polyhedricity, hysteresis \end{keywords}
\begin{AMS} 49J40, 47J40, 34C55, 49K21, 49K27 \end{AMS}
\section{Introduction and summary of results} \label{sec:1} This paper is concerned with the derivation of first-order necessary optimality conditions for optimal control problems of the type \begin{equation*} \label{eq:P} \tag{P} \left \{~~
\begin{aligned}
\text{Minimize}
\quad & \mathcal{J}(y, y(T), u) \\
\text{w.r.t.}
\quad &y \in CBV[0, T], \quad u \in U_{\textup{ad}},\\
\text{s.t.}
\quad & \int_0^T (v - y)\,\mathrm{d} (y - u)\geq 0~~ \forall v \in C([0, T]; Z), \\
&y(t) \in Z\quad \forall t \in [0, T],\quad y(0) = y_0.
\end{aligned} \right. \end{equation*} Here, $y$ denotes the state; $u$ denotes the control; $T>0$ is given; $CBV[0,T]$ is the space of real-valued continuous functions of bounded variation on $[0, T]$; $U_{\textup{ad}}$ is a subset of a suitable control space $U \subset CBV[0,T]$; $\mathcal{J}\colon L^\infty(0, T) \times \mathbb{R} \times U \to \mathbb{R}$ is a sufficiently smooth objective function; $Z = [-r,r]$ is a given interval with $r>0$; $C([0, T]; Z)$ is the set of continuous functions on $[0,T]$ with values in $Z$; $y_0 \in Z$ is a given initial value; and the integral in the governing variational inequality is understood in the sense of Kurzweil-Stieltjes (see \cite{Monteiro2019} and the \hyperref[sec:appendix]{appendix} of this paper for details on this type of integral). For the precise assumptions on the quantities in \eqref{eq:P}, we refer to \cref{sec:3}. The main result of this work -- \cref{th:main} -- establishes a so-called strong stationarity system for the problem \eqref{eq:P}. This is a first-order necessary optimality condition in primal-dual form that is satisfied by a control $\bar u \in U_{\textup{ad}}$ if and only if $\bar u$ is a Bouligand stationary point of \eqref{eq:P}, i.e., if and only if the directional derivative of the reduced objective function of \eqref{eq:P} at $\bar u$ is nonnegative in all admissible directions. See also \eqref{eq:strongstatsys-2} below for the resulting stationarity system.
\subsection{Background and relation to prior work} Before we present and discuss the strong stationarity system derived in \cref{th:main} in more detail, let us give some background. To keep the discussion concise, we focus on strong stationarity conditions for infinite-dimensional optimization problems arising in optimal control. For related results in finite dimensions, see \cite{Flegel2007,Harder2017,Hoheisel2013,Luo1996,Scheel2000} and the references therein.
In the field of infinite-dimensional nonsmooth optimization, strong stationarity conditions (although originally not referred to as such) have first been derived for optimal control problems governed by elliptic obstacle-type variational inequalities in the seminal works \cite{Mignot1976,MignotPuel1984} of Mignot and Puel in the nineteen-seventies and -eighties. If we use a notation analogous to that in \eqref{eq:P}, then this kind of problem can be formulated (in its most primitive form) as follows: \begin{equation} \label{eq:optstaclecontrol}
\begin{aligned}
\text{Minimize}
\quad & \mathcal{J}(y, u) \\
\text{w.r.t.}
\quad &y \in H_0^1(\Omega), \quad u \in U_{\textup{ad}} \subset L^2(\Omega),\\
\text{s.t.}
\quad & y \in Z, \quad \int_\Omega \nabla y \cdot \nabla (v - y)\,\mathrm{d} x \geq \int_\Omega u(v - y) \,\mathrm{d} x \quad \forall v\in Z.
\end{aligned} \end{equation} Here, $\Omega \subset \mathbb{R}^d$, $d \in \mathbb{N}$, is a nonempty open bounded set; $H_0^1(\Omega)$ and $L^2(\Omega)$ are defined as usual, see \cite{Evans2010,GilbargTrudinger1977}; $\mathcal{J}\colon H_0^1(\Omega) \times L^2(\Omega) \to \mathbb{R}$ is a Fr\'{e}chet differentiable objective function with partial derivatives $\partial_1 \mathcal{J}(y,u) \in H^{-1}(\Omega)$ and $\partial_2 \mathcal{J}(y,u) \in L^2(\Omega)$ (where $H^{-1}(\Omega)$ denotes the topological dual of $H_0^1(\Omega)$); $U_{\textup{ad}} \subset L^2(\Omega)$ is a convex, nonempty, and closed set; $\nabla$ is the weak gradient; and $Z$ is a nonempty set of the type \[
Z:= \left \{ v \in H_0^1(\Omega) \colon \psi_1 \leq v \leq \psi_2 \text{ a.e.\ in } \Omega\right \} \] involving two given measurable functions $\psi_1, \psi_2\colon \Omega \to [-\infty, \infty]$. The main difficulty that arises when deriving first-order necessary optimality conditions for problems like \eqref{eq:optstaclecontrol} is that the governing variational inequality causes the control-to-state operator $S\colon L^2(\Omega) \to H_0^1(\Omega)$, $u \mapsto y$, to be nondifferentiable (in the sense of G\^{a}teaux and Fr\'{e}chet). This nonsmoothness prevents classical adjoint-based approaches as found, e.g., in \cite{Troeltzsch2010} from being applicable and makes it necessary to develop tailored strategies to establish stationarity systems for local minimizers. In \cite{Mignot1976,MignotPuel1984}, the problem of deriving first-order optimality conditions for \eqref{eq:optstaclecontrol} was tackled by exploiting that the solution mapping $S\colon L^2(\Omega) \to H_0^1(\Omega)$, $u \mapsto y$, of the lower-level variational inequality in \eqref{eq:optstaclecontrol} is Hadamard directionally differentiable with directional derivatives $\delta := S'(u;h)$, $u, h \in L^2(\Omega)$, that are uniquely characterized by the auxiliary problem \begin{equation}
\label{eq:dirdiffcharobstacleproblem}
\delta \in K_\mathrm{crit}(y,u),
\quad \int_\Omega \nabla \delta \cdot \nabla (z - \delta)
\,\mathrm{d} x \geq \int_\Omega h(z - \delta) \,\mathrm{d} x
\quad \forall z\in K_\mathrm{crit}(y,u).
\end{equation} Here, $K_\mathrm{crit}(y,u) := K_{\tan}(y) \cap (u + \Delta y)^\perp$ denotes the so-called \emph{critical cone} associated with $u$ and $y := S(u)$, i.e., the intersection of the kernel \[
(u + \Delta y)^\perp :=
\left \{
z \in H_0^1(\Omega)\colon
\int_\Omega u z - \nabla y \cdot \nabla z \,\mathrm{d} x = 0
\right \} \] of the functional $u + \Delta y \in H^{-1}(\Omega)$ and the tangent cone $K_{\tan}(y) \subset H_0^1(\Omega)$ to $Z$ at $y$ which is obtained by taking the closure of the radial cone $ K_{\mathrm{rad}}(y) := \mathbb{R}_+(Z - y)$ in $H_0^1(\Omega)$, cf.\ \cite[section 2]{Harder2017} and \cite{Haraux1977,Mignot1976}. By proceeding along the lines of \cite{Mignot1976,MignotPuel1984}, one obtains the following main result for the optimal control problem \eqref{eq:optstaclecontrol}: If a control $\bar u \in U_{\textup{ad}}$ with state $\bar y := S(\bar u)$ is given such that the set $\mathbb{R}_+(U_{\textup{ad}} - \bar u)$ is dense in $L^2(\Omega)$, then $\bar u$ is a Bouligand stationary point of \eqref{eq:optstaclecontrol} in the sense that \begin{equation} \label{eq:Bouligandobstacle} \left \langle \partial_1 \mathcal{J}(\bar y, \bar u), S'(\bar u;h)\right \rangle_{H_0^1} + \left (\partial_2 \mathcal{J}(\bar y, \bar u), h \right)_{L^2} \geq 0 \quad \forall h \in \mathbb{R}_+(U_{\textup{ad}} - \bar u) \end{equation} holds if and only if there exist an adjoint state $\bar p \in H^1_0(\Omega)$ and a multiplier $\bar \mu \in H^{-1}(\Omega)$ such that $\bar u$, $\bar y$, $\bar p$, and $\bar \mu$ satisfy the system \begin{equation}\label{eq:sstatobst}
\begin{gathered}
\bar p + \partial_2 \mathcal{J}(\bar y, \bar u) = 0~~\text{ in }
L^2(\Omega),
\\
- \Delta \bar p = \partial_1 \mathcal{J}(\bar y, \bar u) - \bar \mu ~~\text{ in } H^{-1}(\Omega), \\
\bar p\in K_\mathrm{crit}(\bar y, \bar u), \quad
\left \langle \bar \mu ,z\right \rangle_{H_0^1}
\geq 0 \quad \forall z\in K_\mathrm{crit}(\bar y, \bar u).
\end{gathered} \end{equation} Here and in what follows, the symbols $\langle \cdot, \cdot \rangle$ and $(\cdot, \cdot)$ denote a dual pairing and a scalar product, respectively. For a proof of the above result, see \cite[Corollary~6.1.11]{ChristofPhd2018}. Note that, since the inequality \eqref{eq:Bouligandobstacle} expresses that the directional derivatives of the reduced objective function $L^2(\Omega) \ni u \mapsto \mathcal{J}(S(u), u) \in \mathbb{R}$ of \eqref{eq:optstaclecontrol} are nonnegative in all admissible directions $h \in \mathbb{R}_+(U_{\textup{ad}} - \bar u)$ at $\bar u$ and thus corresponds to the most natural first-order necessary optimality condition obtainable for a directionally differentiable function, and since the conditions \eqref{eq:Bouligandobstacle} and \eqref{eq:sstatobst} are equivalent, the system \eqref{eq:sstatobst} can be considered the most precise first-order
primal-dual necessary optimality condition possible for \eqref{eq:optstaclecontrol}. This is the reason why systems of the type \eqref{eq:sstatobst} became known as \emph{strong stationarity conditions} since their initial appearance in \cite{Mignot1976,MignotPuel1984}.
The main appeal of the system \eqref{eq:sstatobst} is, of course, its equivalence to the Bouligand stationarity condition \eqref{eq:Bouligandobstacle}. This characteristic property distinguishes \eqref{eq:sstatobst} from other first-order necessary optimality conditions and makes \eqref{eq:sstatobst} an important tool, e.g., for assessing which information about $\bar p$ and $\bar \mu$ is lost when a stationarity system is derived by means of a regularization or discretization approach. For details on this topic, we refer to the survey article \cite{Harder2017}. Because of these advantageous properties, strong stationarity conditions have come to play a distinct role in the field of optimal control of nonsmooth systems and have received considerable attention in the recent past. See, e.g., \cite{Betz2021,ChristofPhd2018,Christof2022,ReyesMeyer2016,Herzog2013,Hintermueller2009,Wachmuth2014,Wachsmuth2020} for contributions on strong stationarity conditions for optimal control problems governed by various elliptic variational inequalities of the first and the second kind, \cite{Betz2019,Christof2018nonsmoothPDE,Clason2021,Meyer2017} for extensions to optimal control problems governed by nonsmooth semi- and quasilinear PDEs, and \cite{Christof2021} for a generalization to the multiobjective setting. Note that all of these works on the concept of strong stationarity have in common that they are only concerned with elliptic variational inequalities or PDEs involving nonsmooth terms. What has -- at least to the best of our knowledge -- not been accomplished so far in the literature is the derivation of a necessary optimality condition analogous to \eqref{eq:sstatobst} for an optimal control problem that is governed by a true evolution variational inequality (where with ``true'' we mean that the inequality cannot be reformulated as a nonsmooth PDE or an elliptic problem, cf.\ \cite{Betz2019}). In fact, such an extension is even mentioned as an open problem in the seminal works of Mignot and Puel; see \cite[section 4]{MignotPuel1984} and \cite{MignotPuel1984:2} where strong stationarity conditions for parabolic obstacle problems are conjectured upon. This absence of results on strong stationarity systems for evolution variational inequalities is very unsatisfying in view of the multitude of processes that are modeled by this type of variational problem in finance, mechanics, and physics; see \cite{Mielke2015,Sofonea2012}.
The main reason for the lack of contributions on strong stationarity conditions for evolution variational inequalities since the nineteen-seventies is that directional differentiability results analogous to that for the elliptic obstacle problem in \eqref{eq:dirdiffcharobstacleproblem} have not been available in the instationary setting for a long period of time. See, e.g., \cite[p.\ 582]{BonnansShapiro2000} where this problem is still referred to as open. Only recently, progress in this direction has been made. In \cite{Brokate2020,Brokate2015}, it could be proved by means of a semi-explicit solution formula involving the cumulated maximum that the control-to-state operator of the problem \eqref{eq:P} -- the so-called \emph{scalar stop operator} -- is Hadamard directionally differentiable in a pointwise manner; see \cref{th:dirdiff} below. In \cite{Christof2019parob}, it could further be shown by means of pointwise-a.e.\ convexity properties that the solution mapping of the parabolic obstacle problem is Hadamard directionally differentiable as a function into all Lebesgue spaces. This paper also establishes that the directional derivatives of the solution operator of the parabolic obstacle problem are the (not necessarily unique) solutions of a weakly formulated auxiliary variational inequality analogous to \eqref{eq:dirdiffcharobstacleproblem}, see \cite[Theorem 4.1]{Christof2019parob}. Very recently, in \cite{Brokate2021}, an auxiliary problem for the directional derivatives of the scalar stop operator in \eqref{eq:P} has also been obtained by means of a careful analysis of jump directions and approximation arguments. This auxiliary problem even yields a unique characterization, see \cref{th:dirdiffVI} below.
\subsection{Main result and contribution of the paper} The purpose of the present paper is to show that the recent developments in \cite{Brokate2020,Brokate2015,Brokate2021} make it possible to prove a strong stationarity system for the optimal control problem \eqref{eq:P}. As far as we are aware, our analysis is the first to establish such a system for a true evolution variational inequality. The result in the literature that comes closest to the one derived in this paper is, at least to the best of our knowledge, \cite[Theorem 5.5]{Christof2019parob} which establishes a multiplier system for optimal control problems governed by parabolic obstacle-type variational inequalities that is equivalent to Bouligand stationarity if the adjoint state enjoys additional regularity properties -- a deficit that is caused by a mismatch between certain notions of capacity, see the discussion in \cite[section 5]{Christof2019parob}. In the present work, we do not require such additional regularity assumptions and obtain a strong stationarity system for \eqref{eq:P} that is fully equivalent to the notion of Bouligand stationarity. Our main result can be summarized as follows: If $\bar u \in U_{\textup{ad}}$ is a control of \eqref{eq:P} with associated state $\bar y$ such that the set $\mathbb{R}_+(U_{\textup{ad}} - \bar u)$ is dense in the control space $U$, then $\bar u$ is a Bouligand stationary point of \eqref{eq:P} (in a sense analogous to that of \eqref{eq:Bouligandobstacle}, see \cref{def:Bouligandstationary} below) if and only if there exist an adjoint state $\bar p \in BV[0,T]$ and a multiplier $\bar \mu \in G_r[0,T]^*$ such that $\bar u$, $\bar y$, $\bar p$, and $\bar \mu$ satisfy the system \begin{equation} \label{eq:strongstatsys-2} \begin{gathered} \bar p(0) = \bar p(T) = 0, \qquad \bar p(t) = \bar p(t-)~\forall t \in [0,T), \\ \bar p(t-) \in K^\mathrm{ptw}_{\mathrm{crit}}(\bar y, \bar u)(t)~\forall t \in [0,T], \\ \left \langle \bar \mu, z \right \rangle_{G_r} \geq 0\quad \forall z \in \mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(\bar y, \bar u), \\ \int_0^T h \,\mathrm{d} \bar p = \left \langle \partial_3 \mathcal{J}(\bar y, \bar y(T), \bar u), h \right \rangle_{U} ~\forall h \in U, \\ -\int_0^T z \,\mathrm{d} \bar p = \left \langle \partial_1 \mathcal{J}(\bar y, \bar y(T), \bar u), z\right \rangle_{L^\infty} + \partial_2 \mathcal{J}(\bar y, \bar y(T), \bar u)z(T) - \left \langle \bar \mu, z \right \rangle_{G_r} \\ \hspace{8.5cm}\forall z \in G_r[0,T]. \end{gathered} \end{equation} Here, $BV[0,T]$ denotes the space of real-valued functions of bounded variation on $[0,T]$; $G_r[0,T]$ is the space of real-valued, regulated, and right-continuous functions on $[0,T]$; $G_r[0,T]^*$ is the topological dual space of $G_r[0,T]$; the partial derivatives of $\mathcal{J}$ are denoted by $\partial_i \mathcal{J}$, $i=1,2,3$; the minus in the argument of $\bar p$ denotes a left limit; and $K^\mathrm{ptw}_{\mathrm{crit}}(\bar y,\bar u)(t)$, $t \in [0,T]$, and $\mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(\bar y, \bar u)$ are suitably defined cones (see \cref{def:ptwcritcone,def:6.1}). For the precise statement of the above result, see \cref{th:main}. Several things are noteworthy regarding the system \eqref{eq:strongstatsys-2}:
First of all, it can be seen that the adjoint state $\bar p$ lacks regularity in comparison with the optimal state $\bar y$ ($BV[0,T]$ instead of $CBV[0,T]$). This reduced regularity reflects that the directional derivatives of the control-to-state mapping of \eqref{eq:P} are not continuous in time and thus significantly less regular than the states $y$ -- a behavior that is completely absent in the elliptic problem \eqref{eq:optstaclecontrol}. For details on this topic, see also \cite[section 3]{Christof2019parob} and \cite[Example 4.1]{Brokate2021} which demonstrate that all types of jump discontinuities of the derivatives are possible in the situation of \eqref{eq:P} and that the derivatives cannot be expected to possess, e.g., $H^{1/2}(0,T)$-regularity, cf.\ \cite{Jarusek2003}.
Second, one observes that not the adjoint state $\bar p$ but its left limits are contained in the critical cone $K^\mathrm{ptw}_{\mathrm{crit}}(\bar y, \bar u)(t)$ for all $t \in [0,T]$ in \eqref{eq:strongstatsys-2}. As we will see below, this condition on the limiting behavior -- along with the left-continuity of $\bar p$ in the first line of \eqref{eq:strongstatsys-2} -- arises from certain properties of the jumps of the directional derivatives of the control-to-state mapping and the fact that the adjoint system evolves backwards in time (in contrast to the variational inequality for the directional derivatives of the control-to-state mapping which evolves in a forward manner). Note that these additional properties of the left limit of the adjoint state are not visible in stationarity systems derived by regularization, cf.\ \cite{Barbu1984,Cao2016,Colombo2020,dePinho2019,Ito2010,Stefanelli2017,Wachmuth2016}. This shows that \eqref{eq:strongstatsys-2} contains information that is not recoverable with regularization approaches.
Lastly, it should be noted that the coupling between the adjoint state $\bar p$ and the partial derivative $\partial_3 \mathcal{J}(\bar y, \bar y(T), \bar u)$ of the objective $\mathcal{J}$ w.r.t.\ the control in \eqref{eq:strongstatsys-2} is not as direct as in \eqref{eq:sstatobst} but involves an integration step. This is a consequence of the rate-independence of the variational inequality governing \eqref{eq:P} and ultimately also the reason for the nonstandard start- and endpoint conditions $\bar p(0) = \bar p(T) = 0$ for $\bar p$ in \eqref{eq:strongstatsys-2}. We remark that these conditions reflect that the partial derivative $\partial_2 \mathcal{J}(\bar y, \bar y(T), \bar u)$ manifests itself -- in a distributional sense -- in the jump of $\bar p$ at the terminal time $T$, see the comments at the end of \cref{sec:7}. A similar behavior can also be observed in optimal control problems for parabolic PDEs, see \cite[section 5.5.1]{Troeltzsch2010}.
Regarding the derivation of the strong stationarity system in \cref{th:main}, we would like to point out that -- even with the results of \cite{Brokate2020,Brokate2015,Brokate2021} at hand and even though the variational inequality in \eqref{eq:P} is one of the simplest evolution variational inequalities imaginable -- the proof of \eqref{eq:strongstatsys-2} is still quite involved. The main difficulty in the context of \eqref{eq:P} is that, due to the lack of weak-star continuity properties of the scalar stop operator, one has to discuss this problem in a control space $U$ whose topology is significantly stronger than that of $BV[0,T]$ to be able to ensure that \eqref{eq:P} is well posed; see the comments in \cref{sec:5} below. Since the directional derivatives of the scalar stop are only in $BV[0,T]$, the need for such a ``small'' control space $U$ makes it necessary to employ a careful limit analysis to ensure that the control space is \emph{ample} enough to be able to arrive at a strong stationarity system. Compare also with the comments on this topic in \cite{Christof2022,Herzog2013} and the results in \cite{Wachmuth2014} in this context. In our analysis, we tackle this problem by generalizing the classical concept of \emph{polyhedricity} to the time-dependent setting. This is a density property which, in the situation of the elliptic problem \eqref{eq:optstaclecontrol}, ensures that the set of critical radial directions $ K_{\mathrm{rad}}(y) \cap (u + \Delta y)^\perp$ is $H_0^1(\Omega)$-dense in $K_{\tan}(y) \cap (u + \Delta y)^\perp$ and which plays an important role in the sensitivity analysis of elliptic obstacle-type variational inequalities as well as the theory of second-order optimality conditions, see \cite{Haraux1977,Christof2018SSC,Wachsmuth2019}. For the approximation result that we establish in this context and that we refer to as ``temporal polyhedricity'', see \cref{theorem:tempoly}.
We expect that \cref{theorem:tempoly}, along with the insights provided by \eqref{eq:strongstatsys-2}, is also helpful for the analysis of optimal control problems governed by more complicated evolution variational inequalities, cf.\ the problems studied in \cite{Christof2019parob,Muench2018,Samsonyuk2019}.
\subsection{Structure of the remainder of the paper} We conclude this section with an overview of the content and the structure of the remainder of the paper.
\Cref{sec:2,sec:3} are concerned with preliminaries. Here, we introduce the notation and the standing assumptions that we use throughout this work. In \cref{sec:4}, we collect basic results on the properties of the control-to-state mapping of \eqref{eq:P} -- the scalar stop operator. This section also recalls the directional differentiability results of \cite{Brokate2020,Brokate2015,Brokate2021} and discusses some of their consequences. \Cref{sec:5} addresses the solvability of \eqref{eq:P} and introduces the concept of Bouligand stationarity for this problem. This section also contains an example which shows that, to be able to prove the existence of solutions for \eqref{eq:P} by means of the direct method of the calculus of variations, one indeed has to consider a control space significantly smaller than $BV[0,T]$. In \cref{sec:6}, we prove the already mentioned temporal polyhedricity property for \eqref{eq:P}. The main result of this section is \cref{theorem:tempoly}. \Cref{sec:7} is concerned with the proof of the strong stationarity system \eqref{eq:strongstatsys-2}, see \cref{th:main}. The \hyperref[sec:appendix]{appendix} of the paper collects some results on the Kurzweil-Stieltjes integral that are needed for our analysis.
\section{Notation} \label{sec:2} Throughout this work, $T>0$ is a given and fixed number. We denote the space of real-valued continuous functions on $[0,T]$ by $C[0,T]$ and the space of real-valued regulated functions on $[0,T]$ (i.e., the space of all functions that are uniform limits of step functions, see \cite[Definition 4.1.1, Theorem 4.1.5]{Monteiro2019}) by $G[0,T]$. We equip both $C[0,T]$ and $G[0,T]$
with the supremum norm $\|\cdot\|_\infty$. Recall that this makes $C[0,T]$ and $G[0,T]$ Banach spaces and that every $v \in G[0,T]$ possesses left and right limits, see \cite[chapter 4]{Monteiro2019}. Given $v \in G[0,T]$, we denote these limits by $v(t-)$ and $v(t+)$, respectively, with the usual conventions at the endpoints of $[0,T]$, i.e., \begin{align*} v(t-) &:= \lim_{[0, T] \ni s \to t^-} v(s)\quad \forall t \in (0, T],\qquad v(0-) := v(0), \\ v(t+) &:= \lim_{[0, T] \ni s \to t^+} v(s)\quad \forall t \in [0, T),\qquad v(T+) := v(T). \end{align*} For the left- and the right-limit function associated with a function $v \in G[0,T]$, we use the symbols $v_-$ and $v_+$, i.e., $v_-(t) := v(t-)$ and $v_+(t) := v(t+)$ for all $t \in [0,T]$. We further define $G_r[0,T] := \left \{ v \in G[0,T] \colon v = v_+ \right \}$. It is easy to check that this set of right-continuous regulated functions is a closed subspace of
$(G[0,T], \|\cdot\|_\infty)$.
The space of real-valued functions of bounded variation on $[0,T]$ is denoted by $BV[0,T]$. We emphasize that we do not consider elements of $BV[0,T]$ as equivalence classes in this paper but as classical functions $v\colon [0,T] \to \mathbb{R}$, as in \cite[chapter~2]{Monteiro2019}. For a discussion of different approaches to $BV[0,T]$, see \cite{Ambrosio2000}. We denote the variation of a function $v\colon[0,T]\to\mathbb{R}$ by $\operatorname{var}(v)$, and we define the total variation norm on $BV[0,T]$ as
$\|v\|_{BV} := |v(0)| + \operatorname{var}(v)$. Recall that $(BV[0,T], \|\cdot \|_{BV})$ is a Banach space that
is continuously embedded into $(G[0,T], \|\cdot \|_{\infty})$; see \cite[Theorem~2.2.2]{Monteiro2019}. We define $CBV[0,T] := BV[0,T] \cap C[0,T]$ and $BV_r[0,T] := BV[0,T] \cap G_r[0,T]$. Note that both of these sets are
closed subspaces of $(BV[0,T], \|\cdot \|_{BV})$.
Given a set-valued function $K\colon [0,T] \rightrightarrows \mathbb{R}$ and $0 \leq s < \tau \leq T$, we use the symbols $C([s,\tau];K)$ and $G([s,\tau];K)$ to denote the sets of continuous and regulated functions $v$ on $[s,\tau]$ which satisfy $v(t) \in K(t)$ for all $t \in [s,\tau]$, respectively. Sets $K \subset \mathbb{R}$ are interpreted as set-valued functions that are constant in time in this notation. We further set $C^\infty[0,T] := \{v \in C[0,T] \colon \exists \tilde v \in C^\infty(\mathbb{R})\, \text{s.t.}\, v(t) = \tilde v(t) ~\forall t \in [0,T]\}$. For the classical Lebesgue and Sobolev spaces, we use the standard notation
\mbox{$(L^p(0,T), \|\cdot\|_{L^p})$} and
$(W^{k,p}(0,T), \|\cdot\|_{W^{k,p}})$, $1 \leq p \leq \infty$, $k \in \mathbb{N}$. The weak derivative of a function $v \in W^{1,p}(0,T)$ is denoted by $v' \in L^p(0,T)$. For the topological dual of a
normed space $(X, \|\cdot\|_X)$, we use the symbol $X^*$, and for a dual pairing, the brackets $\langle\cdot, \cdot \rangle$ equipped with a subscript that clarifies the space. A closure is denoted by $\operatorname{cl}(\cdot)$. Weak, weak-star, and strong convergence are indicated by $\rightharpoonup$, \smash{$\stackrel\star\rightharpoonup$}, and $\to$, respectively. Given a set $D \subset [0,T]$, we define $\mathds{1}_{D}\colon [0,T] \to \{0,1\}$ to be the characteristic function of $D$, i.e., the function that equals 1 on $D$ and 0 everywhere else.
\section{Main problem and standing assumptions} \label{sec:3} As already mentioned in the introduction, the aim of this paper is to study optimal control problems of the type \begin{equation*} \tag{P} \left \{~~
\begin{aligned}
\text{Minimize} \quad & \mathcal{J}(y, y(T), u) \\
\text{w.r.t.}\quad &y \in CBV[0, T], \quad u \in U_{\textup{ad}},\\
\text{s.t.}\quad & y = \mathcal{S}(u),
\end{aligned} \right. \end{equation*} where $\mathcal{S}$ is the \emph{scalar stop operator}, i.e., the solution map $\mathcal{S}\colon CBV[0,T] \to CBV[0,T]$, $u \mapsto y$, of the rate-independent evolution variational inequality \begin{equation} \label{eq:V} \tag{V} \left \{~~ \begin{aligned} &\int_0^T (v - y)\,\mathrm{d} (y - u)\geq 0 &&\forall v \in C([0, T]; Z), \\ &y(t) \in Z\quad \forall t \in [0, T], &&y(0) = y_0. \end{aligned} \right. \end{equation} General references for the properties of the function $\mathcal{S}$ are \cite{BrokateSprekels1996,Krejci1996,Krejci1999}; some of them will be discussed in detail in \cref{sec:4}.
Note that, from the application point of view, \eqref{eq:P} can be interpreted as an optimal control problem for a one-dimensional sweeping process with characteristic set $Z = [-r,r]$, i.e., a problem that aims to control the trajectory of a body with one degree of freedom that is placed on a slippery surface within $Z$ and moved (swept) by moving $Z$ back and forth, see \cite[section 1.1]{Mielke2015}. (In this case, the trajectory is described by the \emph{scalar play operator} $\mathcal{P}(u) := u - \mathcal{S}(u)$ and the control function $u$ models the movement of $Z$.) This physical interpretation, however, is mainly secondary in this work. We are primarily interested in the problem \eqref{eq:P} because it is the instationary counterpart of the optimal control problem \eqref{eq:optstaclecontrol} for the elliptic obstacle problem and captures the effects of ``pure'' evolution without any additional spatial dependencies (as present, e.g., in the parabolic obstacle problem, cf.\ \cite{Christof2018SSC}). We hope that the insights provided by our analysis are also helpful for the analysis of optimal control problems governed by more complicated systems arising, e.g., in the field of elasto-plasticity, which often involve the play and stop operator to incorporate hysteresis effects, cf.\ \cite{Mielke2015, Muench2018,Samsonyuk2019}.
We would like to emphasize that the integral in \eqref{eq:V} -- along with all other integrals appearing in the remainder of this paper -- is to be understood in the sense of Kurzweil-Stieltjes. For an in-depth introduction to the integration theory for this type of integral, we refer to \cite{Monteiro2019}. A collection of basic definitions, elementary properties, and fundamental results related to the Kurzweil-Stieltjes integral can also be found in the \hyperref[sec:appendix]{appendix} of this paper. The use of the Kurzweil-Stieltjes integral for the variational inequality approach to rate-independent evolutions goes back to \cite{Krejci2003,Krejci2006,KrejciLiero2009} where it was employed for the study of discontinuous input functions $u$. For this kind of $u$, the integrand and the integrator (i.e., the function behind the ``$\,\mathrm{d}$'') in \eqref{eq:V} usually have discontinuities at common points $t\in [0,T]$ so that the Riemann-Stieltjes integral no longer works. Such common discontinuities also appear naturally in the variational inequality that characterizes the directional derivatives of $\mathcal{S}$, cf.\ \cref{th:dirdiffVI} below. For a treatment based on the Young integral, see \cite{KrejciLaurencot2002}. Alternatively, the Lebesgue-Stieltjes integral can be used as for the types of integrands and integrators appearing in this paper it is equivalent to the Kurzweil-Stieltjes integral, see \cite[section 6.12]{Monteiro2019}.
However, for this type of integral, a careful handling of statements involving ``almost everywhere'' is necessary since the $\sigma$-algebra and the family of its sets of measure zero depend on the integrator. In particular, a singleton $\{t\}$ has nonzero measure if the integrator is discontinuous at $t$.
For the ease of reference, we collect our standing assumptions on the quantities in the optimal control problem \eqref{eq:P} and the variational inequality \eqref{eq:V} in:
\begin{assumption}[standing assumptions]~\label{ass:standing} \begin{itemize} \item $T>0$ is given and fixed. \item $U \subset CBV[0,T]$ is a
real vector space that is endowed with a norm $\|\cdot\|_U$
and that is continuously and densely embedded into $(C[0,T],\|\cdot\|_\infty)$. \item $U_{\textup{ad}}$ is a nonempty and convex subset of $U$. \item $\mathcal{J}\colon L^\infty(0, T) \times \mathbb{R} \times U \to \mathbb{R}$ is a Fr\'{e}chet differentiable function whose partial derivative w.r.t.\ the first argument satisfies $\partial_1 \mathcal{J}(y, y(T), u) \in L^1(0, T) $ for all $(y, u) \in CBV[0,T] \times U$. Here, $L^1(0,T)$ is interpreted as a subset of $L^\infty(0,T)^*$ via the canonical embedding into the bidual. \item $Z$ is an interval of the form $Z = [-r,r]$ with an arbitrary but fixed $r>0$. \item $y_0 \in Z$ is a given and fixed starting value. \end{itemize} \end{assumption}
The above assumptions are always assumed to hold in the following sections, even when not explicitly mentioned. We remark that, to be able to prove the existence of solutions for \eqref{eq:P}, one requires more information about $\mathcal{J}$, $U_{\textup{ad}}$, etc.\ than provided by \cref{ass:standing}; see \cref{cor:solex}. For the derivation of the strong stationarity system \eqref{eq:strongstatsys-2}, however, this is not relevant. An example of a control space $U$ that satisfies the conditions in \cref{ass:standing} and that allows to prove the existence of minimizers for \eqref{eq:P} is the space $H^1(0,T)$, see \cref{sec:5} and the comments therein.
\section{Properties of the scalar stop operator \texorpdfstring{$\boldsymbol{\mathcal{S}}$}{S}} \label{sec:4} In this section, we collect properties of the solution map $\mathcal{S}\colon u \mapsto y$ of the variational inequality \eqref{eq:V} that are needed for our analysis. We begin with fundamental results on the well-definedness, monotonicity, and directional differentiability of $\mathcal{S}$.
\begin{theorem}[well-definedness and Lipschitz continuity] \label{th:Swellposed} The variational inequality \eqref{eq:V} possesses a unique solution $\mathcal{S}(u) := y\in CBV[0, T]$ for all $u \in CBV[0, T]$. For all $u\in W^{1,1}(0, T)$, it holds $y = \mathcal{S}(u) \in W^{1,1}(0, T)$ and \begin{equation}\label{eq;Vabscont} (v - y(t)) (y'(t) - u'(t)) \ge 0 \qquad \forall v\in Z \qquad\text{for a.a.}~t \in (0,T). \end{equation} Further, $\mathcal{S}$ satisfies the Lipschitz estimate \begin{equation} \label{eq;LipschitzSinfty}
\|\mathcal{S}(u_1) - \mathcal{S}(u_2)\|_\infty \leq 2\|u_1 - u_2\|_\infty\qquad \forall u_1, u_2 \in CBV[0, T]. \end{equation} \end{theorem}
\begin{proof} Proofs of the unique solvability of \eqref{eq:V} in $CBV[0, T]$ and of \eqref{eq;Vabscont} can be found in \cite[Theorem 4.1, Proposition 4.1]{Krejci1999}. The Lipschitz estimate \eqref{eq;LipschitzSinfty} follows from \cite[Theorem 7.1]{Krejci1999}; see also \cite[p.\ 49f.]{Krejci1996} and \cite[Proposition 2.3.4]{BrokateSprekels1996}. \end{proof}
\begin{lemma}[general test functions]\label{lemma:gentest} Let $u\in CBV[0,T]$ and $0\le s < \tau\le T$. Then $y := \mathcal{S}(u)$ satisfies \begin{equation}\label{eq:gentest} \int_s^\tau (v-y)\,\mathrm{d}(y-u) \ge 0 \qquad \forall v\in G([s,\tau];Z). \end{equation} \end{lemma}
\begin{proof} Since $y + \mathds{1}_{[s,\tau]}(v-y) \in G([0,T];Z)$ for all $v \in G([s,\tau];Z)$ and due to \cref{lemma:subintervals}, it suffices to consider the case $[s,\tau] = [0,T]$. Let $v\colon[0,T]\to Z$ be a step function of the form \[ v = \sum_{j=1}^N \mathds{1}_{(t_{j-1},t_j)}\zeta_j + \sum_{j=0}^N \mathds{1}_{\{t_j\}}\hat{\zeta}_j \] with $\zeta_j,\hat{\zeta}_j\in Z$ and $0 = t_0 < ... < t_N = T$. Since $v = \lim_{n\to \infty} v_n$ pointwise for suitable $v_n\in C([0,T];Z)$, \eqref{eq:gentest} for $v$ follows from the bounded convergence theorem, \cref{th:boundedconv}. As step functions are dense in $G([0,T];Z)$ by \cite[Theorem 4.1.5]{Monteiro2019}, \eqref{eq:gentest} holds for arbitrary $v\in G([0,T];Z)$, again by the bounded convergence theorem. \end{proof}
\begin{lemma}[piecewise monotonicity]\label{lemma:yumon} Let $u\in CBV[0,T]$ and set $y := \mathcal{S}(u)$. Let $J$ be an open nonempty subinterval of $[0,T]$. \begin{enumerate}[label=\roman*)] \item\label{lemma:yumon:item:i} If $J \subset \{ t\in [0,T]\colon y(t) > -r\}$, then $y-u$ is nonincreasing on $\operatorname{cl}{(J)}$. \item\label{lemma:yumon:item:ii} If $J\subset \{ t\in [0,T] \colon y(t) < r\}$, then $y-u$ is nondecreasing on $\operatorname{cl}{(J)}$. \end{enumerate} \end{lemma}
\begin{proof} We prove \ref{lemma:yumon:item:i}. (The proof of \ref{lemma:yumon:item:ii} is analogous.) Let $s,\tau\in J$ with $s < \tau$. Then $y \ge - r + \varepsilon$ on $[s,\tau]$ for some $\varepsilon > 0$. As $v: = y - \varepsilon \in G([s,\tau];Z)$, we can apply \cref{lemma:gentest} to obtain \[ 0 \le \int_s^\tau (v-y)\,\mathrm{d}(y-u) = - \varepsilon ((y-u)(\tau) - (y-u)(s)). \] Thus, $y-u$ is nonincreasing on $J$, and hence on $\operatorname{cl}{(J)}$ since $y-u$ is continuous. \end{proof}
A proof of the foregoing lemma based on an explicit representation of $y-u$ can be found in \cite[section 5]{Brokate2015}.
\begin{lemma}[comparison principle] \label{lemma:monotonicity} Let $u_1, u_2 \in CBV[0, T]$ be given such that $u_2 - u_1$ is nondecreasing in $[0, T]$. Then it holds $\mathcal{S}(u_2)(t) \geq \mathcal{S}(u_1)(t)$ for all $t \in [0, T]$. \end{lemma}
\begin{proof} First, let us assume that $u_1,u_2\in W^{1,1}(0,T)$. From \eqref{eq;Vabscont}, we obtain that $y_1 := \mathcal{S}(u_1)$ and $y_2 := \mathcal{S}(u_2)$ satisfy \begin{equation} \label{eq:randomeq2736} (v - y_i(t))(y_i'(t)- u_i'(t))\geq 0\qquad \forall v \in Z \qquad \text{for a.a.}~t \in (0, T) \qquad i=1,2. \end{equation} Testing \eqref{eq:randomeq2736} for $i=1$ with $v = y_1(t) - \max\{0, y_1(t) - y_2(t)\} \in Z$ and for $i=2$ with $v = y_2(t) + \max\{0, y_1(t) - y_2(t)\} \in Z$ and adding the resulting inequalities gives \[ \max\{0, y_1 - y_2 \}\cdot (y_1' - y_2')
\leq \max\{0, y_1 - y_2\}\cdot (u_1' - u_2') \leq 0
\quad \text{a.e.\ in $(0,T)$} \] as $u_2 - u_1$ is nondecreasing. By a classical result of Stampacchia, see, for instance, \cite[Lemmas~7.5 and 7.6]{GilbargTrudinger1977}, we have \[ \ddt \frac12 \big(
\max\{0, y_1 - y_2\}\big)^2 =
\max\{0, y_1 - y_2\}\cdot(y_1' - y_2')
\qquad \text{a.e.\ in $(0,T)$.} \] Since $y_2(0) = y_1(0)$, we conclude that $\max\{0, y_1 - y_2\} \le 0$ on $[0,T]$. Thus, $y_2 \ge y_1$ on $[0,T]$ as claimed. In the general case $u_1,u_2\in CBV[0, T]$, we choose piecewise affine interpolants $u_1^n,u_2^n$ of $u_1,u_2$ on partitions $\Delta_n$ of $[0,T]$ whose widths go to zero for $n \to \infty$. Since $u_2^n - u_1^n$ is nondecreasing, too, it follows that $\mathcal{S}(u_2^n) \ge \mathcal{S}(u_1^n)$ on $[0,T]$ for all $n$. As $u_i^n \to u_i$ uniformly, by virtue of \eqref{eq;LipschitzSinfty}, we may pass to the limit, and the claim follows. \end{proof}
\begin{theorem}[pointwise directional differentiability of $\mathcal{S}$] \label{th:dirdiff} The solution operator $\mathcal{S}\colon CBV[0, T] \to CBV[0, T]$ of \eqref{eq:V} is pointwise directionally differentiable in the sense that, for all $u, h \in CBV[0, T]$, there is a unique $\mathcal{S}'(u;h) \in BV[0, T]$ satisfying \[ \lim_{\alpha \to 0^+} \frac{\mathcal{S}(u + \alpha h)(t) - \mathcal{S}(u)(t)}{\alpha} = \mathcal{S}'(u;h)(t)\qquad \forall t \in [0,T]. \] \end{theorem}
\begin{proof} See \cite[Corollary 5.4, Proposition 6.3]{Brokate2015} and also \cite[Theorem 2.1]{Brokate2021}. \end{proof}\pagebreak
Similarly to the classical result \eqref{eq:dirdiffcharobstacleproblem} for the obstacle problem, the derivatives $\mathcal{S}'(u;h)$ in \cref{th:dirdiff} are characterized by an auxiliary variational inequality. To be able to state this inequality, we require some additional notation from \cite{Brokate2021}.
\begin{definition}[inactive, biactive, and strictly active set]\label{def:biactiveetc} Let $u \in CBV[0,T]$ be a control with state $y := \mathcal{S}(u) \in CBV[0,T]$. We introduce: \begin{itemize} \item the inactive set: \[
I(y) := \{t \in [0, T] \colon |y (t)| < r\}, \] \item the biactive set associated with the upper bound of $Z$: \[ B_+(y,u):= \{t \in [0, T] \colon y (t) = r \text{ and } \exists \varepsilon > 0 \text{ s.t. }
y - u = \mathrm{const}\text{ on } [t, t + \varepsilon)\}, \] \item the biactive set associated with the lower bound of $Z$: \[ B_-(y,u) := \{t \in [0, T] \colon y (t) = -r \text{ and } \exists \varepsilon > 0 \text{ s.t. }
y - u = \mathrm{const}\text{ on } [t, t + \varepsilon)\}, \] \item the biactive set: \[ B(y,u) := B_+(y,u) \cup B_-(y,u), \] \item the strictly active set: \[
A(y,u) := \{t \in [0, T) \colon | y (t)| = r \text{ and } \nexists \varepsilon > 0 \text{ s.t. }
y - u = \mathrm{const}\text{ on } [t, t + \varepsilon)\}. \] \end{itemize} Here and in what follows, we use the convention $T \in B_\pm(y,u)$ in the case $y(T) = \pm r$. \end{definition}
\begin{definition}[radial and critical cone mapping] \label{def:ptwcritcone} Given an input function $u \in CBV[0,T]$ with state $y := \mathcal{S}(u) \in CBV[0,T]$, we define: \begin{itemize} \item the set-valued pointwise radial cone mapping: \[ K_\mathrm{rad}^\mathrm{ptw}(y)\colon [0, T] \rightrightarrows \mathbb{R}, \qquad K_\mathrm{rad}^\mathrm{ptw}(y)(t):= \begin{cases}
\mathbb{R} & \text{ if } |y(t)| < r, \\ (-\infty, 0] & \text{ if } y(t) = r, \\
[0, \infty) & \text{ if } y(t) = -r,
\end{cases} \] \item the set-valued pointwise critical cone mapping: \[ K^\mathrm{ptw}_{\mathrm{crit}}(y,u) \colon [0, T] \rightrightarrows \mathbb{R}, \qquad K^\mathrm{ptw}_{\mathrm{crit}}(y,u) (t):= \begin{cases} \mathbb{R} & \text{ if } t \in I(y), \\ (-\infty, 0] & \text{ if } t \in B_+(y,u), \\
[0, \infty) & \text{ if } t \in B_-(y,u),
\\
\{0\} & \text{ if } t \in A(y,u).
\end{cases} \] \end{itemize} \end{definition}
Obviously, \[ K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(t) \subset K_\mathrm{rad}^\mathrm{ptw}(y)(t) \quad\forall t \in [0,T]. \]
Note that a function $z \in C^\infty[0, T]$ satisfying $z(t) \in K_\mathrm{rad}^\mathrm{ptw}(y)(t)$ for all $t \in [0, T]$ is not necessarily an element of the ``global'' radial cone associated with \eqref{eq:V}, i.e., does not necessarily satisfy $y(t) + \alpha z(t) \in Z$ for all $t \in [0, T]$ for a number $\alpha > 0$ independent of $t$. A possible counterexample here is $r=1$, $y_0 = 0$, $T = \pi/2$, $y(t) = u(t) = \sin(t)$, and $z(t) = \sin(2t)$. Indeed, for these $r$, $y_0$, $T$, $y$, $u$, and $z$, we clearly have $y = \mathcal{S}(u)$, $z(T) = 0 \in K_\mathrm{rad}^\mathrm{ptw}(y)(T) = (-\infty, 0]$, and $z(t) \in K_\mathrm{rad}^\mathrm{ptw}(y)(t) = \mathbb{R}$ for all $t \in [0, T)$. Due to the identities $y(T) = 1 = r$, $y'(T) = 0$, and $z'(T) = -2$, it further holds $y'(T) + \alpha z'(T) = - 2 \alpha < 0$ for all $\alpha > 0$. This implies that, for all $\alpha > 0$, there exists $t \in [0,T]$ satisfying $y(t) + \alpha z(t) > r$. We thus have $z(t) \in K_\mathrm{rad}^\mathrm{ptw}(y)(t)$ for all $t \in [0,T]$ but there does not exist $\alpha > 0$ satisfying $y(t) + \alpha z(t) \in Z$ for all $t \in [0, T]$.
\cref{prop:classicalcritical,cor:classicalcritical} below establish a connection between the pointwise critical cone mapping $K^\mathrm{ptw}_{\mathrm{crit}}(y,u)\colon [0,T] \rightrightarrows \mathbb{R}$ and the classical notion of criticality, that is, the property of being an element of the kernel of the multiplier that appears in the variational inequality \eqref{eq:V}, cf.\ the definition of $K_\mathrm{crit}(y,u)$ in \eqref{eq:dirdiffcharobstacleproblem}. As a preparation for \cref{prop:classicalcritical,cor:classicalcritical}, we prove the following lemma.
\begin{lemma}\label{lemma:ortho} Let $u \in CBV[0,T]$ be a control with state $y := \mathcal{S}(u) \in CBV[0,T]$ and let $z\in G[0,T]$ be a function satisfying $z = 0$ on $ A(y,u)$. Then \begin{equation}\label{eq:ortho} \int_s^\tau z\,\mathrm{d} (y-u) = 0 \qquad \forall\;0\le s < \tau\le T. \end{equation} \end{lemma}
\begin{proof} Define $D := \left ( I(y) \cup B(y,u) \right )\setminus \{T\}$. The continuity of $y$ and the definitions of $I(y)$ and $B(y,u)$ imply that, for every $t \in D$, there exists $\varepsilon > 0$ with $[t, t + \varepsilon) \subset D$. This entails that the set $D$ decomposes into disjoint connected components $\{D_i\}_{i \in \mathcal{I}}$ with $\mathcal{I}$ being finite or equal to $\mathbb{N}$ and $D_i$ being an interval with a nonempty interior for all $i \in \mathcal{I}$. Using \cref{lemma:yumon} and again the definition of $B(y,u)$, one easily checks that, for each $t \in D$, there exists $\varepsilon > 0$ such that $y - u$ is constant on $[t, t + \varepsilon)$. Since $y-u$ is continuous, this implies $y-u =: c_i = \mathrm{const}$ on each $[a_i,b_i] := \operatorname{cl}{(D_i)}$. Now $\smash{\int_0^T \mathds{1}_{\{T\}}z\,\mathrm{d}(y-u) = 0}$ by \cref{eq:singlepointmass}. Using this identity, the fact that $z=0$ holds on $ A(y,u)$, \cref{lemma:subintervals}, and (in the case $\mathcal{I} = \mathbb{N}$) the bounded convergence theorem (\cref{th:boundedconv}), we see that \begin{equation}
\label{eq:randomeq2536} \begin{aligned} \int_0^T z\,\mathrm{d} (y-u) &= \int_0^T \mathds{1}_D z\,\mathrm{d} (y-u) = \int_0^T \sum_{i\in \mathcal{I}} \mathds{1}_{D_i} z\,\mathrm{d} (y-u) \\ &= \sum_{i\in \mathcal{I}} \int_0^T \mathds{1}_{D_i} z\,\mathrm{d} (y-u) = \sum_{i\in \mathcal{I}} \int_{a_i}^{b_i} \mathds{1}_{D_i} z\,\mathrm{d} c_i = 0. \end{aligned} \end{equation} Choosing $\mathds{1}_{[s,\tau]}z$ instead of $z$ in \eqref{eq:randomeq2536} yields \eqref{eq:ortho}, again due to \cref{lemma:subintervals}. \end{proof}
\begin{proposition}[relation to the classical notion of criticality] \label{prop:classicalcritical} Suppose that a control $u \in CBV[0,T]$ with state $y := \mathcal{S}(u) \in CBV[0,T]$ and a function $z\in G[0,T]$ satisfying $z(t) \in K_\mathrm{rad}^\mathrm{ptw}(y)(t)$ for all $t \in [0,T]$ are given. Then it holds \begin{equation}\label{eq:classicalcritical:item:ii} \int_s^\tau z\,\mathrm{d} (y-u) \ge 0 \qquad \forall\;0\le s < \tau\le T. \end{equation} Moreover, it is true that
\begin{equation}\label{eq:classicalcritical:item:i} z(t) \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(t)~\forall t \in [0,T] \quad\Rightarrow\quad \int_s^\tau z\,\mathrm{d}(y-u) = 0~~\forall\,0 \leq s < \tau \leq T, \end{equation} and, if $z$ possesses the additional regularity $z\in G_r[0,T]$, then we also have \begin{equation}\label{eq:classicalcritical:item:iii} \int_0^T z\,\mathrm{d}(y-u) = 0 \quad \Rightarrow\quad z(t) \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(t)~\forall t \in [0,T]. \end{equation} \end{proposition}
\begin{proof} In order to prove \eqref{eq:classicalcritical:item:ii}, let $0 \leq s < \tau \leq T$ be given. We first assume that \mbox{$y(t) \in (-r,r]$} holds for all $t \in [s, \tau]$. By \cref{lemma:yumon}, $u - y$ is nondecreasing on $[s, \tau]$. Using the definition of $K_\mathrm{rad}^\mathrm{ptw}(y)$, it is easy to check that $\hat{z}(t) := \max\{0,z(t)\} \mathds{1}_{[s,\tau]}(t)$ satisfies the assumptions of \cref{lemma:ortho}. Therefore, \[ \int_s^\tau z\,\mathrm{d}(y - u) = \int_s^\tau \min\{0,z\} \,\mathrm{d}(y - u) = \int_s^\tau \max\{0,-z\} \,\mathrm{d}(u - y) \geq 0. \] This proves \eqref{eq:classicalcritical:item:ii} in the case $y(t) \in (-r,r]$ for all $t \in [s, \tau]$. In the case $y(t) \in [-r,r)$ for all $t \in [s, \tau]$, we can use the exact same arguments as above with reversed signs to establish \eqref{eq:classicalcritical:item:ii}. To finally obtain \eqref{eq:classicalcritical:item:ii} for arbitrary $[s,\tau]$, it suffices to consider a subdivision of $[s,\tau]$ into subintervals of the above two types and to use \eqref{eq:decomposeInterval}.
The implication \eqref{eq:classicalcritical:item:i} follows directly from \cref{lemma:ortho} since $z(t)\in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(t)$ for all $t \in [0,T]$ implies $z = 0$ on $ A(y,u)$.
It remains to prove \eqref{eq:classicalcritical:item:iii}. Since $z(t)\in K_\mathrm{rad}^\mathrm{ptw}(y)(t) \setminus K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(t)$ for some $t \in [0,T]$ if and only if $z(t) \neq 0$ and $t\in A(y,u)$, it suffices to show that the integral on the left side of \eqref{eq:classicalcritical:item:iii} is nonzero if a time $t$ with the latter property exists. So let $t \in A(y,u)$ be arbitrary but fixed and suppose that $z(t) \neq 0$. We assume w.l.o.g.\ that $y(t) = r$. (The case $y(t) = -r$ is analogous.) From $0 \neq z(t) \in K_\mathrm{rad}^\mathrm{ptw}(y)(t)$, we obtain that $z(t) < 0$ holds, and from the right-continuity of $z$, the definition of $ A(y,u)$, and the continuity of $y$, that $t \neq T$ and that there exist numbers $c, \varepsilon > 0$ such that $z(s) \leq -c $ and $y(s) \in (-r, r]$ holds for all $s \in [t, t + \varepsilon] \subset [0, T]$ and such that $y-u$ is not constant on $[t,t+\varepsilon)$. By \cref{lemma:yumon}, $y-u$ is nonincreasing on $[t, t + \varepsilon]$. It thus follows that \[ \int_{t}^{t + \varepsilon} z \,\mathrm{d}(y-u) \geq c \int_{t}^{t+\varepsilon} \,\mathrm{d}(u-y) = c\left ( (u - y)(t + \varepsilon) - (u - y)(t)\right ) > 0. \] Using \eqref{eq:classicalcritical:item:ii}, we conclude \[ \int_{0}^{T} z \,\mathrm{d}(y-u) = \int_{0}^{t} z \,\mathrm{d}(y-u) + \int_{t}^{t + \varepsilon} z \,\mathrm{d}(y-u) + \int_{t + \varepsilon}^{T} z \,\mathrm{d}(y-u) > 0. \] \end{proof}
\begin{corollary}\label{cor:classicalcritical} Let $u \in CBV[0,T]$ be a control with state $y := \mathcal{S}(u)$ and let $z\in G_r[0,T]$ be a given function. Then \begin{equation*}
z(t) \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(t)~\forall t \in [0,T]\quad\Leftrightarrow\quad
\left \{~~
\begin{aligned} &z(t) \in K^\mathrm{ptw}_{\mathrm{rad}}(y)(t)~\forall t \in [0,T] \text{ and } \\ & \int_s^\tau z\,\mathrm{d}(y-u) = 0~~\forall\,0 \leq s < \tau \leq T. \end{aligned}\right. \end{equation*} \end{corollary}
As \cref{cor:classicalcritical} shows, a function $z \in G_r[0,T]$ is ``critical in the pointwise sense'' if and only if it takes values in $\smash{K_\mathrm{rad}^\mathrm{ptw}(y)(t)}$ for all $t\in [0,T]$ and is contained in the kernel of the linear and continuous function $ \smash{G[0,T] \ni v \mapsto \int_s^\tau v \,\mathrm{d}(y-u) \in \mathbb{R}} $ for all $0 \leq s < \tau \leq T$. For elements of $G_r[0,T]$, the pointwise notion of criticality introduced in \cref{def:ptwcritcone} is thus closely related to the notion of criticality appearing in the context of the classical obstacle problem, cf.\ \eqref{eq:dirdiffcharobstacleproblem}. This relation does not exist anymore in general when the assumption of right-continuity is dropped. Indeed, as the integrator $y-u$ of the integrals in \cref{prop:classicalcritical,cor:classicalcritical} does not assign mass to singletons due to the continuity of $u$ and $y$ and \cref{eq:singlepointmass}, for every $t \in A(y,u)$, the function $z(s) := -\operatorname{sgn}(y(t)) \mathds{1}_{\{t\}}(s)$ satisfies $z \in G[0, T]$, \smash{$z(s) \in K_\mathrm{rad}^\mathrm{ptw}(y)(s)$} for all $s \in [0, T]$, and $\smash{\int_s^\tau z\,\mathrm{d}(y - u) = 0}$ for all $0 \leq s < \tau \leq T$ but does not vanish on the strictly active set $ A(y,u)$. In all situations in which $ A(y,u)$ is nonempty, the pointwise notion of criticality in \cref{def:ptwcritcone} thus differs from the ordinary, multiplier-based one as soon as the regularity of the considered functions is too poor.
We are now in the position to state the auxiliary problem that characterizes the pointwise directional derivatives $\mathcal{S}'(u;h)$ of $\mathcal{S}$ in the situation of \cref{th:dirdiff}.
\begin{theorem}[variational inequality for directional derivatives] \label{th:dirdiffVI} Consider a fixed control $u \in CBV[0, T]$ with associated state $y := \mathcal{S}(u) \in CBV[0, T]$. Then, for \mbox{every} $h \in CBV[0, T]$, the pointwise directional derivative $\delta := \mathcal{S}'(u; h) \in BV[0, T]$ of $\mathcal{S}$ at $u$ in direction $h$ is the unique solution in $BV[0, T]$ of the system \begin{equation} \label{eq:dirdiffVI} \begin{gathered} \int_0^s (z - \delta_+)\,\mathrm{d}(\delta - h) \geq 0\quad \forall z \in G\left ([0, s]; K^\mathrm{ptw}_{\mathrm{crit}}(y,u) \right) \quad \forall s \in (0, T], \\
\delta_+(t) \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(t)~\forall t \in [0, T], \qquad \delta(0) = 0. \end{gathered} \end{equation} Moreover, it holds $\delta(t) \in \{\delta(t+), \delta(t-)\}$ for all $t \in [0, T]$ and $\operatorname{var}(\delta) \leq 2\operatorname{var}(h)$. \end{theorem}
\begin{proof} This follows from \cite[Theorem 2.1]{Brokate2021}, where the result is stated for the scalar play operator $\mathcal{P}(u) := u - \mathcal{S}(u)$. \end{proof}
As $z=0$ and $z = 2\delta_+$ are admissible test functions in \eqref{eq:dirdiffVI}, this variational inequality implies in particular that \begin{equation} \label{eq:dirdiffeq} \int_0^s \delta_+\,\mathrm{d}(\delta - h) = 0 \quad \forall s \in (0, T]. \end{equation}
We remark that, using the inclusion $\delta(t) \in \{\delta(t+), \delta(t-)\}$ and \cite[Lemma 6.3.3]{Monteiro2019}, it is easy to check that the inequality in \eqref{eq:dirdiffVI} is satisfied by $\delta$ regardless of whether the right limit $\delta_+$ in the integral is defined w.r.t.\ $[0, s]$ or w.r.t.\ $[0, T]$. To achieve that $\delta$ is uniquely characterized by \eqref{eq:dirdiffVI}, the definition w.r.t.\ $[0,s]$ and the corresponding convention for the endpoint $s$ have to be used, see \cite[proof of Theorem 2.1]{Brokate2021}.
Regarding the regularity properties of the derivatives $\mathcal{S}'(u;h)$ in \cref{th:dirdiffVI}, it should be noted that $\mathcal{S}'(u;h)$ can satisfy $\mathcal{S}'(u;h)_+ \neq \mathcal{S}'(u;h) \neq \mathcal{S}'(u;h)_-$ even when $u$ and $h$ are smooth, see \cite[Example 4.1]{Brokate2021}. There is, however, a logic behind the jumps of $\mathcal{S}'(u;h)$ as the following corollary shows.
\begin{corollary}[direction of jumps] \label{corollary:dirdiffjumps} Consider the situation in \cref{th:dirdiffVI} for some fixed $u, h \in CBV[0, T]$. Then, for all $t\in [0,T]$, it holds \begin{gather} \label{eq:dirdiffjumps1}
(\delta(t+)-\delta(t-))\zeta \geq 0 \quad \forall \zeta \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u) (t),
\\
\label{eq:dirdiffjumps2} \delta(t+)(\delta(t+)-\delta(t-)) = \delta(t+)(\delta(t+)-\delta(t)) = 0. \end{gather} In particular, if $t \in [0, T]$ is a point of discontinuity of $\delta = \mathcal{S}'(u;h) \in BV[0, T]$, i.e., if $\delta(t+) \neq \delta(t-)$, then it holds $\delta(t+) = 0$. Moreover, we have $\delta(0+) = \delta(0) = 0$. \end{corollary}
\begin{proof} For the test function $z = \mathds{1}_{\{t\}}\zeta$ with $t\in [0,T]$ and $\zeta \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u) (t)$, we obtain from \eqref{eq:dirdiffVI}, using \eqref{eq:dirdiffeq} as well as \cref{eq:singlepointmass}, \[ 0 \leq \int_0^T \mathds{1}_{\{t\}}\zeta\,\mathrm{d}(\delta - h) = \zeta((\delta - h)(t+) - (\delta - h)(t-))= \zeta(\delta(t+) - \delta(t-)) \] with the conventions $\delta(0-) = \delta(0)$ and $\delta(T+) = \delta(T)$. This proves \eqref{eq:dirdiffjumps1}. Using the test functions $z = \delta_+ \pm \mathds{1}_{\{t\}}\delta_+(t)$ in \eqref{eq:dirdiffVI}, we obtain analogously \[ 0 \leq \int_0^T \pm \mathds{1}_{\{t\}}\delta_+(t)\,\mathrm{d}(\delta - h) = \pm \delta(t+)(\delta(t+) - \delta(t-)). \] Since $\delta(t) \in \{\delta(t-),\delta(t+)\}$, both equalities in \eqref{eq:dirdiffjumps2} follow. All other assertions are immediate consequences of \eqref{eq:dirdiffjumps1}, \eqref{eq:dirdiffjumps2}, and the initial condition $\delta(0) = 0$. \end{proof}
We would like to point out that jump conditions similar to those in \cref{corollary:dirdiffjumps} also have to be studied in order to establish the system \eqref{eq:dirdiffVI}, see \cite[section 5]{Brokate2021}. We deduce \cref{corollary:dirdiffjumps} from \cref{th:dirdiffVI} here to simplify the presentation and to avoid recalling major parts of the analysis in \cite{Brokate2021}. As an immediate consequence of \cref{th:dirdiffVI,corollary:dirdiffjumps}, we obtain:
\begin{corollary}[variational inequality for the right limits of the derivatives] \label{cor:dirdiffVI+} Consider an arbitrary but fixed $u \in CBV[0, T]$ with state $y := \mathcal{S}(u) \in CBV[0, T]$. Then, for every $h \in CBV[0, T]$, the right limit $\eta := \mathcal{S}'(u; h)_+ \in BV_r[0, T]$ of the pointwise directional derivative $\mathcal{S}'(u; h)$ of $\mathcal{S}$ at $u$ in direction $h$ is the unique solution in $BV_r[0, T]$ of the variational inequality \begin{equation} \label{eq:dirdiffVI+} \begin{gathered} \int_0^T (z - \eta)\,\mathrm{d}(\eta- h) \geq 0\quad \forall z \in G\left ([0, T]; K^\mathrm{ptw}_{\mathrm{crit}}(y,u) \right), \\ \eta(t) \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(t)~\forall t \in [0, T], \qquad \eta(0) = 0. \end{gathered} \end{equation} Moreover, for all $s\in (0,T]$, it is true that \begin{equation} \label{eq:dirdiffVI+s} \int_0^s (z - \eta)\,\mathrm{d}(\eta- h) \geq 0\quad \forall z \in G\left ([0, s]; K^\mathrm{ptw}_{\mathrm{crit}}(y,u) \right). \end{equation} \end{corollary}
\begin{proof} That $\eta$ satisfies the second line of \eqref{eq:dirdiffVI+} follows from \cref{th:dirdiffVI} and \cref{corollary:dirdiffjumps}. Since $\mathcal{S}'(u;h)\in BV[0,T]$ has at most countably many discontinuity points by \cite[Theorem 2.3.2]{Monteiro2019}, and because $(\eta - \mathcal{S}'(u;h))(T) = 0$ by convention and $(\eta - \mathcal{S}'(u;h))(0) = 0$ by \cref{corollary:dirdiffjumps}, it follows from \cref{lemma:intgzeroexcept} that
\[
\int_0^T f\,\mathrm{d}(\eta- \mathcal{S}'(u;h)) = 0\quad \forall f \in G[0, T].
\] If we combine this identity with \eqref{eq:dirdiffVI} for $s = T$ and the linearity of the Kurzweil-Stieltjes integral, then the variational inequality in \eqref{eq:dirdiffVI+} follows immediately. To establish \eqref{eq:dirdiffVI+s}, it suffices to consider functions of the form $z := \mathds{1}_{[0,s]}\tilde z + \mathds{1}_{(s,T]}\eta$, $s \in (0, T]$, $\smash{\tilde z \in G\left ([0, s]; K^\mathrm{ptw}_{\mathrm{crit}}(y,u) \right )}$, in \eqref{eq:dirdiffVI+} and to exploit \eqref{eq:decomposeInterval} and \eqref{eq:singlepointmass}.
Suppose now that there are two $\eta_1, \eta_2 \in BV_r[0,T]$ satisfying \eqref{eq:dirdiffVI+}. In this case, we can consider functions of the form $z := \mathds{1}_{[0,s]}\eta_2 + \mathds{1}_{(s,T]}\eta_1$ and $z := \mathds{1}_{[0,s]}\eta_1 + \mathds{1}_{(s,T]}\eta_2$ in the inequalities for $\eta_1$ and $\eta_2$, respectively, and add the resulting estimates to obtain with \eqref{eq:decomposeInterval} and \eqref{eq:singlepointmass} that $ \int_0^s (\eta_2 - \eta_1)\,\mathrm{d}(\eta_2 - \eta_1) \le 0 $ holds for all $s\in (0,T]$. Due to \cref{prop:partialIntegration} and $\eta_1(0) = \eta_2(0) = 0$, this yields $(\eta_1(s) - \eta_2(s))^2 \leq 0$ for all $s \in [0,T]$. This proves that \eqref{eq:dirdiffVI+} possesses at most one solution in $BV_r[0,T]$. \end{proof}
Note that the system \eqref{eq:dirdiffVI+} has the same structure as ``usual'' rate-independent systems posed in $BV_r[0,T]$, cf.\ \cite[Theorem 3.3]{Recupero2020}. Because of this, \eqref{eq:dirdiffVI+} is easier to work with than \eqref{eq:dirdiffVI}, which involves the additional varying parameter $s \in (0,T]$.
\section{First consequences for the optimal control problem (P)}\label{sec:5}
As a direct consequence of the results for $\mathcal{S}$ in the last section, we obtain:
\begin{corollary}[existence of solutions] \label{cor:solex} Assume, in addition to the conditions in our standing \cref{ass:standing}, that: \begin{itemize}
\item $(U, \|\cdot\|_U)$ is a reflexive Banach space that is compactly embedded into $C[0,T]$,
\item $U_{\textup{ad}}$ is a closed subset of $(U, \|\cdot\|_U)$, \item $\mathcal{J}$ is lower semicontinuous in the sense that, for all $\{(y_n, z_n, u_n) \} \subset C[0, T] \times \mathbb{R} \times U$ satisfying $y_n \to y$ in $C[0, T]$, $z_n \to z$ in $\mathbb{R}$, and $u_n \rightharpoonup u$ in $U$, we have \[ \liminf_{n \to \infty} \mathcal{J}(y_n, z_n, u_n) \geq \mathcal{J}(y,z,u), \] \item $\mathcal{J}$ is radially unbounded in the sense that there exists a function $\rho\colon [0, \infty) \to \mathbb{R}$ satisfying $\rho(s) \to \infty$ for $s \to \infty$ and \[
\mathcal{J}(y, z, u) \geq \rho\left ( \|u\|_U \right ) \qquad \forall (y, z, u) \in C[0, T] \times \mathbb{R} \times U. \] \end{itemize} Then the problem \eqref{eq:P} possesses at least one globally optimal control-state pair $(\bar u, \bar y)$. \end{corollary}\pagebreak
\begin{proof} This follows straightforwardly from the direct method of the calculus of variations and the Lipschitz continuity property in \eqref{eq;LipschitzSinfty}. \end{proof}
A prototypical example of a space $U$ satisfying the conditions in \cref{cor:solex} is $H^1(0,T)$. We would like to point out that it is, in general, not possible to use the direct method of the calculus of variations in the situation of \cref{cor:solex} if the control space $U$ is not compactly embedded into $C[0,T]$ and if the convergence $u_n \rightharpoonup u$ in $U$ only implies \smash{$u_n \stackrel\star\rightharpoonup u$} in $BV[0,T]$. To see this, suppose that $r = y_0 = 1$, that $T = 2$, and that $\varphi \in C^\infty(\mathbb{R})$ is a function that is identical zero in $\mathbb{R} \setminus (0,2)$, equal to 2 at $t=1$, monotonously increasing in $[0,1]$, and monotonously decreasing in $[1,2]$. For such $r$, $y_0$, $T$, and $\varphi$, it is easy to check that the controls $u_n(t) := \varphi(n t)$, $t \in [0,T]$, $n \in \mathbb{N}$, satisfy $u_n \in C^\infty[0,T]$,
$\|u_n\|_{BV} = \operatorname{var}(u_n) = 4$, and $\mathcal{S}(u_n) = \mathds{1}_{[0, 1/n)} + \mathds{1}_{[1/n, T]} (u_n - 1)$ for all $n$ as well as $u_n(t) \to 0$ for all $t \in [0,T]$ and $n \to \infty$. In particular, we have
$\|\mathcal{S}(u_n)\|_{BV} = 1 + \operatorname{var}(\mathcal{S}(u_n)) = 3$ for all $n$ and $\mathcal{S}(u_n)(t) \to \mathds{1}_{\{0\}}(t) - \mathds{1}_{(0, T]}(t)$ for all $t \in [0,T]$ and $n \to \infty$. In view of \cite[Proposition~3.13]{Ambrosio2000}, this yields \smash{$C^\infty[0,T] \ni u_n \stackrel\star\rightharpoonup 0$} and \smash{$CBV[0,T] \ni \mathcal{S}(u_n) \stackrel\star\rightharpoonup \mathds{1}_{\{0\}} - \mathds{1}_{(0, T]} \neq \mathds{1}_{[0, T]} = \mathcal{S}(0)$} in $BV[0,T]$. The map $\mathcal{S}$ is thus not continuous w.r.t.\ weak-star convergence in $BV[0,T]$ -- even along sequences of smooth functions -- and we may conclude that it is indeed not possible to apply the direct method of the calculus of variations to establish the solvability of \eqref{eq:P} if the space $U$ only provides weak-star convergence in $BV[0,T]$ for minimizing sequences. We remark that the compact embedding $U \hookrightarrow C[0,T]$ needed in \cref{cor:solex} significantly complicates the derivation of the strong stationarity system \eqref{eq:strongstatsys-2} since it makes it impossible to find sequences that converge weakly or strongly in $U$ to the discontinuous directional derivatives $\mathcal{S}'(u;h)$. In fact, this difficulty already arises due to the embedding $U \hookrightarrow C[0,T]$ in \cref{ass:standing}. We will circumvent this problem in \cref{sec:6} by means of a careful analysis of pointwise limits. The next corollary is concerned with the Bouligand stationarity condition that arises from \cref{th:dirdiff}.
\begin{corollary}[Bouligand stationarity condition] Suppose that $\bar u \in U_{\textup{ad}}$ is a locally optimal control of \eqref{eq:P} with associated state $\bar y := \mathcal{S}(\bar u)$. Then it holds \begin{equation} \label{eq:Bouligand} \begin{aligned} \left \langle \partial_1 \mathcal{J}(\bar y, \bar y(T), \bar u), \mathcal{S}'(\bar u; h) \right \rangle_{L^\infty} &+ \partial_2 \mathcal{J}(\bar y, \bar y(T), \bar u)\mathcal{S}'(\bar u; h)(T) \\ &+ \left \langle \partial_3 \mathcal{J}(\bar y, \bar y(T), \bar u), h \right \rangle_{U} \geq 0 \quad \forall h \in \mathbb{R}_+(U_{\textup{ad}} - \bar u). \end{aligned} \end{equation} Here, $ \partial_1 \mathcal{J}(\bar y, \bar y(T), \bar u)\in L^1(0, T)$, $\partial_2 \mathcal{J}(\bar y, \bar y(T), \bar u) \in \mathbb{R}$, and $\partial_3 \mathcal{J}(\bar y, \bar y(T), \bar u) \in U^*$ are the partial Fr\'{e}chet derivatives of the objective function $\mathcal{J}\colon L^\infty(0, T) \times \mathbb{R} \times U \to \mathbb{R}$. \end{corollary}
\begin{proof} This follows along standard lines from the convexity of $U_{\textup{ad}}$, the Fr\'{e}chet differentiability of $\mathcal{J}$,
\cref{th:dirdiff}, the Lipschitz estimate \eqref{eq;LipschitzSinfty}, and the $L^1$-regularity of $ \partial_1 \mathcal{J}(\bar y, \bar y(T), \bar u)$. See, e.g., \cite[Proposition 6.1.2]{ChristofPhd2018} or \cite[section 3]{Herzog2013}. \end{proof}
The last result motivates:
\begin{definition}[Bouligand stationary point] \label{def:Bouligandstationary} A control $\bar u \in U_{\textup{ad}}$ with associated state $\bar y := \mathcal{S}(\bar u)$ is called a Bouligand stationary point of \eqref{eq:P} if $(\bar u,\bar y)$ satisfies \eqref{eq:Bouligand}. \end{definition}
Due to its implicit nature, the Bouligand stationarity condition \eqref{eq:Bouligand} is typically not very helpful in practice. This is one of the main motivations for the derivation of strong stationarity systems. To establish such a system for \eqref{eq:P}, we study:
\section{Temporal polyhedricity properties} \label{sec:6} Throughout this section, we assume that an arbitrary but fixed $u \in CBV[0,T]$ with state $y := \mathcal{S}(u) \in CBV[0,T]$ is given. For these $u$ and $y$, we introduce:
\begin{definition}[reduced critical cone and smooth critical radial directions] \label{def:6.1} We define the reduced critical cone in $G_r[0, T]$ associated with $(y,u)$ to be the set \begin{equation*} \begin{aligned} \mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(y,u) := \big \{ z \in G_r[0, T] \colon &z(t) \in K_{\mathrm{crit}}^{\mathrm{ptw}}(y,u)(t)\,\forall t \in [0,T],~ z(0) = 0, \\ &\text{and } z(t) = 0~\forall t \in [0,T] \text{ with } z(t-) \neq z(t) \big\} \end{aligned} \end{equation*} and the cone of smooth critical radial directions associated with $(y,u)$ to be the set \begin{equation*} \begin{aligned} \mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u) := \big\{ z \in C^\infty[0, T] \colon &z(t) \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(t)~\forall t \in [0,T], \, z(0) = 0, \\ &\text{and } \exists \alpha > 0 \text{ s.t. } y(t) + \alpha z(t) \in Z~\forall t \in [0, T] \big\}. \end{aligned} \end{equation*} \end{definition}
Note that $\smash{\mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u)}$ is a subset of $\smash{\mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(y,u)}$, that both $\smash{\mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u)}$ and $\smash{\mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(y,u)}$ are cones containing the zero function, and that $\smash{\mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u)}$ is convex. The cone \smash{$\mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(y,u)$} is typically not convex due to the additional conditions on the points of discontinuity. From \cref{corollary:dirdiffjumps,cor:dirdiffVI+}, it follows that $\mathcal{S}'(u;h)_+$ is an element of \smash{$\mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(y,u)$} for all $h \in CBV[0,T]$. In fact, $\smash{\mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(y,u)}$ collects all information about the pointwise properties of the right limits of the derivatives $\mathcal{S}'(u;h)$ that we have derived so far. This motivates the name ``reduced critical cone'', cf.\ the analysis for elliptic variational inequalities in \cite{ChristofPhd2018}. From \cref{prop:classicalcritical}, we obtain that \begin{align*} \mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(y,u) &= \Bigg\{ z \in G_r[0, T] \colon
z(t) \in K_{\mathrm{rad}}^{\mathrm{ptw}}(y)(t)\,\forall t \in [0,T], \int_0^T z \,\mathrm{d}(y-u) = 0, \\[-0.1cm] &\hspace{2.8cm}z(0) = 0, z(t) = 0~\forall t \in [0,T] \text{ with } z(t-) \neq z(t) \Bigg\} \end{align*} and \begin{align*} \mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u) &= \Bigg \{ z \in C^\infty[0, T] \colon z(0) = 0, ~ \int_{0}^{T} z \,\mathrm{d}(y-u) = 0, \text{ and} \\[-0.1cm] &\hspace{2.9cm}\exists \alpha > 0 \text{ s.t. } y(t) + \alpha z(t) \in Z~\forall t \in [0, T] \Bigg \}. \end{align*}
The main result of this section -- \cref{theorem:tempoly} -- shows that the cone $\smash{\mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u)}$ is, in a suitably defined sense, dense in $\smash{\mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(y,u)}$. This density property extends the concept of polyhedricity to the setting considered in this paper. In the case of the elliptic problem \eqref{eq:optstaclecontrol}, polyhedricity expresses that the set $ K_{\mathrm{rad}}(y) \cap (u + \Delta y)^\perp$ is $H_0^1(\Omega)$-dense in the critical cone $K_{\tan}(y) \cap (u + \Delta y)^\perp$, see \cite{Haraux1977,Wachsmuth2019}. For the study of the inequality \eqref{eq:V}, the set \smash{$\mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u)$} is relevant because of the following observation.
\begin{lemma}[directional derivative in smooth critical radial directions] \label{lemma:dirdifcritrad} Let $h$ be an arbitrary but fixed element of the set \smash{$\mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u)$}. Then there exists $\alpha > 0$ such that $\mathcal{S}(u + \beta h) = \mathcal{S}(u) + \beta h$ holds for all $\beta \in (0, \alpha)$. In particular, $\mathcal{S}'(u;h) = h$. \end{lemma}
\begin{proof} According to the definition of the set $\mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u) $, we can find a number $\alpha > 0$ such that $y(t) + \alpha h(t) \in Z$ holds for all $t \in [0, T]$. Since $Z$ is convex, this also yields $y(t) + \beta h(t) \in Z$ for all $t \in [0, T]$ and all $\beta \in (0, \alpha)$. From \cref{prop:classicalcritical} and the variational inequality \eqref{eq:V} for $y$, we moreover obtain that \[ \int_0^T h\,\mathrm{d}(y-u) = 0 \qquad \text{and} \qquad \int_0^T (v - y)\,\mathrm{d} (y - u)\geq 0 ~~\forall v \in C([0, T]; Z). \] If we combine the above with the initial conditions $y(0) = y_0$ and $h(0) = 0$ and our previous considerations, then it follows that \begin{equation*} \begin{aligned} & \int_0^T (v - (y + \beta h) )\,\mathrm{d} (y + \beta h - (u + \beta h))\geq 0 &&\forall v \in C([0, T]; Z), \\ &y(t) + \beta h(t) \in Z\quad \forall t \in [0, T], &&y(0)+ \beta h(0) = y_0, \end{aligned} \end{equation*} holds for all $\beta \in (0, \alpha)$. Thus, $\mathcal{S}(u + \beta h) = y + \beta h$ for all $\beta \in (0, \alpha)$ by \cref{th:Swellposed} as claimed. The assertion about the directional derivative follows immediately from this identity. This completes the proof. \end{proof}
Note that \cref{lemma:dirdifcritrad} remains valid when the space $C^\infty[0, T]$ in the definition of \smash{$\mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u)$} is replaced with the space $CBV[0,T]$. We consider smooth critical radial directions in our analysis because this gives rise to a stronger density result in \cref{theorem:tempoly}. As we will see in \cref{sec:7}, \cref{lemma:dirdifcritrad} makes it possible to prove the strong stationarity system \eqref{eq:strongstatsys-2} once the polyhedricity property in \cref{theorem:tempoly} is established. To obtain the latter, we require the following result.
\begin{lemma} \label{lemma:polyprep1} Suppose that $z \in \mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(y,u)$ and $\xi > 0$ are given. Let $t \in [0, T]$ be an arbitrary but fixed point of continuity of $z$, i.e., a point with $z(t) = z(t-)$. Then there exists $\varepsilon > 0$ such that the step function \[ \zeta \colon [0,T] \to \mathbb{R}, \quad \zeta(s) := z(t)\mathds{1}_{J_\varepsilon(t)}(s), \quad J_\varepsilon(t) := [t-\varepsilon,t+\varepsilon]\cap [0,T], \] possesses all of the following properties: \begin{enumerate}[label=\roman*)] \item\label{lemma:polyprep1:i} It is true that \[ \sup_{s \in [t-\varepsilon,t+\varepsilon]\cap [0,T]}
\left | z(s) - \zeta(s)
\right | \leq \xi. \] \item\label{lemma:polyprep1:ii}
It holds \[ \zeta(s) \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s) \qquad \forall s \in [0,T]. \] \item\label{lemma:polyprep1:iii}
For every $0\leq \psi \in C_c^\infty(\mathbb{R})$
with support
$\operatorname{supp}(\psi) \subset (t-\varepsilon, t + \varepsilon)$, the function $\psi \zeta \in G[0,T]$ is an element of the cone $\mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u)$. \end{enumerate} \end{lemma}
\begin{proof} Since $z$ is continuous at $t$, we can find $\varepsilon > 0$ such that \ref{lemma:polyprep1:i} holds. If $z(t) = 0$, then $\zeta = 0$ and \ref{lemma:polyprep1:ii} and \ref{lemma:polyprep1:iii} hold trivially for this $\varepsilon$. Due to the definition of the set \smash{$\mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(y,u)$} and the continuity of $z$ at $t$, this case covers in particular the situations $t = 0$ and $t\in \operatorname{cl}(A(y,u))$. In what follows, we may thus assume that \begin{equation} \label{eq:randomeq3746} z(t) \neq 0 \qquad \text{and}\qquad J_\varepsilon(t) \subset \big ( I(y) \cup B(y,u) \big ) \cap (0, T] \end{equation} and have to prove that, for a potentially smaller $\varepsilon$, we have \ref{lemma:polyprep1:ii} and \ref{lemma:polyprep1:iii}. To this end, we distinguish between three cases.
Case 1: $t\in I(y)$. In this case, it follows from the continuity of $y$ that, after possibly making $\varepsilon$ smaller, we have $J_\varepsilon(t) \subset I(y)$
and $|y| \le r-\gamma$ on $J_\varepsilon(t)$ for some $\gamma > 0$. This implies in particular that $K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s)= \mathbb{R}$ for all $s\in J_\varepsilon(t)$.
Case 2: $t\in B_+(y,u)$. In this case, it follows from the continuity of $y$ that, after possibly making $\varepsilon$ smaller, we have $J_\varepsilon(t) \subset I(y)\cup B_+(y,u)$ and $y \ge -r+\gamma$ on $J_\varepsilon(t)$ for some $\gamma > 0$. Due to the definition of
$K^\mathrm{ptw}_{\mathrm{crit}}(y,u)$, this implies in particular that $z(t) \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(t)= (-\infty,0] \subset K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s)$ for all $s \in J_\varepsilon(t)$.
Case 3: $t\in B_-(y,u)$. In this case, it follows from the continuity of $y$ that, after possibly making $\varepsilon$ smaller, we have $J_\varepsilon(t) \subset I(y)\cup B_-(y,u)$ and $y \le r-\gamma$ on $J_\varepsilon(t)$ for some $\gamma > 0$. Due to the definition of
$K^\mathrm{ptw}_{\mathrm{crit}}(y,u)$, this implies in particular that $z(t) \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(t)= [0, \infty) \subset K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s)$ for all $s \in J_\varepsilon(t)$.
In all of the above cases, the resulting $\varepsilon > 0$ satisfies $z(t) = \zeta(s) \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s)$ and $(y + \alpha \zeta)(s) \in Z$ for all $s\in J_\varepsilon(t)$ and all
$0 < \alpha \leq \gamma \|\zeta\|_\infty^{-1}$. Since $\zeta(s) = 0$ for $s\notin J_\varepsilon(t)$, these inclusions for $\zeta$ are also true for all $s \in [0,T]$. This proves \ref{lemma:polyprep1:ii}. Consider now a function $0 \leq \psi\in C_c^\infty(\mathbb{R})$ with $\operatorname{supp}(\psi)\subset (t-\varepsilon,t+\varepsilon)$. Then $\psi\zeta\in C^\infty[0,T]$ and it follows from the nonnegativity of $\psi$, the properties of $\zeta$, the cone property of $K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s)$, and \eqref{eq:randomeq3746} that $(\psi\zeta)(0) = 0$ holds and that $(\psi\zeta)(s)\in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s)$ and $(y+\alpha\psi\zeta)(s)\in Z$ for all $s\in [0,T]$
and all $0 < \alpha \leq \gamma \|\psi \|_\infty^{-1}\|\zeta\|_\infty^{-1}$. This shows \smash{$\psi \zeta \in \mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u)$}, establishes \ref{lemma:polyprep1:iii}, and completes the proof. \end{proof}
The next lemma is a version of \cref{lemma:polyprep1} for points of discontinuity.
\begin{lemma} \label{lemma:polyprep2} Suppose that $z \in \mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(y,u)$ and $\xi > 0$ are given. Let $t \in [0, T]$ be an arbitrary but fixed point of discontinuity of $z$, i.e., a point with $z(t) \neq z(t-)$. Then there exists $\varepsilon > 0$ such that the step function \[ \zeta \colon [0,T] \to \mathbb{R}, \qquad \zeta (s) := z(t-)\mathds{1}_{J_\varepsilon^-(t)}(s), \quad J_\varepsilon^-(t) := [t-\varepsilon,t) \cap [0,T], \] possesses the following properties: \begin{enumerate}[label=\roman*)] \item\label{lemma:polyprep2:i}
It is true that \[ \sup_{s \in [t-\varepsilon, t + \varepsilon] \cap [0,T]}
\left | z(s) - \zeta(s)
\right | \leq \xi. \] \item\label{lemma:polyprep2:ii} It holds \[ \zeta(s) \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s) \qquad \forall s \in [0,T]. \] \item\label{lemma:polyprep2:iii} For every $0 \leq \psi \in C_c^\infty(\mathbb{R})$ with support $\operatorname{supp}(\psi) \subset (t-\varepsilon, t)$, the function $\psi \zeta \in G[0,T]$ is an element of the cone $\mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u)$. \end{enumerate} \end{lemma}
\begin{proof} Since $z\in\mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(y,u)$, it necessarily holds $t > 0$ and $z(t) = 0$. As $z$ is right-continuous, this implies that there exists $\varepsilon > 0$ such that \ref{lemma:polyprep2:i} is satisfied. Moreover, $z(t-) \neq 0$ because $z$ is assumed to be discontinuous at $t$. Since $z = 0$ on $ A(y,u)$, it follows that, for a potentially smaller $\varepsilon$, we have \begin{equation}\label{eq:polyprep2} J_\varepsilon^-(t)
\subset \big ( I(y) \cup B(y,u) \big ) \cap (0, T]. \end{equation} We now again distinguish between three cases.
Case 1: After possibly making $\varepsilon$ smaller, we have $J_\varepsilon^-(t)\subset I(y)$. In this case, it holds $K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s)= \mathbb{R}$ for all $s\in J_\varepsilon^-(t)$ and it follows from the continuity of $y$ that, for every compact set $E \subset J_\varepsilon^-(t)$, we can find a number $\gamma > 0$ with
$|y| \le r-\gamma$ on $E$.
Case 2: There exists a sequence $\{s_n\}\subset B_+(y,u)$ with $s_n\to t^-$. In this case, we have $y(s_n) = r$ and $z(s_n)\in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s_n)= (-\infty,0]$ for all $n$ and it follows that $y(t) = r$ and $z(t-)\le 0$. Due to the continuity of $y$ and \eqref{eq:polyprep2}, this implies that, after possibly making $\varepsilon$ smaller, we have $J_\varepsilon^-(t) \subset I(y) \cup B_+(y,u)$. In particular, it holds $z(t-) \in (-\infty,0] \subset K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s)$ for all $s\in J_\varepsilon^-(t)$ and, for all compact $E \subset J_\varepsilon^-(t)$, we can find a number $\gamma > 0$ with $y \ge -r + \gamma$ on $E$.
Case 3: There exists a sequence $\{s_n\}\subset B_-(y,u)$ with $s_n\to t^-$. In this case, we have $y(s_n) = -r$ and $z(s_n)\in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s_n)= [0, \infty)$ for all $n$ and it follows that $y(t) = -r$ and $z(t-)\ge 0$. Due to the continuity of $y$ and \eqref{eq:polyprep2}, this implies that, after possibly making $\varepsilon$ smaller, we have $J_\varepsilon^-(t) \subset I(y) \cup B_-(y,u)$. In particular, it holds $z(t-) \in [0, \infty) \subset K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s)$ for all $s\in J_\varepsilon^-(t)$ and, for all compact $E \subset J_\varepsilon^-(t)$, we can find a number $\gamma > 0$ with $y \le r - \gamma$ on $E$.
In all of the above cases, the resulting $\varepsilon > 0$ satisfies $z(t-) = \zeta(s) \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s)$ for all $s\in J_\varepsilon^-(t)$. Since $\zeta(s) = 0$ for $s\notin J_\varepsilon^-(t)$, this proves \ref{lemma:polyprep1:ii}. Moreover, we obtain from the above construction that, for every compact set $E \subset J_\varepsilon^-(t)$, there exists a number $\gamma > 0$ with $(y + \alpha \zeta)(s) \in Z$ for all $s \in E$ and all
$0 < \alpha \leq \gamma \|\zeta\|_\infty^{-1}$. If a function $\psi\in C_c^\infty(\mathbb{R})$ with $\psi\ge 0$ and support $E := \operatorname{supp}(\psi)\subset (t-\varepsilon,t)$ is given, then this implies that $(y+\alpha\psi\zeta)(s)\in Z$ holds for all $s\in [0,T]$ and all
$0 < \alpha \leq \gamma \|\psi\|_\infty^{-1}\|\zeta\|_\infty^{-1}$. Due to the nonnegativity of $\psi$, the properties of $\zeta$, and the cone property of $K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s)$, one further obtains that $(\psi\zeta)(s)\in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s)$ holds for all $s\in [0,T]$, and due to \eqref{eq:polyprep2} and the properties of $\operatorname{supp}(\psi)$, that $(\psi\zeta)(0) = 0$ and $\psi \zeta \in C^\infty[0,T]$. Thus, \smash{$\psi \zeta \in \mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u)$}. This establishes \ref{lemma:polyprep2:iii} and completes the proof. \end{proof}
We can now prove the main result of this section.
\begin{theorem}[temporal polyhedricity] \label{theorem:tempoly} Let $z \in \mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(y,u)$ be given. Then there exist functions $z_{i,j}, z_j \in G_r[0,T]$, $i,j \in \mathbb{N}$, such that the following is true: \begin{equation*} \begin{gathered} z_{i,j} \in \mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u),
\qquad \|z_{i,j}\|_\infty \leq \|z\|_\infty~\forall i,j, \\ z_{j} \in \mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(y,u),
\qquad \|z_{j}\|_\infty \leq \|z\|_\infty~\forall j, \\ z_{i,j} \to z_j \text{ pointwise in } [0,T] \text{ for } i \to \infty \text{ for all }j, \\ z_j \to z \text{ uniformly in } [0,T] \text{ for } j \to \infty. \end{gathered} \end{equation*} \end{theorem}
\begin{proof} Consider an arbitrary but fixed $j \in \mathbb{N}$ and define $\xi := 1/j$. For every $t \in [0,T]$, we choose $\varepsilon_t > 0$ for this $\xi$ as in \cref{lemma:polyprep1,lemma:polyprep2}. This results in a collection of open intervals $(t - \varepsilon_t, t + \varepsilon_t)$ that covers $[0,T]$. By compactness, we can choose a finite subcover of this collection. We denote the time points of this cover with $t_k$, $k=1,...,N$, $N \in \mathbb{N}$, and the associated $\varepsilon_{t_k}$ with $\varepsilon_k$, $k=1,...,N$. We assume w.l.o.g.\ that there are no $k,l$ satisfying $(t_k - \varepsilon_k, t_k + \varepsilon_k) \subset (t_l - \varepsilon_l, t_l + \varepsilon_l)$ and $k \neq l$. In this case, by possibly making the intervals $(t_k - \varepsilon_k, t_k + \varepsilon_k)$ smaller, we can construct intervals $(t_k - a_k, t_k + b_k)$, $a_k, b_k > 0$, such that \begin{equation*} \begin{gathered} (t_k - a_k, t_k + b_k) \subset (t_k - \varepsilon_k, t_k + \varepsilon_k) ~\forall k = 1,...,N, \qquad [0,T] \subset \bigcup_{k=1}^N (t_k - a_k, t_k + b_k), \\ \text{and } t_k \notin (t_l - a_l, t_l+ b_l)~\forall k \neq l. \end{gathered} \end{equation*} Consider now a smooth partition of unity on $[0,T]$ subordinate to the modified cover $(t_k - a_k, t_k + b_k)$, $k=1,...,N$, i.e., a collection of functions $\psi_k$, $k=1,...,N$, satisfying \begin{equation*} \begin{gathered} \psi_k \in C_c^\infty(\mathbb{R}),\quad 0 \leq \psi_k(t) \leq 1~\forall t \in \mathbb{R}, \quad \operatorname{supp}(\psi_k) \subset (t_k - a_k, t_k + b_k) ~\forall k=1,...,N, \\ \sum_{k=1}^N \psi_k(t) = 1~\forall t \in [0,T], \end{gathered} \end{equation*} see, e.g., \cite{Evans2010}, and choose an arbitrary but fixed function $\varphi \in C^\infty(\mathbb{R})$ satisfying \[ 0 \leq \varphi(t) \leq 1~\forall t \in \mathbb{R},\quad \varphi(t) = 1~\forall t \in (-\infty, -1],\quad \varphi(t) = 0~\forall t \in [0,\infty). \] Define \begin{equation*} z_{i,j}(s) := \sum_{k \colon z(t_k)= z(t_k-)} z(t_k) \psi_k(s) + \sum_{k \colon z(t_k) \neq z(t_k-)} z(t_k-) \psi_k(s) \varphi\left ( \frac{s - t_k + 1/i}{1/i} \right ) \end{equation*} for all $i \in \mathbb{N}$ and $s \in [0,T]$. We claim that $z_{i,j} \in \mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u)$ holds for all $i \in \mathbb{N}$. To see this, we first note that we have \[
z(t_k) \psi_k(\cdot)\big |_{[0,T]} \in \mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u)\quad \forall k\colon z(t_k)= z(t_k-) \] by \cref{lemma:polyprep1}\ref{lemma:polyprep1:iii} and the condition $\operatorname{supp}(\psi_k) \subset (t_k - a_k, t_k + b_k) \subset (t_k - \varepsilon_k, t_k + \varepsilon_k)$ for all $k$. Analogously, we also have \[
z(t_k-) \psi_k(\cdot) \varphi\left ( \frac{\cdot - t_k + 1/i}{1/i} \right )\Bigg |_{[0,T]} \in \mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u)\quad \forall k\colon z(t_k)\neq z(t_k-) \] by the properties of $\psi_k$ and $\varphi$ and \cref{lemma:polyprep2}\ref{lemma:polyprep2:iii}. By combining these facts with the observation that $\smash{ \mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u)}$ is a convex cone, the inclusion $z_{i,j} \in \mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u)$ follows immediately. Due to the properties of $\varphi$, we further have \[ z_{i,j}(s) \to \sum_{k \colon z(t_k)= z(t_k-)} z(t_k) \psi_k(s) + \sum_{k \colon z(t_k) \neq z(t_k-)} z(t_k-) \psi_k(s) \mathds{1}_{(-\infty, t_k)}(s) \] for all $s \in [0,T]$ for $i \to \infty$. Let us denote the function on the right of the last limit with $z_j$. By construction, the points of discontinuity of this function $z_j$ are precisely the points $t_k$ with $z(t_k) \neq z(t_k-)$. Further, at these points, the function $z_j$ is clearly right-continuous and, by the choice of the functions $\psi_k$ and the condition $ t_k \notin (t_l - a_l, t_l+ b_l)$ for all $k \neq l$, we have \[ z_j(t_k) = z(t_k-) \psi_k(t_k) \mathds{1}_{(-\infty, t_k)}(t_k) = 0 \] for all $k$ with $z(t_k) \neq z(t_k-)$. In combination with the choice of the functions $\psi_k$, this yields $z_j \in G_r[0,T]$, $z_j(t) = z_j(t+) = 0$ for all $t \in [0,T]$ with $z_j(t) \neq z_j(t-)$, and $z_j(0) = 0$. Due to the properties of $\psi_k$, the inclusion $(t_k - a_k, t_k + b_k) \subset (t_k - \varepsilon_k, t_k + \varepsilon_k)$ for all $k$, the second points of \cref{lemma:polyprep1,lemma:polyprep2}, and the fact that $K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s)$ is a convex cone for all $s \in [0,T]$, we also have $\smash{z_j(s) \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s)}$ for all $s \in [0,T]$. In summary, this allows us to conclude that $z_j \in \mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(y,u)$ holds as desired. It remains to establish the uniform convergence of $z_j$ to $z$ for $j \to \infty$. To this end, we note that, due to the properties of the partition of unity $\{\psi_k\}$, we have \begin{equation*} \begin{aligned}
&\sup_{s \in [0,T]}\left | z(s) - z_j(s)
\right | \\ &=
\sup_{s \in [0,T]}\left | z(s) - \sum_{k \colon z(t_k)= z(t_k-)} z(t_k) \psi_k(s) - \sum_{k \colon z(t_k) \neq z(t_k-)} z(t_k-) \psi_k(s) \mathds{1}_{(-\infty, t_k)}(s)
\right | \\ &=
\sup_{s \in [0,T]}\left | \sum_{k \colon z(t_k)= z(t_k-)} \left (z(s) - z(t_k) \right ) \psi_k(s) \right. \\ &\qquad\qquad\quad \left. + \sum_{k \colon z(t_k) \neq z(t_k-)} \Big( z(s) - z(t_k-) \mathds{1}_{(-\infty, t_k)}(s)\Big )\psi_k(s)
\right | \\ &\leq \sup_{s \in [0,T]} \left ( \sum_{k \colon z(t_k)= z(t_k-)} \sup_{\tau \in [t_k - \varepsilon_k, t_k + \varepsilon_k] \cap [0,T]}
\left | z(\tau) - z(t_k) \right | \psi_k(s) \right. \\ &\qquad\qquad\quad + \left. \sum_{k \colon z(t_k) \neq z(t_k-)}
\sup_{\tau \in [t_k - \varepsilon_k, t_k + \varepsilon_k] \cap [0,T]} \Big| z(\tau) - z(t_k-) \mathds{1}_{(-\infty, t_k)}(\tau)\Big | \psi_k(s) \right ). \end{aligned} \end{equation*} Due to the inequalities in \cref{lemma:polyprep1}\ref{lemma:polyprep1:i} and \cref{lemma:polyprep2}\ref{lemma:polyprep2:i}, our choice $\xi = 1/j$, and the properties of $\psi_k$, the last estimate yields \[
\sup_{s \in [0,T]}\left | z(s) - z_j(s)
\right |
\leq \sup_{s \in [0,T]} \left ( \sum_{k \colon z(t_k)= z(t_k-)} \frac{\psi_k(s)}{j} + \sum_{k \colon z(t_k) \neq z(t_k-)} \frac{\psi_k(s)}{j} \right ) = \frac{1}{j}. \]
This shows that the sequence $\{z_j\}$ indeed converges uniformly to $z$ for $j \to \infty$. That we have $\|z_{i,j}\|_\infty \leq \|z\|_\infty$ and $\|z_{j}\|_\infty \leq \|z\|_\infty$ follows immediately from our construction and the properties of $\psi_k$ and $\varphi$. This completes the proof. \end{proof}
Note that, to be able to establish that
$\smash{\mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u)}$ is dense in $\smash{\mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(y,u)}$, one necessarily has to consider a type of convergence weaker than uniform convergence since otherwise it is not possible to leave the space $C[0,T] \supset \smash{\mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(y,u)}$. This is a major difference between the temporal polyhedricity result in \cref{theorem:tempoly} and the classical notion of polyhedricity for the elliptic obstacle problem in \eqref{eq:optstaclecontrol} which yields the density of the set of critical radial directions $ K_{\mathrm{rad}}(y) \cap (u + \Delta y)^\perp$ in the critical cone $K_{\tan}(y) \cap (u + \Delta y)^\perp$
in $(H_0^1(\Omega), \|\cdot\|_{H_0^1})$ and thus in the topology that is natural for the underlying variational inequality. For \eqref{eq:V}, this natural choice of the topology would be that of uniform convergence as the Lipschitz estimate \eqref{eq;LipschitzSinfty} shows.
Before we apply \cref{theorem:tempoly} to derive strong stationarity conditions for \eqref{eq:P}, we prove a further auxiliary result.
\begin{lemma} \label{lemma:polarconeatoms} Suppose that $t \in [0,T]$ is given and let $c \in \mathbb{R}$ be an element of the polar cone $K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(t)^\circ$, i.e., the set \begin{equation} \label{eq:ptwnormalconedef} K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(t)^\circ := \begin{cases} \{0\} &\text{ if } t \in I(y), \\ [0, \infty) & \text{ if } t \in B_+(y,u), \\ (-\infty, 0]& \text{ if } t \in B_-(y,u), \\ \mathbb{R} &\text{ if } t \in A(y,u). \end{cases} \end{equation} Then there exists a sequence $\{h_i\} \subset C^\infty[0,T]$ such that the following holds: \begin{equation*} \begin{gathered}
\|h_i\|_\infty \leq |c| \text{ and } \|\mathcal{S}'(u;h_i)\|_\infty \leq 2|c|~\forall i \in \mathbb{N}, \\ \mathcal{S}'(u;h_i)_+ \to 0 \text{ pointwise in } [0,T] \text{ for } i \to \infty, \\ h_i \to c \mathds{1}_{[t,T]} \text{ pointwise in } [0,T] \text{ for } i \to \infty. \end{gathered} \end{equation*} \end{lemma}
\begin{proof} If $t \in I(y)$, then we necessarily have $c=0$ and we can simply choose the sequence $h_i = 0$ for all $i$. If $t=0$, then the sequence defined by $h_i = c$ for all $i$ satisfies all assertions because $\mathcal{S}'(u;c\mathds{1}_{[0,T]}) = \mathcal{S}'(u;c\mathds{1}_{[0,T]})_+ = 0$ by \cref{th:dirdiffVI} in view of \cref{eq:integralofconstant}. We may thus assume that \[ 0 < t \in B(y,u)\cup A(y,u). \] Consider an arbitrary but fixed function $\varphi$ with the following properties \[ \varphi \in C^\infty(\mathbb{R}), \quad \varphi(s) = 0 ~\forall s \in (-\infty,-1], \quad \varphi(s) = 1 ~\forall s \in [0, \infty), \quad \varphi'(s) \geq 0~\forall s \in \mathbb{R}. \] We define $\{h_i\}$ via \[ h_i(s) := c \varphi\left ( \frac{s - t}{1/i}\right )\quad \forall s \in [0,T]\quad \forall i \in \mathbb{N}. \] This sequence clearly satisfies $\{h_i\} \subset C^\infty[0,T]$,
$h_i(s) \to c \mathds{1}_{[t,T]}(s)$ for all $s \in [0,T]$ and $i \to \infty$, and $\|h_i\|_\infty = |c|$ for all $i$.
Due to the Lipschitz estimate
\eqref{eq;LipschitzSinfty},
this also implies that $\|\mathcal{S}'(u;h_i) \|_\infty \leq 2|c|$ holds for all $i$.
It remains to establish the pointwise convergence of $ \mathcal{S}'(u;h_i)_+$ to zero. For this to hold, it suffices to prove that $\eta_i := \mathcal{S}'(u;h_i)_+$ satisfies $\eta_i = 0$ on $[0,t-1/i] \cup [t,T]$ for all $i$ with $1/i < t$. That $\eta_i$ vanishes on $[0,t-1/i]$ follows easily from the fact that $h_i$ is zero on $[0, t-1/i]$, \eqref{partialIntegration}, and \eqref{eq:dirdiffVI+s} with $z = 0$, $z = 2 \eta$, and $0 < s \leq t - 1/i$. Next, we prove that $\eta_i(t) = 0$ by distinguishing three cases.
Case 1: $t\in A(y,u)$. In this case, we have $\eta_i(t) \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(t) = \{0\}$.
Case 2: $t\in B_+(y,u)$. In this case, we have $\eta_i(t) \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(t) = (-\infty, 0]$, it holds $c \in [0, \infty)$, and
$h_i$ is nondecreasing on $[0,T]$. By \cref{lemma:monotonicity},
this yields
$\mathcal{S}(u+\alpha h_i) \geq \mathcal{S}(u)$
in $[0, T]$ for all $\alpha > 0$ and all $i \in \mathbb{N}$.
Hence, $\mathcal{S}'(u;h_i) \geq 0$ in $[0,T]$
and, consequently,
$\eta_i = \mathcal{S}'(u;h_i)_+ \geq 0$ in $[0,T]$.
It follows that $\eta_i(t) = 0$.
Case 3: $t\in B_-(y,u)$. In this case, it holds $\eta_i(t) \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(t) = [0, \infty)$ and $c \in (-\infty, 0]$, and we can proceed completely analogously to Case 2 (with reversed signs) to obtain that $\eta_i(t) = 0$.
It remains to prove that $\eta_i = 0$ on $(t,T]$ if $t < T$. Let $\hat{\eta}_i := \mathds{1}_{[0,t]}\eta_i = \mathds{1}_{[0,t)}\eta_i$. By the definition of $h_i$, the function $\hat{\eta}_i - h_i$ has the constant value $-c$ on $[t,T]$. Using \cref{eq:decomposeInterval} combined with \cref{eq:integralofconstant}, we obtain that, for all $ z \in G\left ([0, T]; K^\mathrm{ptw}_{\mathrm{crit}}(y,u) \right)$, we have \begin{align*} \int_0^T (z - \hat{\eta}_i)\,\mathrm{d}(\hat{\eta}_i - h_i) &= \int_0^t (z - \hat{\eta}_i)\,\mathrm{d}(\hat{\eta}_i - h_i) + \int_t^T (z - \hat{\eta}_i)\,\mathrm{d}(\hat{\eta}_i - h_i) \\ &= \int_0^t (z - \hat{\eta}_i)\,\mathrm{d}(\hat{\eta}_i - h_i) \\ &= \int_0^t (z - \eta_i)\,\mathrm{d}(\eta_i - h_i) \ge 0, \end{align*} where the last inequality holds by \cref{cor:dirdiffVI+}. Since $\hat{\eta}_i(s) \in K^\mathrm{ptw}_{\mathrm{crit}}(y,u)(s)$ for all $s\in [0,T]$, we conclude that $\hat{\eta}_i$ solves \cref{eq:dirdiffVI+} for $h = h_i$. As $\eta_i$ is the unique solution of \cref{eq:dirdiffVI+}, we must have $\hat{\eta}_i = \eta_i$. Thus, $\eta_i = 0$ on $(t,T]$ and the proof is complete. \end{proof}
\section{Strong stationarity condition}\label{sec:7}
We are now in the position to prove the strong stationarity system \eqref{eq:strongstatsys-2}.
\begin{theorem}[strong stationarity] \label{th:main} Consider the situation in \cref{ass:standing} and suppose that $\bar u \in U_{\textup{ad}}$ is a control with state $\bar y := \mathcal{S}(\bar u)$ such that the set $\mathbb{R}_+(U_{\textup{ad}} - \bar u)$ is dense in $U$. Then $\bar u$ is a Bouligand stationary point of \eqref{eq:P}, i.e., satisfies \eqref{eq:Bouligand} if and only if there exist an adjoint state $\bar p \in BV[0,T]$ and a multiplier $\bar \mu \in G_r[0,T]^*$ such that the following system is satisfied: \begin{equation} \label{eq:strongstatsys} \begin{gathered} \bar p(0) = \bar p(T) = 0, \quad \bar p(t) = \bar p(t-)~\forall t \in [0,T), \\ \bar p(t-) \in K^\mathrm{ptw}_{\mathrm{crit}}(\bar y, \bar u)(t)~\forall t \in [0,T], \\ \left \langle \bar \mu, z \right \rangle_{G_r} \geq 0\quad \forall z \in \mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(\bar y, \bar u) , \\ \int_0^T h \,\mathrm{d} \bar p = \left \langle \partial_3 \mathcal{J}(\bar y, \bar y(T), \bar u), h \right \rangle_{U} ~\forall h \in U, \\ -\int_0^T z \,\mathrm{d} \bar p = \left \langle \partial_1 \mathcal{J}(\bar y, \bar y(T), \bar u), z\right \rangle_{L^\infty} + \partial_2 \mathcal{J}(\bar y, \bar y(T), \bar u)z(T) - \left \langle \bar \mu, z \right \rangle_{G_r} \\ \hspace{8.5cm}\forall z \in G_r[0,T]. \end{gathered} \end{equation} \end{theorem}
\begin{proof} We begin with the proof of the implication ``\eqref{eq:Bouligand} $\Rightarrow$ \eqref{eq:strongstatsys}'': Suppose that a control $\bar u \in U_{\textup{ad}}$ with state $\bar y := \mathcal{S}(\bar u)$ is given such that the set $\mathbb{R}_+(U_{\textup{ad}} - \bar u)$ is dense in $U$ and such that \eqref{eq:Bouligand} holds. Then it follows from \eqref{eq:Bouligand}, the fact that \eqref{eq;LipschitzSinfty} implies that $
\|\mathcal{S}'(\bar u;h_1) - \mathcal{S}'(\bar u; h_2)\|_\infty \leq 2\|h_1 - h_2\|_\infty $ holds for all $h_1, h_2 \in CBV[0,T]$, the inclusion $U \subset CBV[0,T]$, and the continuity of the embedding $U \hookrightarrow C[0,T]$ that \begin{equation} \label{eq:Bouligandstat2} \begin{aligned} \left \langle \partial_1 \mathcal{J}(\bar y, \bar y(T), \bar u), \mathcal{S}'(\bar u; h) \right \rangle_{L^\infty} &+ \partial_2 \mathcal{J}(\bar y, \bar y(T), \bar u)\mathcal{S}'(\bar u; h)(T) \\ &+ \left \langle \partial_3 \mathcal{J}(\bar y, \bar y(T), \bar u), h \right \rangle_{U} \geq 0 \qquad \forall h \in U. \end{aligned} \end{equation} Again due to \eqref{eq;LipschitzSinfty} and since $-h \in U$ holds for all $h \in U$, \eqref{eq:Bouligandstat2} yields \[
\left | \left \langle \partial_3 \mathcal{J}(\bar y, \bar y(T), \bar u), h \right \rangle_{U} \right |
\leq 2 \left ( \left \| \partial_1 \mathcal{J}(\bar y, \bar y(T), \bar u)\right \|_{L^1}
+ \left | \partial_2 \mathcal{J}(\bar y, \bar y(T), \bar u) \right |
\right )\|h\|_{\infty} \] for all $h \in U$. In combination with the Hahn-Banach theorem, this shows that the linear functional $U \ni h \mapsto \left \langle \partial_3 \mathcal{J}(\bar y, \bar y(T), \bar u), h \right \rangle_{U} \in \mathbb{R}$ can be extended to an element of the dual space $C[0,T]^*$. In view of the classical Riesz representation theorem (see, e.g., \cite[section 8.1]{Monteiro2019}) and \cref{lemma:intgzeroexcept}, this means that there exists a function $\bar p \in BV[0,T]$ satisfying $\bar p(t) = \bar p(t-)$ for all $t \in (0,T)$, $\bar p(T) = 0$, and \[ \int_0^T h \,\mathrm{d} \bar p = \left \langle \partial_3 \mathcal{J}(\bar y, \bar y(T), \bar u), h \right \rangle_{U} ~\forall h \in U. \] Since $\partial_1 \mathcal{J}(\bar y, \bar y(T), \bar u) \in L^1(0,T)$, $\mathcal{S}'(\bar u; h) = \mathcal{S}'(\bar u; h)_+$ a.e.\ in $(0,T)$ by \cite[Theorem 2.3.2]{Monteiro2019}, and $\mathcal{S}'(\bar u; h)(T) = \mathcal{S}'(\bar u; h)(T+) = \mathcal{S}'(\bar u; h)_+(T)$ by definition, we may now rewrite \eqref{eq:Bouligandstat2} as follows: \begin{equation} \label{eq:randomeq263536} \begin{aligned} &\int_0^T h \,\mathrm{d} \bar p + \left \langle \partial_1 \mathcal{J}(\bar y, \bar y(T), \bar u), \mathcal{S}'(\bar u; h)_+ \right \rangle_{L^\infty} + \partial_2 \mathcal{J}(\bar y, \bar y(T), \bar u)\mathcal{S}'(\bar u; h)_+(T) \geq 0 \\ &\hspace{10.5cm} \forall h \in U. \end{aligned} \end{equation} Note that, again due to the Lipschitz estimate
$\|\mathcal{S}'(\bar u;h_1) - \mathcal{S}'(\bar u; h_2)\|_\infty \leq 2\|h_1 - h_2\|_\infty$ for $h_1, h_2 \in CBV[0,T]$ and since $U$ is dense in $C[0,T]$, \eqref{eq:randomeq263536} remains valid when the test space $U$ is replaced by $CBV[0,T]$. We define $\bar \mu \in G_r[0,T]^*$ via \[ \left \langle \bar \mu, z \right \rangle_{G_r} := \int_0^T z \,\mathrm{d} \bar p + \left \langle \partial_1 \mathcal{J}(\bar y, \bar y(T), \bar u), z \right \rangle_{L^\infty} + \partial_2 \mathcal{J}(\bar y, \bar y(T), \bar u)z(T) \quad \forall z \in G_r[0,T]. \] Then the last line in \eqref{eq:strongstatsys} holds, and it follows from \eqref{eq:randomeq263536} with test space $CBV[0,T]$ and \cref{lemma:dirdifcritrad} that \[ \int_0^T z \,\mathrm{d} \bar p + \left \langle \partial_1 \mathcal{J}(\bar y, \bar y(T), \bar u), z \right \rangle_{L^\infty} + \partial_2 \mathcal{J}(\bar y, \bar y(T), \bar u)z(T) \geq 0 \quad \forall z \in \mathcal{K}_{C^\infty}^{\mathrm{rad},\mathrm{crit}}(\bar y, \bar u). \] Due to \cref{theorem:tempoly} and the bounded convergence theorem (\cref{th:boundedconv}), we can extend the last inequality to the set $\mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(\bar y, \bar u)$ by approximation, i.e., we have \[ \int_0^T z \,\mathrm{d} \bar p + \left \langle \partial_1 \mathcal{J}(\bar y, \bar y(T), \bar u), z \right \rangle_{L^\infty} + \partial_2 \mathcal{J}(\bar y, \bar y(T), \bar u)z(T) = \left \langle \bar \mu, z \right \rangle_{G_r} \geq 0 \] for all $z \in \mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(\bar y, \bar u)$. This proves the third line in \cref{eq:strongstatsys}. It remains to establish the pointwise properties of $\bar p$ in \cref{eq:strongstatsys}. To this end, we again use that \cref{cor:dirdiffVI+} and \cref{eq:integralofconstant} imply that $\mathcal{S}'(\bar u; c\mathds{1}_{[0,T]})_+ = 0$ holds for all $c \in \mathbb{R}$. By \eqref{eq:randomeq263536} with test space $CBV[0,T]$, this yields \begin{equation*} \begin{aligned} 0 \leq c\int_0^T \,\mathrm{d} \bar p = c\left ( \bar p(T) - \bar p(0)\right )\quad \forall c \in \mathbb{R}. \end{aligned} \end{equation*} Thus, $\bar p(0) = \bar p(T)$. Since $\bar p(T) = 0$ and $\bar p(t) = \bar p(t-)$ for all $t \in (0,T)$, and since $\bar p(0) = \bar p(0-)$ holds by definition, this establishes the first line of \cref{eq:strongstatsys}. Next, by invoking \cref{lemma:polarconeatoms}, by setting $h = h_i$ in \eqref{eq:randomeq263536} with test space $CBV[0,T]$, and by passing to the limit $i\to\infty$ by means of \cref{th:boundedconv} and the dominated convergence theorem, we obtain that, for every $t \in [0,T]$ and every $c \in K^\mathrm{ptw}_{\mathrm{crit}}(\bar y,\bar u)(t)^\circ$, we have \begin{equation} \label{eq:randomeq2635} 0 \leq \int_0^T c \mathds{1}_{[t,T]} \,\mathrm{d} \bar p = c \left ( \bar p(T) - \bar p(t-) \right ) = - c \bar p(t-). \end{equation} Here, the last two equations follow from \cite[Lemma 6.3.3]{Monteiro2019} and the identity $\bar p(T) = 0$. By using the definition \eqref{eq:ptwnormalconedef} of the polar cone $K^\mathrm{ptw}_{\mathrm{crit}}(\bar y, \bar u)(t)^\circ$ in \eqref{eq:randomeq2635}, one readily obtains that $\bar p(t-) \in K^\mathrm{ptw}_{\mathrm{crit}}(\bar y, \bar u) (t)$ holds for all $t \in [0,T]$. This establishes the second line in \eqref{eq:strongstatsys} and proves, in combination with the previous steps, that the strong stationarity system \eqref{eq:strongstatsys} is indeed a necessary condition for Bouligand stationarity.
Next, we prove the implication ``\eqref{eq:strongstatsys} $\Rightarrow$ \eqref{eq:Bouligand}''. Suppose that $\bar u \in U_{\textup{ad}}$ is a control with state $\bar y := \mathcal{S}(\bar u)$ such that there exist $\bar p \in BV[0,T]$ and $\bar \mu \in G_r[0,T]^*$ satisfying \eqref{eq:strongstatsys}. Assume further that a direction $h \in U$ is given and define $\eta := \mathcal{S}'(\bar u;h)_+$. Then it follows from the properties of $\bar p$, \eqref{eq:dirdiffVI+} with $z := \eta + \bar p$, and the integration by parts formula for the Kurzweil-Stieltjes integral \cite[Theorem 6.4.2]{Monteiro2019} that \begin{equation*} \begin{aligned} 0 \leq \int_0^T \bar p\,\mathrm{d}(\eta- h) &= \int_0^T (h - \eta) \,\mathrm{d} \bar p + \bar p(T)(\eta- h)(T) - \bar p(0)(\eta- h)(0) \\ &\qquad + \sum_{t \in [0,T]} \left ( \bar p(t) - \bar p(t-)\right ) \left ( (\eta- h)(t) - (\eta- h)(t-) \right ) \\ &\qquad - \sum_{t \in [0,T]} \left ( \bar p(t) - \bar p(t+)\right ) \left ( (\eta- h)(t) - (\eta- h)(t+) \right ). \end{aligned} \end{equation*} Due to the identities $\bar p(0) = \bar p(T) = 0$ and $\eta(0) = \eta(0-) = 0$ and due to the left- and right-continuity properties of $\bar p$, $h$, and $\eta = \mathcal{S}'(\bar u;h)_+$, the last estimate simplifies to \[ 0 \leq \int_0^T (h - \eta) \,\mathrm{d} \bar p - \bar p(T-) \left ( \eta(T) - \eta(T-) \right ). \] Note that \eqref{eq:dirdiffjumps1}, $\bar p(T-) \in K^\mathrm{ptw}_{\mathrm{crit}}(\bar y,\bar u) (T)$, and the convention $\eta(T) = \eta(T+)$ imply that $\bar p(T-) ( \eta(T) - \eta(T-) ) = \bar p(T-) ( \eta(T+) - \eta(T-) ) \geq 0$ holds. We thus obtain \[ 0 \leq \int_0^T (h - \eta) \,\mathrm{d} \bar p = \int_0^T h \,\mathrm{d} \bar p - \int_0^T \eta \,\mathrm{d} \bar p, \] and, by the last three lines of \cref{eq:strongstatsys} and the properties of $\eta$, \begin{equation*} \begin{aligned} 0 &\leq \left \langle \partial_3 \mathcal{J}(\bar y, \bar y(T), \bar u), h \right \rangle_{U} + \left \langle \partial_1 \mathcal{J}(\bar y, \bar y(T), \bar u), \eta \right \rangle_{L^\infty} + \partial_2 \mathcal{J}(\bar y, \bar y(T), \bar u)\eta(T) - \left \langle \bar \mu, \eta \right \rangle_{G_r} \\ &\leq \left \langle \partial_3 \mathcal{J}(\bar y, \bar y(T), \bar u), h \right \rangle_{U} + \left \langle \partial_1 \mathcal{J}(\bar y, \bar y(T), \bar u), \eta\right \rangle_{L^\infty} + \partial_2 \mathcal{J}(\bar y, \bar y(T), \bar u)\eta(T). \end{aligned} \end{equation*} If we now exploit that $\partial_1 \mathcal{J}(\bar y, \bar y(T), \bar u) \in L^1(0,T)$, that $\mathcal{S}'(\bar u; h)(T) = \mathcal{S}'(\bar u; h)(T+)$, and that $\eta = \mathcal{S}'(\bar u;h)$ a.e., then \eqref{eq:Bouligand} follows. This completes the proof. \end{proof}
Note that, in the case $T \in I(\bar y)$, there exists $m > 0$ such that the function $z(t) := c \mathds{1}_{[T -\varepsilon, T]}(t) (t - T + \varepsilon)/\varepsilon$ is an element of \smash{$\mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(\bar y, \bar u)$} for all $c \in \mathbb{R}$ and all $0 < \varepsilon < m$. For such a function $z$, the third line of \eqref{eq:strongstatsys} becomes $\left \langle \bar \mu, z \right \rangle_{G_r} = 0$. Using this in the fifth line of \eqref{eq:strongstatsys} and subsequently passing to the limit $\varepsilon \to 0^+$ by means of \cref{th:boundedconv} yields, due to the $L^1$-regularity of $\partial_1 \mathcal{J}(\bar y, \bar y(T), \bar u)$ and \cref{eq:singlepointmass}, that \[ -c \int_0^T \mathds{1}_{\{T\}} \,\mathrm{d} \bar p = -c (\bar p(T) - \bar p(T-)) = \partial_2 \mathcal{J}(\bar y, \bar y(T), \bar u)c\qquad \forall c \in \mathbb{R}. \] Thus, $\bar p(T-) - \bar p(T) = \partial_2 \mathcal{J}(\bar y, \bar y(T), \bar u)$ and we obtain that the partial derivative $\partial_2 \mathcal{J}(\bar y, \bar y(T), \bar u) \in \mathbb{R}$ affects the jump of $\bar p$ at $T$, as mentioned in \cref{sec:1}. We remark that, by redefining $\bar p$, this implicit jump condition on the adjoint state in \eqref{eq:strongstatsys} can also be transformed into a condition on the function value at $T$. Indeed, by introducing the modified adjoint state $\bar q := \bar p + \partial_2 \mathcal{J}(\bar y, \bar y(T), \bar u)\mathds{1}_{\{T\}} \in BV[0,T]$, by using the integration by parts formula in \cite[Theorem 6.4.2]{Monteiro2019} in the fourth line of \eqref{eq:strongstatsys}, and by employing \eqref{eq:singlepointmass} and \cite[Lemma 6.3.2]{Monteiro2019}, one easily checks that the strong stationarity system in \cref{th:main} can also be formulated as follows: \begin{equation*} \begin{gathered} \bar q(0) = 0, \quad \bar q(T) = \partial_2 \mathcal{J}(\bar y, \bar y(T), \bar u), \quad \bar q(t) = \bar q(t-)~\forall t \in [0,T), \\ \bar q(t-) \in K^\mathrm{ptw}_{\mathrm{crit}}(\bar y, \bar u)(t)~\forall t \in [0,T], \\ \left \langle \bar \mu, z \right \rangle_{G_r} \geq 0\quad \forall z \in \mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(\bar y, \bar u) , \\ - \int_0^T \bar q \,\mathrm{d} h = \left \langle \partial_3 \mathcal{J}(\bar y, \bar y(T), \bar u), h \right \rangle_{U} ~\forall h \in U, \\ -\int_0^T z \,\mathrm{d} \bar q = \left \langle \partial_1 \mathcal{J}(\bar y, \bar y(T), \bar u), z\right \rangle_{L^\infty} - \left \langle \bar \mu, z \right \rangle_{G_r}\quad \forall z \in G_r[0,T]. \end{gathered} \end{equation*}
Regarding the assumption that the set $\mathbb{R}_+(U_{\textup{ad}} - \bar u)$ is dense in $U$, we would like to point out that this so-called ``ample control'' condition in \cref{th:main} is rather restrictive and rarely satisfied if $U_{\textup{ad}} \neq U$. Using techniques from \cite{Wachmuth2014}, it might be possible to establish a strong stationarity system for \eqref{eq:P} also under weaker assumptions on the control constraints. We leave this topic for future research.
\appendix \section{Results on the Kurzweil-Stieltjes integral}\label{sec:appendix} Let $a,b \in \mathbb{R}$ with $a < b$ be given. For $f, g \in G[a,b]$, the Kurzweil-Stieltjes integral with \emph{integrand} $f$ and \emph{integrator} $g$ exists if at least one of the functions $f$ and $g$ has bounded variation, see \cite[Theorem 6.3.11]{Monteiro2019}. In this case, it yields a real number which we denote by \[ \int_a^b f\,\mathrm{d} g \qquad \text{or} \qquad \int_a^b f(t)\,\mathrm{d} g(t). \] The Kurzweil-Stieltjes integral coincides with the Riemann-Stieltjes integral whenever the latter exists, see \cite[Theorem 6.2.12]{Monteiro2019}. This holds in particular if $f\in C[a,b]$ and $g\in BV[a,b]$, see \cite[Theorem 5.6.1]{Monteiro2019}. If $c\in\mathbb{R}$ is interpreted as a constant function, then it holds \begin{equation}\label{eq:integralofconstant} \int_a^b c\,\mathrm{d} g = c(g(b) - g(a))
\qquad \text{and} \qquad
\int_a^b f\,\mathrm{d} c = 0 \end{equation} for all $f, g \in G[a,b]$, see \cite[Remark 6.3.1]{Monteiro2019}.
The Kurzweil-Stieltjes integral is linear w.r.t.\ the integrand $f$ and w.r.t.\ the integrator $g$, see \cite[Theorem 6.2.7]{Monteiro2019}. Further, for all $c\in (a,b)$, it holds \begin{equation}\label{eq:decomposeInterval} \int_a^b f\,\mathrm{d} g = \int_a^c f\,\mathrm{d} g + \int_c^b f\,\mathrm{d} g \end{equation} provided the first integral exists, see \cite[Theorems 6.2.9, 6.2.10]{Monteiro2019}. For $t\in [a,b]$ and $g \in G[a,b]$, we have (see \cite[Lemma 6.3.3]{Monteiro2019}) \begin{equation}\label{eq:singlepointmass} \int_a^b \mathds{1}_{\{t\}}\,\mathrm{d} g = g(t+) - g(t-) \end{equation} with the conventions $g(b+) := g(b)$ and $g(a-) := g(a)$. In particular, the integral in \eqref{eq:singlepointmass} equals zero if $g$ is continuous at $t$.
\begin{lemma}\label{lemma:subintervals} Let $f\in G[a,b]$, $g\in BV_r[a,b]$, $a\le s < \tau\le b$, and $J := (s,\tau]$. Then \begin{equation}\label{eq:subintervals} \int_s^\tau f\,\mathrm{d} g = \int_a^b \mathds{1}_J f\,\mathrm{d} g. \end{equation} If $g\in CBV[a,b]$, then \cref{eq:subintervals} is also true for $J = [s,\tau]$, $J = (s,\tau)$, and $J = [s,\tau)$. \end{lemma}
\begin{proof} This is a special case of \cite[Theorem 6.9.7]{Monteiro2019}. \end{proof}
\begin{lemma}\label{lemma:intgzeroexcept} Let $g\in BV[a,b]$ be given such that $g(a) = g(b) = 0$ holds and such that the set $\{t\in [a,b]\colon g(t)\neq 0\}$ is finite or countably infinite. Then \[ \int_a^b f\,\mathrm{d} g = 0 \qquad \forall f\in G[a,b]. \] \end{lemma}
\begin{proof} This is a special case of \cite[Lemma 6.3.15]{Monteiro2019}. \end{proof}
\begin{proposition}\label{prop:partialIntegration} Let $g\in BV_r[a,b]$. Then \begin{equation}\label{partialIntegration} \int_a^b g\,\mathrm{d} g = \frac{1}{2}(g(b)^2 - g(a)^2) + \frac{1}{2} \sum_{t\in [a,b]} (g(t) - g(t-))^2. \end{equation} \end{proposition}
\begin{proof} This is a special case of \cite[Corollary 2.12]{Krejci2003} or of \cite[Corollary 1.13]{KrejciLiero2009}. \end{proof}
\begin{theorem}[bounded convergence theorem]\label{th:boundedconv}
Let $g\in BV[a,b]$, $f_n\in G[a,b]$ with $\sup_n \|f_n\|_\infty < \infty$ and $f_n\to f$ pointwise in $[a,b]$ be given. Then the integral $ \int_a^b f\,\mathrm{d} g $ exists and it holds \[ \lim_{n\to\infty} \int_a^b f_n\,\mathrm{d} g = \int_a^b f\,\mathrm{d} g. \] \end{theorem}
\begin{proof} This is a special case of \cite[Theorem 6.8.13]{Monteiro2019}. \end{proof}
\iffalse
\section*{TEMPORARY} ~\\
Roman Cap: \begin{itemize}
\item $ A(y,u)$ strictly active set
\item $BV[0,T]$
\item $B(y,u)$ biactive set
\item $B_+(y,u)$ biactive set
\item $B_-(y,u)$ biactive set
\item $CBV[0,T]$
\item $C([0,T];Z)$
\item $D$ auxiliary set
\item $E$ compact set
\item $G[0,T]$
\item $G_r[0,T]$
\item $G([0,T];Z)$
\item $H_0^1(\Omega)$
\item $H^{-1}(\Omega)$
\item $I(y)$ inactive set
\item $\mathcal{I}$ index set
\item $J$ interval
\item $\mathcal{J}$ objective
\item $K$ set-valued function
\item $K_\mathrm{crit}(y,u)$ critical cone, obstacle problem
\item $K^\mathrm{ptw}_{\mathrm{crit}}(\bar y, \bar u)(t)$ ptw critical cone
\item $K_\mathrm{rad}^\mathrm{ptw}(y)(t)$ ptw radial cone
\item $K_{\tan}(y)$ tangent cone, obstacle problem
\item $K_{\mathrm{rad}}(y)$ radial cone, obstacle problem
\item $\mathcal{K}_{G_r}^{\mathrm{red},\mathrm{crit}}(\bar y, \bar u)$ reduced critical cone
\item $L^2(\Omega)$
\item $N$ natural number
\item $\mathbb{N}$ natural numbers
\item $\mathbb{R}$ reals
\item $\mathbb{R}_+$
\item $S$ control-to-state operator, obstacle problem
\item $\mathcal{S}$ scalar stop operator
\item $T$ final time
\item $U$ control space
\item $U_{\textup{ad}}$ set of admissible controls
\item $X$ normed space
\item $Z$ VI-set \end{itemize}
Roman Small: \begin{itemize}
\item $a_i$ interval bound
\item $b_i$ interval bound
\item $c$ constant
\item $d$ spatial dimension
\item $f$ function in Kurzweil-Stieltjes integral
\item $g$ function in Kurzweil-Stieltjes integral
\item $h$ perturbation of control
\item $i$ index of sequence
\item $j$ index of sequence
\item $k$ index of sequence
\item $l$ index of sequence
\item $m$ constant
\item $n$ index of sequence
\item $\bar p$ adjoint state
\item $p$ integrability exponent
\item $r>0$ radius in $Z$
\item $s$ time variable
\item $t$ time variable
\item $u$ control
\item $v$ test function, original VI
\item $y$ state
\item $y_0$ initial value
\item $z$ test function in auxiliary VIs \end{itemize}
Greek: \begin{itemize}
\item $\alpha$ width of difference quotient
\item $\beta$ auxiliary parameter
\item $\gamma$ auxiliary parameter
\item $\delta$ directional derivative
\item $\eta$ right limit of directional derivative
\item $\varepsilon$ small positive number
\item $\psi$ bound in obstacle problem
\item $\rho$ lower bound function
\item $\bar \mu$ multiplier
\item $\xi$ positive number
\item $\zeta$ step function
\item $\Omega$ domain \end{itemize}
Misc: \begin{itemize}
\item $z(t\pm)$ left/right limit (clarify endpoint conventions)
\item bar for candidate of stationary point etc.
\item all integrals in Kurzweil-Stieltjes sense
\item $\nabla$ spatial gradient
\item $\partial$ partial derivative, always with number
\item $f'(x;h)$ for directional derivative
\item $(\cdot)^\perp$ kernel
\item $(\cdots)^*$ dual space
\item $(\cdot)'$ weak time derivative
\item $\Delta$ Laplacian
\item $\Delta_n$ partition
\item $\langle \cdot, \cdot \rangle$ dual pairing
\item $(\cdot, \cdot)$ scalar product
\item $\mathds{1}$ characteristic function
\end{itemize} \fi
\end{document} | arXiv |
# Overview of the Z-transform and its properties
The Z-transform is a powerful tool in signal processing and is widely used for analyzing and processing signals. It is defined as the continuous analog of the discrete-time Fourier transform (DTFT). The Z-transform is used to convert a discrete-time signal into a continuous-time signal, allowing for easier analysis and manipulation of the signal.
The Z-transform is defined as:
$$X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$$
where $x[n]$ is the discrete-time signal, $z$ is the complex variable, and $X(z)$ is the continuous-time signal.
The Z-transform has several properties that make it useful in signal processing:
- Linearity: The Z-transform is linear, meaning that the sum of two signals is equal to the sum of their Z-transforms.
- Time-invariance: The Z-transform is time-invariant, meaning that it does not depend on the time shift of the input signal.
- Shift property: The Z-transform has a shift property, which allows for easy analysis of signals that are shifted in time.
- Convolution property: The Z-transform has a convolution property, which allows for easy analysis of signals that are the result of a convolution operation.
## Exercise
Calculate the Z-transform of the following signal:
$$x[n] = \delta[n] + 2\delta[n-1] + 3\delta[n-2]$$
The Z-transform of the given signal can be calculated as follows:
$$X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$$
$$X(z) = \delta[n] z^{-n} + 2\delta[n-1] z^{-(n-1)} + 3\delta[n-2] z^{-(n-2)}$$
$$X(z) = 1 + 2z^{-1} + 3z^{-2}$$
# Chirp signals: definition and properties
A chirp signal is a non-stationary signal that varies its frequency over time. Chirp signals are characterized by a linear frequency sweep, where the frequency increases or decreases at a constant rate. They are often used in radar systems, radio communication systems, and sonar systems.
The equation for a chirp signal is:
$$x[n] = A(t) \cos(2\pi f_0 t + \phi[t])$$
where $A(t)$ is the amplitude function, $f_0$ is the initial frequency, $t$ is time, and $\phi[t]$ is the phase function.
The properties of chirp signals include:
- Non-stationarity: Chirp signals are non-stationary, meaning that their frequency and phase vary over time.
- Linear frequency sweep: Chirp signals have a linear frequency sweep, where the frequency increases or decreases at a constant rate.
- Time-varying amplitude and phase: The amplitude and phase of chirp signals are functions of time, allowing for complex modulation of the signal.
## Exercise
Calculate the Z-transform of the following chirp signal:
$$x[n] = \cos(2\pi f_0 n + \phi[n])$$
where $f_0 = 1$ and $\phi[n] = \frac{2\pi}{T} n$.
The Z-transform of the given chirp signal can be calculated as follows:
$$X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$$
$$X(z) = \sum_{n=-\infty}^{\infty} \cos(2\pi f_0 n + \phi[n]) z^{-n}$$
$$X(z) = \sum_{n=-\infty}^{\infty} \cos(2\pi n + \frac{2\pi}{T} n) z^{-n}$$
$$X(z) = \sum_{n=-\infty}^{\infty} \cos(\frac{2\pi}{T} n) z^{-n}$$
# Constructing chirp signals
Constructing chirp signals involves defining the amplitude and phase functions, as well as the initial frequency and the rate of frequency change. There are several methods for constructing chirp signals, including linear chirp, exponential chirp, and logarithmic chirp.
Linear chirp:
A linear chirp signal is defined by the equation:
$$x[n] = A(t) \cos(2\pi f_0 t + \phi[t])$$
where $A(t) = A_0 + \frac{1}{2} k t$, $f_0$ is the initial frequency, $t$ is time, and $\phi[t] = \frac{2\pi}{T} n$.
Exponential chirp:
An exponential chirp signal is defined by the equation:
$$x[n] = A(t) \cos(2\pi f_0 t + \phi[t])$$
where $A(t) = A_0 + k t$, $f_0$ is the initial frequency, $t$ is time, and $\phi[t] = \frac{2\pi}{T} n$.
Logarithmic chirp:
A logarithmic chirp signal is defined by the equation:
$$x[n] = A(t) \cos(2\pi f_0 t + \phi[t])$$
where $A(t) = A_0 + \frac{k}{2} t \ln(t)$, $f_0$ is the initial frequency, $t$ is time, and $\phi[t] = \frac{2\pi}{T} n$.
## Exercise
Construct a linear chirp signal with an initial frequency of 1 Hz, a rate of frequency change of 2 Hz/s, and a duration of 5 seconds.
To construct a linear chirp signal with the given parameters, we can define the amplitude and phase functions as follows:
$$A(t) = A_0 + \frac{1}{2} k t$$
$$\phi[t] = \frac{2\pi}{T} n$$
where $A_0$ is the initial amplitude, $k$ is the rate of frequency change, $T$ is the duration of the signal, and $n$ is the time index.
# Analyzing chirp signals in the time-frequency domain
Analyzing chirp signals in the time-frequency domain involves using the Z-transform to convert the chirp signal into the frequency domain. This allows for easy analysis and manipulation of the signal's frequency and phase properties.
The Z-transform of a chirp signal can be calculated using the following equation:
$$X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$$
The Z-transform of a chirp signal can be expressed as a product of the amplitude and phase functions:
$$X(z) = A(z) \cdot \Phi(z)$$
where $A(z)$ is the amplitude function and $\Phi(z)$ is the phase function.
## Exercise
Analyze the following chirp signal in the time-frequency domain:
$$x[n] = \cos(2\pi f_0 n + \phi[n])$$
where $f_0 = 1$ and $\phi[n] = \frac{2\pi}{T} n$.
To analyze the given chirp signal in the time-frequency domain, we can calculate the Z-transform of the signal:
$$X(z) = \sum_{n=-\infty}^{\infty} \cos(2\pi f_0 n + \phi[n]) z^{-n}$$
$$X(z) = \sum_{n=-\infty}^{\infty} \cos(\frac{2\pi}{T} n) z^{-n}$$
The Z-transform of the chirp signal can be expressed as a product of the amplitude and phase functions:
$$X(z) = A(z) \cdot \Phi(z)$$
where $A(z)$ is the amplitude function and $\Phi(z)$ is the phase function.
# Applications of the Chirp Z-transform in signal processing
The Chirp Z-transform has numerous applications in signal processing, including:
- Frequency analysis: The Chirp Z-transform allows for easy analysis of signals' frequency content, making it useful for tasks such as spectrum analysis and filtering.
- Time-frequency analysis: The Chirp Z-transform enables the analysis of signals in both the time and frequency domains, making it useful for tasks such as time-frequency analysis and modulation analysis.
- Signal synthesis: The Chirp Z-transform can be used to synthesize signals, making it useful for tasks such as signal generation and waveform design.
- Filter design: The Chirp Z-transform can be used to design linear and nonlinear filters, making it useful for tasks such as noise reduction and signal enhancement.
- Communication systems: The Chirp Z-transform is widely used in communication systems, making it useful for tasks such as signal modulation and demodulation.
## Exercise
Describe an application of the Chirp Z-transform in signal processing.
One application of the Chirp Z-transform in signal processing is in the design of linear and nonlinear filters. By analyzing signals in the frequency domain using the Chirp Z-transform, engineers can easily design filters that attenuate unwanted frequencies and amplify desired frequencies. This allows for efficient signal processing and improved signal quality.
# Linear and nonlinear filtering with the Chirp Z-transform
Linear filtering with the Chirp Z-transform involves designing filters that attenuate unwanted frequencies and amplify desired frequencies. This can be achieved by using the properties of the Chirp Z-transform, such as the shift property and the convolution property.
Nonlinear filtering with the Chirp Z-transform involves designing filters that modify the frequency and phase properties of signals. This can be achieved by using the properties of the Chirp Z-transform, such as the amplitude and phase modulation functions.
## Exercise
Design a linear filter using the Chirp Z-transform that attenuates frequencies below 1 Hz and amplifies frequencies above 2 Hz.
To design a linear filter using the Chirp Z-transform, we can follow these steps:
1. Define the filter's amplitude and phase functions.
2. Calculate the Z-transform of the filter.
3. Apply the filter to the input signal.
The filter's amplitude and phase functions can be defined as follows:
$$A(z) = \begin{cases} 1 & \text{if } f > 2 \\ 0 & \text{if } f < 1 \end{cases}$$
$$\Phi(z) = 0$$
The Z-transform of the filter can be calculated as follows:
$$H(z) = A(z) \cdot \Phi(z)$$
The filter can be applied to the input signal by multiplying the input signal's Z-transform with the filter's Z-transform:
$$Y(z) = X(z) \cdot H(z)$$
# Frequency domain analysis of signals using the Chirp Z-transform
Frequency domain analysis of signals using the Chirp Z-transform involves converting the signals into the frequency domain using the Z-transform. This allows for easy analysis and manipulation of the signals' frequency and phase properties.
Some common frequency domain analysis techniques using the Chirp Z-transform include:
- Spectrum analysis: Analyzing the frequency content of a signal to identify its components and characteristics.
- Time-frequency analysis: Analyzing the time and frequency properties of a signal simultaneously.
- Modulation analysis: Analyzing the modulation properties of a signal, such as its amplitude and phase modulation.
## Exercise
Analyze the following signal in the frequency domain using the Chirp Z-transform:
$$x[n] = \cos(2\pi f_0 n + \phi[n])$$
where $f_0 = 1$ and $\phi[n] = \frac{2\pi}{T} n$.
To analyze the given signal in the frequency domain using the Chirp Z-transform, we can follow these steps:
1. Calculate the Z-transform of the signal.
2. Apply the Z-transform to the signal.
The Z-transform of the signal can be calculated as follows:
$$X(z) = \sum_{n=-\infty}^{\infty} \cos(\frac{2\pi}{T} n) z^{-n}$$
The Z-transform of the signal can be expressed as a product of the amplitude and phase functions:
$$X(z) = A(z) \cdot \Phi(z)$$
The frequency domain analysis of the signal can be performed by analyzing the amplitude and phase functions.
# Algorithms for computing the Chirp Z-transform
There are several algorithms for computing the Chirp Z-transform, including:
- Fast Fourier Transform (FFT): The FFT is a widely used algorithm for computing the DTFT of a signal. By applying the FFT to the discrete-time signal, the Chirp Z-transform can be calculated.
- Discrete-Time Z-transform (DTZT): The DTZT is an algorithm that computes the Z-transform of a discrete-time signal directly. By applying the DTZT to the discrete-time signal, the Chirp Z-transform can be calculated.
- Fast Z-transform (FZT): The FZT is an algorithm that computes the Z-transform of a discrete-time signal using a fast algorithm. By applying the FZT to the discrete-time signal, the Chirp Z-transform can be calculated.
## Exercise
Choose an algorithm for computing the Chirp Z-transform and describe how it works.
One algorithm for computing the Chirp Z-transform is the Fast Fourier Transform (FFT). The FFT is a widely used algorithm for computing the DTFT of a signal. By applying the FFT to the discrete-time signal, the Chirp Z-transform can be calculated.
The FFT algorithm works by dividing the signal into smaller overlapping segments and computing the DTFT of each segment. The DTFTs of the segments are then combined to form the Chirp Z-transform of the signal.
# Comparing the Chirp Z-transform with other transforms in signal processing
The Chirp Z-transform can be compared with other transforms in signal processing, such as the Discrete-Time Fourier Transform (DTFT) and the Discrete-Time Z-transform (DTZT).
Some key differences between the Chirp Z-transform and other transforms include:
- Linearity: The Chirp Z-transform is linear, while the DTFT and DTZT are non-linear.
- Time-invariance: The Chirp Z-transform is time-invariant, while the DTFT and DTZT are time-variant.
- Shift property: The Chirp Z-transform has a shift property, while the DTFT and DTZT do not.
- Convolution property: The Chirp Z-transform has a convolution property, while the DTFT and DTZT do not.
## Exercise
Compare the Chirp Z-transform with another transform in signal processing.
One transform that can be compared with the Chirp Z-transform is the Discrete-Time Fourier Transform (DTFT). The DTFT is a non-linear transform that is time-variant and does not have a shift or convolution property.
The DTFT can be calculated using the following equation:
$$X(\omega) = \int_{-\infty}^{\infty} x[n] e^{-j\omega n} dn$$
where $X(\omega)$ is the DTFT of the signal, $x[n]$ is the discrete-time signal, and $\omega$ is the frequency variable.
# Challenges and future developments in the Chirp Z-transform
Despite its numerous applications in signal processing, the Chirp Z-transform faces several challenges and has potential for future developments:
- Computational complexity: The Chirp Z-transform can be computationally complex, especially for large signals or high-resolution analysis.
- Numerical stability: The Chirp Z-transform can suffer from numerical stability issues, such as round-off errors and overflow.
- Hardware implementation: The Chirp Z-transform can be challenging to implement in hardware, requiring specialized hardware or algorithms.
- Real-time processing: The Chirp Z-transform can be challenging to implement in real-time systems, due to its computational complexity.
Future developments in the Chirp Z-transform may involve:
- Improved algorithms: Developing more efficient and accurate algorithms for computing the Chirp Z-transform.
- Hardware acceleration: Developing hardware accelerators or specialized circuits for implementing the Chirp Z-transform.
- Real-time processing: Developing algorithms and hardware for real-time implementation of the Chirp Z-transform.
- Applications in machine learning: Exploring the use of the Chirp Z-transform in machine learning applications, such as signal classification and feature extraction.
## Exercise
Discuss a potential future development in the Chirp Z-transform.
One potential future development in the Chirp Z-transform is the exploration of its use in machine learning applications. By transforming signals into the frequency domain, the Chirp Z-transform can be used as a feature extractor for machine learning algorithms. This can potentially lead to improved signal classification and feature extraction, as well as the development of new applications in areas such as speech recognition and image analysis. | Textbooks |
\begin{document}
\large \begin{center} {\bf Compositionality of the Runge-Kutta Method} \end{center}
\normalsize
\begin{abstract}
In Spivak \cite{spivak}, dynamical systems are described in terms of their inputs and outputs in a pictorial way using an operad of wiring diagrams. Each dynamical system is a box with certain inputs and outputs, and multiple dynamical systems are linked together using wiring diagrams, which describe how the outputs of one dynamical system to the inputs of another. By describing dynamical systems in this way, we can decompose a large dynamical system as a collection of smaller, simpler dynamical systems linked together. Of course, this decomposition is only useful if we can work with these smaller, simpler dynamical systems instead of the larger one. In his paper, Spivak shows that we can perform Euler's method on these smaller systems and still get the same results as working on the larger one. In this paper, we extend his results to prove that we can do something similar with the Runge-Kutta method. However, we need to modify the framework used in Spivak's paper to account for the fact that the Runge-Kutta method requires multiple steps, unlike Euler's method. To better describe these systems, we define wiring diagrams as objects of a double-category and dynamical systems in terms of double functors, giving a categorical description of this approximation method.
\end{abstract}
\section{Introduction} \label{introduction}
Dynamical systems are everywhere, from the cells in our bodies to the computers which we work on. Each of these machines take inputs, modify an internal state, and then give an output. For a computer, the inputs could be user inputs on a keyboard, which then modify the binary state held in the hard drive and memory, and the output could be the display on the monitor. In addition, by using the outputs of and providing the inputs for other dynamical systems, dynamical systems can become interconnected through a network which as a whole represents one larger dynamical system, such as the internet, which is a dynamical system composed of many interconnected computers. When working with such a network, often it would be much easier to work with the individual dynamical systems and then combine the results instead of composing all the dynamical systems first and then working with that large system.
In Spivak's work \cite{spivak}, he provides a framework for describing dynamical systems and their interconnections through wiring diagrams. A dynamical system with input set $A$ and output set $B$ is described as a tuple $(S,f^{rdt},f^{upd})$, with $S$ the state set, $f^{rdt}: S \to B$ a readout function turning the state to an output, and $f^{upd}: A \times S \to S$ an update function changing the state depending on the input. For a discrete dynamical system, we let our state set be any set, and we define $f^{upd}$ by taking an input and state and give back the next state which should result. For a continuous dynamical system, we let our state set be a real vector space and define $f^{upd}$ so that it takes the input and state and returns the infinitesimal change (like a derivative) which should result. Often, we will describe a dynamical system as a diagram \[ \begin{tikzcd} A \times S \arrow[r] & S \arrow[r] & B \end{tikzcd} \] with the arrows representing the update and readout functions.
In his paper, Spivak proves that Euler's method works in a natural way when converting continuous dynamical systems to discrete dynamical systems so that we can work with the individual dynamical systems before composing instead of having to work with the much larger combined dynamical system. We would like to extend that result to the Runge-Kutta method, but unfortunately, the discrete dynamical system format above does not capture the four-step approach that Runge-Kutta uses. In every iteration of Runge-Kutta, there are four computations performed, each depending on all the previous results of the iteration, but this iterative cycle of four steps is not captured in the original definition of a discrete dynamical system, as it only goes step by step and does not remember any states besides the previous one. In addition, Spivak describes his continuous dynamical systems using the manifold structure of a real vector space, but for Runge-Kutta, it is the vector space properties which are more important. As a result, we need to describe dynamical systems differently to more accurately capture this information. In particular, we introduce a second type of morphism which describes a dynamical system working in line with another dynamical system. An iterative four-step dynamical system can be thought of as a dynamical system which lines up with the cyclic four-step dynamical system $C_4$ shown below. \[ \begin{tikzcd} 1 \arrow[r] & 2 \arrow[d] \\ 4 \arrow[u] & \arrow[l] 3 \end{tikzcd} \] To incorporate both types of morphisms, we describe our dynamical systems using double categories and double functors and describe Runge-Kutta as a double transformation. To capture how dynamical systems can be combined, we also impose a monoidal product to describe dynamical systems being put side-by-side without interacting with each other, similar to what is described in Spivak's work, and we prove that our double functors and double transformations preserve this monoidal structure.
The next section, Section \ref{definitions}, defines what double categories, double functors, and double transformations are, along with their monoidal versions. To understand these definitions, however, the reader needs to know the definitions of categories, functors, and natural transformations, in addition to the monoidal forms of each of these concepts. Spivak's paper describes continuous systems using a functor $\CS$ and discrete dynamical systems using a functor $\DS$ from the category of wiring diagrams to the category of sets. In Section \ref{dynamical}, we define the double category of wiring diagrams $\mathscr{W}$ and describe our continuous dynamical systems through a double functor $\CS: \mathscr{W} \to \Prof$ and our four-step dynamical systems through a double functor $\DS/C_4 : \mathscr{W} \to \Prof$, where $\Prof$ is the double category of profunctors. We then describe a monoidal product to represent combining dynamical systems together and show that our double category and double functors preserve this product. Finally, in Section \ref{runge-kutta}, we describe Runge-Kutta as a double transformation between $\CS$ and $\DS/C_4$ and prove the following theorem. \begin{restatable}{thm}{rkthm} The Runge-Kutta method is a monoidal double transformation between $\CS$ and $\DS/ C_4$. \end{restatable}
\section{Definitions} \label{definitions}
We begin by defining double categories, double functors, and double transformations, using the works of Shulman \cite{shulman:1}\cite{shulman:2}. As mentioned in the introduction, double categories allow us to define two types of morphisms for our objects.
\begin{definition} A \textit{(pseudo) double category} $\mbb{D}$ consists of a ``category of objects $\mbb{D}_0$'' and a ``category of arrows'' $\mbb{D}_1$, with structure functors \[ U: \mbb{D}_0 \to \mbb{D}_1 \] \[ L,R: \mbb{D}_1 \to \mbb{D}_0 \] \[ \odot: \mbb{D}_1 \times_{\mbb{D}_0} \mbb{D}_1 \to \mbb{D}_1 \] where the pullback is over
\begin{center} \begin{tikzcd} \mbb{D}_1 \arrow{r}{R} & \mbb{D}_0 & \arrow[l,swap,"L"] \mbb{D}_1 \end{tikzcd} \end{center} These structure functors must satisfy the relations
\begin{center} \begin{tabular}{rl} $L(U_A) = A$ &$R(U_A) = A$\\ $L(M \odot N) = L(M)$ &$R(M \odot N) = R(N)$ \end{tabular} \end{center} and are equipped with natural transformations \[ a: (M \odot N) \odot P \to M \odot (N \odot P) \] \[ l: U_{L(M)} \odot M \to M \] \[ r: M \odot U_{R(M)} \to M \] such that $L(a)$, $R(a)$, $L(l)$, $R(l)$, $L(r)$, $R(r)$ are all identities and the following diagrams commute:
(Triangle) \begin{center} \begin{tikzcd} (x \odot U_{Rx}) \odot y \arrow[rr,"a"] \arrow[dr,"r \odot 1"'] && x \odot (U_{Ly} \odot y) \\ & x \odot y \arrow[ur,"1 \odot l"'] & \end{tikzcd} \end{center}
(Pentagon) \begin{center} \begin{tikzcd}
& (w \odot x) \odot (y \odot z) \arrow[dr,"a"]\\ ((w \odot x) \odot y) \odot z \arrow[ur,"a"] \arrow[d,"a"']& & (w \odot (x \odot (y \odot z))) \\ (w \odot (x \odot y)) \odot z \arrow[rr,"a"'] && w \odot ((x \odot y) \odot z) \arrow[u,"a"'] \end{tikzcd} \end{center} \end{definition}
\begin{definition} Let $\mbb{D}$ and $\mbb{E}$ be double categories. A \textit{lax double functor} $F: \mbb{D} \to \mbb{E}$ consists of the following: \begin{itemize} \item Functors $F_0: \mbb{D}_0 \to \mbb{E}_0$ and $F_1: \mbb{D}_1 \to \mbb{E}_1$ such that $L \circ F_1 = F_0 \circ L$ and $R \circ F_1 = F_0 \circ R$.
\item Natural Transformations $F_{\odot} : F_1 M \odot F_1 N \to F_1(M \odot N)$ and $F_U : U_{F_0 -} \to F_1(U_{-})$, whose components are globular , and for which the following diagrams commute. \end{itemize}
(Associativity) \begin{center} \begin{tikzcd} (F_1(x) \odot F_1(y)) \odot F_1(z) \arrow[d,"F_{\odot} \odot id"'] \arrow[r,"a"] & F_1(x) \odot (F_1(y) \odot F_1(z)) \arrow[d,"id \odot F_{\odot}"]\\ F_1(x \odot y) \odot F_1(z) \arrow[d,"F_\odot"'] & F_1(x) \odot F_1(y \odot z) \arrow[d,"F_\odot"]\\ F_1((x \odot y) \odot z) \arrow[r,"F_1(a)"'] & F_1(x \odot (y \odot z)) \end{tikzcd} \end{center}
(Unitality) \begin{center} \begin{tikzcd} U_{L \circ F_1x} \odot F_1(x) \arrow[d,"l"'] \arrow[r,"F_u \odot id"] & F_1(U_{Lx}) \odot F_1x \arrow[d,"F_\odot"]\\ F_1(x) & \arrow[l,"F_1(l)"] F_1(U_{Lx} \circ x) \end{tikzcd} \begin{tikzcd} F_1(x) \odot U_{F_0 \circ Rx} \arrow[d,"r"'] \arrow[r,"id \odot F_U"] & F_1(x) \odot F_1(U_{Rx}) \arrow[d,"F_\odot"]\\ F_1(x) & \arrow[l,"F_1(r)"] F_1(x \odot U_{Rx}) \end{tikzcd} \end{center}
If $F_U$ is an isomorphism, then $F$ is called a normal double functor. \end{definition}
\begin{definition} A \textit{double transformation} between two lax double functors $\alpha: F \Rightarrow G: \mbb{D} \to \mbb{E}$ consists of natural transformations $\alpha_0: F_0 \Rightarrow G_0: \mbb{D}_0 \to \mbb{E}_0$ and $\alpha_1 : F_1 \Rightarrow G_1: \mbb{D}_1 \to \mbb{E}_1$ such that $L(\alpha_M) = \alpha_{LM}$ and $R(\alpha_M) = \alpha_{RM}$ and
\[ \begin{tikzcd} FA \arrow[drr,phantom,"F_{\odot}"] \arrow[d,equal] \arrow[r,tick,"FM"] & FB \arrow[r,tick,"FN"] & FC \arrow[d,equal]\\ FA\arrow[drr,phantom,"\alpha_{M \odot N}"] \arrow[d,"\alpha_A"'] \arrow[rr,tick,"F(M \odot N)"] && FC \arrow[d,"\alpha_C"]\\ GA \arrow[rr,tick,"G(M \odot N)"'] && GC \end{tikzcd} = \begin{tikzcd} FA \arrow[dr,phantom,"\alpha_M"] \arrow[d,"\alpha_A"'] \arrow[r,tick,"FM"] & FB \arrow[dr,phantom,"\alpha_N"] \arrow[d,"\alpha_D"]\arrow[r,tick,"FN"] & FC \arrow[d,"\alpha_C"] \\ GA \arrow[drr,phantom,"G_{\odot}"] \arrow[d,equal] \arrow[r,tick,"GM"] & GB \arrow[r,tick,"GN"] & GC \arrow[d,equal]\\ GA \arrow[rr,tick,"G(M \odot N)"'] && GC \end{tikzcd} \]
\[ \begin{tikzcd} FA \arrow[dr,phantom," F_U"] \arrow[d,equal] \arrow[r,tick,"U_{FA}"] & FA \arrow[d,equal] \\ FA \arrow[dr,phantom,"\alpha_{U_A}"] \arrow[d,"\alpha_A"'] \arrow[r,tick,"F(U_A)"] & FA \arrow[d,"\alpha_A"]\\ GA \arrow[r,tick,"G(U_A)"'] & GA \end{tikzcd} = \begin{tikzcd} FA \arrow[dr,phantom,"U_{\alpha_A}"] \arrow[d,"\alpha_A"'] \arrow[r,tick,"U_{FA}"] & FA \arrow[d,"\alpha_A"]\\ GA \arrow[dr,phantom, "G_U"] \arrow[d,equal] \arrow[r,tick,"U_{GA}"] & GA \arrow[d,equal]\\ GA \arrow[r,tick,"G(U_A)"'] & GA \end{tikzcd} \] \end{definition}
We now describe the monoidal versions of these definitions. The definitions for the monoidal double functor and monoidal double transformation are the same as the definitions Shulman uses for monoidal framed bicategories.
\begin{definition} A \textit{monoidal double category} is a double category equipped with functors $\otimes: \mathbb{D} \times \mathbb{D}$ and $I: * \to \mbb{D}$, and invertible transformations \[ \otimes \circ (\Id \times \otimes) \cong \otimes \circ (\otimes \times \Id) \] \[ \otimes \circ (\Id \times I) \cong \Id \] \[ \otimes \circ (I \times \Id) \cong \Id \] satisfying the usual axioms.
Unpacking this definition more explicitly, we see that a monoidal double category is a double category together with the following structure. \begin{itemize} \item $\mbb{D}_0$ and $\mbb{D}_1$ are both monoidal categories. \item If $I$ is the monoidal unit of $\mbb{D}_0$, then $U_I$ is the monoidal unit of $\mbb{D}_I$. \item The functors $L$ and $R$ are strict monoidal, i.e. $L(M \otimes N) = LM \otimes LN$ and $R(M \otimes N) = RM \otimes RN$ and $L$ and $R$ also preserve the associativity and unit constraints. \item We have globular isomorphisms \[ \mathfrak{x}: (M_1 \otimes N_1) \odot (M_2 \otimes N_2) \xrightarrow{\cong} (M_1 \odot M_2) \otimes (N_1 \odot N_2) \] and \[ \mathfrak{u}: U_{A \otimes B} \xrightarrow{\cong} (U_A \otimes U_B) \] such that the following diagrams commute: \begin{center} \begin{tikzcd} ((M_1 \otimes N_1) \odot (M_2 \otimes N_2)) \odot (M_3 \otimes N_3)\arrow[d] \arrow[r] & ((M_1 \odot M_2) \otimes (N_1 \odot N_2)) \odot (M_3 \otimes N_3) \arrow[d]\\ (M_1 \otimes N_1) \odot ((M_2 \otimes N_2) \odot (M_3 \otimes N_3)) \arrow[d] & ((M_1 \odot M_2) \odot M_3) \otimes ((N_1 \odot N_2) \odot N_3) \arrow[d] \\ (M_1 \otimes N_1) \odot ((M_2 \odot M_3) \otimes (N_2 \odot N_3)) \arrow[r] & (M_1 \odot (M_2 \odot M_3)) \otimes (N_1 \odot (N_2 \odot N_3)) \end{tikzcd}
\begin{tikzcd} (M \times N) \odot U_{C \otimes D} \arrow[d]\arrow[r] & (M \otimes N) \circ (U_C \otimes U_D) \arrow[d]\\ M \otimes N & \arrow[l] (M \otimes U_C) \otimes (N \odot U_D) \end{tikzcd}
\begin{tikzcd} U_{A \otimes B} \odot (M \otimes N) \arrow[r] \arrow[d]& (U_A \otimes U_B) \odot (M \otimes N) \arrow[d] \\ M \otimes N & \arrow[l] (U_A \odot M) \otimes (U_B \odot N) \end{tikzcd} \end{center} (these arise from the constraint data from the pseudo double functor $\otimes$). \item The following diagrams commute, expressing that the associativity isomorphism for $\otimes$ is a transformation of double categories.
\begin{center} \begin{tikzcd} ((M_1 \otimes N_1) \otimes P_1) \odot ((M_2 \otimes N_2) \otimes P_2) \arrow[d]\arrow[r] & (M_1 \otimes (N_1 \otimes P_1)) \odot (M_2 \otimes (N_2 \otimes P_2)) \arrow[d]\\ ((M_1 \otimes N_1) \odot (M_2 \otimes N_20) \otimes (P_1 \odot P_2) \arrow[d]& (M_1 \odot M_2) \otimes ((N_1 \otimes P_1) \odot (N_2 \otimes P_2)) \arrow[d]\\ ((M_1 \otimes M_2) \otimes (N_1 \odot N_2)) \otimes (P_1 \odot P_2) \arrow[r] & (M_1 \odot M_2) \otimes ((N_1 \odot N_2) \otimes (P_1 \odot P_2)) \end{tikzcd}
\begin{tikzcd} U_{(A \otimes B) \otimes C} \arrow[r] \arrow[d] & U_{A \otimes (B \otimes C)} \arrow[d] \\ U_{A \otimes B} \otimes U_C \arrow[d] & U_A \otimes U_{B \otimes C} \arrow[d] \\ (U_A \otimes U_B) \otimes U_C \arrow[r] & U_A \otimes (U_B \otimes U_C) \end{tikzcd} \end{center}
\item The following diagrams commute, expressing that the unit isomorphisms for $\otimes$ are transformations of double categories.
\begin{center} \begin{tikzcd} (M \otimes U_I) \odot (N \otimes U_I) \arrow[r] \arrow[d] & (M \odot N) \otimes (U_I \odot U_I) \arrow[d]\\ M \odot N & \arrow[l] (M \odot N) \otimes U_I \end{tikzcd} \begin{tikzcd} U_{ A \otimes I} \arrow[r] \arrow[rd] & U_A \otimes U_I \arrow[d]\\
& U_A \end{tikzcd}
\begin{tikzcd} (U_I \otimes M) \odot (U_I \otimes N) \arrow[r] \arrow[d] & (U_I \odot U_I) \otimes (M \odot N) \arrow[d] \\ M \odot N & \arrow[l] U_I \otimes (M \odot N) \end{tikzcd} \begin{tikzcd} U_{I \otimes A} \arrow[dr] \arrow[r] & U_I \otimes U_A \arrow[d] \\
& U_A \end{tikzcd} \end{center} \end{itemize} \end{definition}
A good example of a monoidal double category is $\Prof$, \textit{the double category of categories and profunctors}. One can check that imposing the Cartesian product as the monoidal product gives us a monoidal double category.
\begin{definition} A \textit{lax monoidal double functor} between monoidal double categories $\mbb{D},\mbb{E}$ consists of the following structure and properties. \begin{itemize} \item A lax double functor $F: \mbb{D} \to \mbb{E}$. \item The structure of a lax monoidal functor on $F_0$ and $F_1$. \item Equalities $LF_1 = F_0L$ and $RF_1 = F_0 R$ of lax monoidal functors. \item The composition constraints for the lax double functor $F$ are monoidal natural transformations. \end{itemize} \end{definition}
\begin{definition} A \textit{monoidal double transformation} is a double transformation such that $\alpha_0$ and $\alpha_1$ are monoidal natural transformations. \end{definition}
\section{Dynamical Systems as a Double Functor} \label{dynamical}
We can now describe our wiring boxes and wiring diagrams as a double category, allowing us to describe one morphism using wiring diagrams and another morphism using commuting diagrams which describe a wiring box acting in step with another.
\begin{definition} We define $\mathscr{W}$, \textit{the double category of wiring diagrams}, by the following categories $\mathscr{W}_0$ and $\mathscr{W}_1$. $\mathscr{W}_0$ is the category of wiring diagrams. The objects are ordered pairs $(A,B)$ with $A,B \in \mathbf{Set}$ and the morphisms are wiring diagrams.
\begin{center} \begin{tikzcd} (A,B) \arrow[r,"\varphi"] & (C,D) \end{tikzcd} \end{center}
\[ \varphi^{in}: C \times B \to A \] \[ \varphi^{out}: B \to D \] The composition of wiring diagrams $\varphi$ and $\psi$ \begin{center} \begin{tikzcd} (A,B) \arrow[r,"\varphi"] & (C,D) \arrow[r,"\psi"] & (E,F) \end{tikzcd} \end{center} is the wiring diagram with $(\psi \circ \varphi)^{in} : E \times B \to A$ given by \begin{center} \begin{tikzcd} E \times B \arrow[r,"id \times \Delta"] &[5pt] E \times B \times B \arrow[r,"id \times \varphi^{out} \times id"] &[30pt] E \times D \times B \arrow[r,"\psi^{in} \times id"] &[15pt] C \times B \arrow[r,"\varphi^{in}"] & A \end{tikzcd} \end{center} and $(\psi \circ \varphi)^{out} : B \to F$ given by \begin{center} \begin{tikzcd} B \arrow[r,"\varphi^{out}"] & D \arrow[r,"\psi^{out}"] & F \end{tikzcd} \end{center}
We define $\mathscr{W}_1$ to be the following category. The objects are ordered pairs $(f,g)$ where $f: A \to A'$ and $g: B \to B'$ with $L(f,g) = (A,B)$ and $R(f,g) = (A',B')$. We define a morphism $(f,g)$ to $(f',g')$ to be a pair of wiring diagrams $(\varphi_1,\varphi_2)$ such that the following diagrams commute.
\begin{center} \begin{tikzcd} B_1 \arrow[d,"\varphi_1^{out}"'] \arrow[r,"g"] & B_2 \arrow[d,"\varphi_2^{out}"]\\ B_1' \arrow[r,"g'"'] & B_2' \end{tikzcd} \begin{tikzcd} A_1' \times B_1 \arrow[d,"\varphi_1^{in}"'] \arrow[r,"f' \times g"] & A_2' \times B_2 \arrow[d,"\varphi_2^{in}"]\\ A_1 \arrow[r,"f"'] & A_2 \end{tikzcd} \end{center} The resulting 2-cell can be described by the following box. \begin{center} \begin{tikzcd} (A_1,B_1) \arrow[d,"\varphi_1"'] \arrow[r,tick,"{(f,g)}"] & (A_2,B_2) \arrow[d,"\varphi_2"] \\ (A_1',B_1') \arrow[r,tick,"{(f',g')}"'] & (A_2',B_2') \end{tikzcd} \end{center}
Then we let the composition of two morphisms in $\mathscr{W}_1$ be given by the following equality. \[ \begin{tikzcd} (A_1',B_1') \arrow[d,"\varphi'_1"'] \arrow[r,"{(f',g')}"] & (A_2',B_2') \arrow[d,"\varphi_2'"] \\ (A_2',B_2') \arrow[r,"{(f'',g'')}"'] & (A_2'',B_2'') \end{tikzcd} \circ \begin{tikzcd}
(A_1,B_1) \arrow[d,"\varphi_1"'] \arrow[r,"{(f,g)}"] & (A_2,B_2) \arrow[d,"\varphi_2"]\\ (A_1',B_1') \arrow[r,"{(f',g')}"'] & (A_2',B_2') \end{tikzcd} = \begin{tikzcd} (A_1,B_1) \arrow[d,"\varphi_1' \circ \varphi_1"'] \arrow[r,"{(f,g)}"] & (A_2,B_2) \arrow[d,"\varphi_2' \circ \varphi_2"] \\ (A_1'',B_1'') \arrow[r,"{(f'',g'')}"'] & (A_2'',B_2'') \end{tikzcd} \] The latter term is a valid morphism since we have the following commuting diagrams.
\begin{center} \begin{tabular}{cc} \begin{tikzcd} B_1 \arrow[d,"\varphi_1^{out}"'] \arrow[r,"g"] & B_2 \arrow[d,"\varphi_2^{out}"] \\ B_1' \arrow[d,"{\varphi''_1}^{out}"']\arrow[r,"g'"] & B_2' \arrow[d,"{\varphi'_2}^{out}"] \\ B_1'' \arrow[r,"g''"'] & B_2'' \end{tikzcd} & \begin{tikzcd} A_1'' \times B_1 \arrow[d,"id \times \Delta"'] \arrow[r,"f'' \times g"] & A_2'' \times B_2 \arrow[d,"id \times \Delta"]\\ A_1'' \times B_1 \times B_1 \arrow[d,"id \times \varphi_1^{out} \times id"'] \arrow[r,"f'' \times g \times g"] & A_2'' \times B_2 \times B_2 \arrow[d,"id \times \varphi_2^{out} \times id"]\\ A''_1 \times B_1' \times B_1 \arrow[d,"{\varphi_1'}^{in} \times id"'] \arrow[r,"f'' \times g' \times g"] & A_2'' \times B_2' \times B_2 \arrow[d,"{\varphi_2'}^{in} \times id"] \\ A'_1 \times B_1 \arrow[d,"\varphi_1^{in}"] \arrow[r,"f' \times g"'] & A'_2 \times B_2 \arrow[d,"\varphi_2^{in}"] \\ A_1 \arrow[r,"f"'] & A_2 \end{tikzcd} \end{tabular} \end{center}
We define horizontal composition for objects by $(f',g') \odot (f,g) = (f' \circ f,g' \circ g)$ and define horizontal composition for morphisms by taking the unique morphism having the required left and right values. One can check that the identity natural transformations work for $a,l,r$. \end{definition}
Now we impose the following monoidal product to represent the combination of two wiring boxes.
\[ \left(\begin{tikzcd} (A_1,B_1) \arrow[r,"{(f,g)}"] & (A_2,B_2) \end{tikzcd}\right) \otimes \left(\begin{tikzcd} (A_1',B_1') \arrow[r,"{(f',g')}"] & (A_2',B_2') \end{tikzcd}\right) \] \[= \begin{tikzcd} (A_1 \times A_1',B_1 \times B_1') \arrow[r,"{(f \times f',g \times g')}"] &[25pt] (A_2 \times A_2',B_2 \times B_2') \end{tikzcd} \]
\[ \begin{tikzcd} (A_1,B_1) \arrow[d,"\varphi_1"] \\ (C_1,D_1) \end{tikzcd} \otimes \begin{tikzcd} (A_2,B_2) \arrow[d,"\varphi_2"] \\ (C_2,D_2) \end{tikzcd} = \begin{tikzcd} (A_1 \times A_2, B_1 \times B_2) \arrow[d,"\varphi_1 \times \varphi_2"]\\ (C_1 \times C_2,D_1 \times D_2) \end{tikzcd} \] In the second equality, the last wiring diagram $\varphi_1 \times \varphi_2$ is the product of wiring diagrams $\varphi_1$ and $\varphi_2$ and is given by the following functions. \[ (\varphi_1 \times \varphi_2)^{out}(s_1 \times s_2) = \varphi_1^{out}(s_1) \times \varphi_2^{out}(s_2) \] \[ (\varphi_1 \times \varphi_2)^{in}(a_1 \times a_2,s_1 \times s_2) = \varphi_1^{in}(a_1,s_1) \times \varphi_2^{in}(a_2,s_2) \] Since the monoidal product of horizontal moprhisms and vertical morphisms act coordinate-wise on the objects, the monoidal product of 2-cells make sense using the monoidal product of the boundaries.
We prove that $\mathscr{W}_0$ forms a monoidal category. Our product forms a functor since \[ (\varphi_2^{out} \circ \varphi_1^{out}) \times (\psi_2^{out} \circ \psi_1^{out}) = (\varphi_2^{out} \times \psi_2^{out}) \circ (\varphi_1^{out} \times \psi_1^{out}) \]
and \begin{multline*} \varphi_1(\varphi_2(A_3,\varphi_1^{out}(B_1)),B_1) \times \psi_1(\psi_2(C_2,\psi_1^{out}(D_1)),D_1) =\\
(\varphi_1 \times \psi_1)(\varphi_2 \times \psi_2)(A_2 \times C_2, \varphi_1 \times \psi_1(B_1 \times D_1)),B_1 \times D_1) \end{multline*} The unit object is $(*,*)$ and the unit morphism is the trivial wiring diagram $(*,*) \to (*,*)$.
$\mathscr{W}_1$ also forms a monoidal category with the following product. \[ \begin{tikzcd} (A_1,B_1) \arrow[r,"{(f_1,g_1)}"] \arrow[d,"\varphi"'] & (C_1,D_1) \arrow[d,"\psi"] \\ (A_2,B_2) \arrow[r,"{(f_2,g_2)}"'] & (C_2,D_2) \end{tikzcd} \otimes \begin{tikzcd} (A_1',B_1') \arrow[r,"{(f_1',g_1')}"] \arrow[d,"\varphi'"'] & (C_1',D_1') \arrow[d,"\psi'"] \\ (A_2',B_2') \arrow[r,"{(f_2',g_2')}"'] & (C_2',D_2') \end{tikzcd}\]\[ = \begin{tikzcd} (A_1 \times A_1',B_1 \times B_1') \arrow[r,"{(f_1 \times f_1',g_1 \times g_1')}"] \arrow[d,"\varphi \times \varphi'"'] &[25pt] (C_1 \times C_1', D_1 \times D_1') \arrow[d,"\psi \times \psi'"] \\ (A_2 \times A_2', B_2 \times B_2') \arrow[r,"{(f_2 \times f_2',g_2 \times g_2')}"'] &[25pt] (C_2 \times C_2', D_2 \times D_2') \end{tikzcd} \] Our product forms a functor since everything works coordinate-wise, and we can take the unit object to be \begin{tikzcd} (*,*) \arrow[r,"{(!,!)}"] & (*,*) \end{tikzcd} and the unit morphism to be the box with the unit objects at the top and bottom and unit wiring diagrams at the left and right. These products on $\mathscr{W}_0$ and $\mathscr{W}_1$ affect $L$ and $R$ in the same way so $L$ and $R$ are strict monoidal functors. Finally, we can take $\mathfrak{x}$ and $\mathfrak{u}$ to be identities and the remaining conditions can be easily checked.
As mentioned in the introduction, for continuous dynamical systems, our state set $S$ is a real vector space. We define a morphism of continuous dynamical systems to be a commuting diagram \[ \begin{tikzcd} A \times S \arrow[d,"id \times m"'] \arrow[r,"f^{in}"] & S \arrow[d,"m"] \arrow[r,"f^{out}"] & B \arrow[d,"id"]\\ A \times T \arrow[r,"g^{in}"'] & T \arrow [r,"g^{out}"'] & B \end{tikzcd} \] such that $m$ is a linear map. If $m$ is an isomorphism, then we can say that the dynamical systems are isomorphic, and from this, we can split dynamical systems with inputs $A$ and $B$ into isomorphism classes.
We define the category $\CS(A,B)$, with the objects being the isomorphism classes of dynamical systems \begin{center} \begin{tikzcd} A \times S \arrow[r,"f^{in}"] & S \arrow[r,"f^{out}"] & B \end{tikzcd} \end{center} where $S$ is a real vector space and given representatives of two isomorphism classes, say with state sets $S$ and $T$, we define a morphism to be a diagram \begin{center} \begin{tikzcd} A \times S \arrow[d,"id \times m"'] \arrow[r,"f^{in}"] & S \arrow[d,"m"] \arrow[r,"f^{out}"] & B \arrow[d,"id"]\\ A \times T \arrow[r,"g^{in}"'] & T \arrow [r,"g^{out}"'] & B \end{tikzcd} \end{center} with $m: S \to T$ a linear map such that the diagram commutes.
We can now define the map $\CS: \mathscr{W} \to \Prof$ to represent our continuous dynamical systems. We define $\CS_0$ by sending $(A,B) \in \mathscr{W}_0$ to the category $\CS(A,B)$. To show that this is a functor, we need to show that each wiring diagram $\varphi: (A,B) \to (A',B')$ gives a functor $CS(A,B) \to CS(A',B')$. The wiring diagram $\varphi$ act on morphisms in $\CS(A,B)$ by the following commuting diagrams.
\begin{center} \begin{tabular}{cc} \begin{tikzcd} A' \times S \arrow[d,"id \times m"'] \arrow[r,"id \times \Delta"] & A' \times S \times S \arrow[d,"id \times m \times m"] \arrow[r,"id \times out \times id"] &[15pt] A' \times B \times S \arrow[d,"id \times id \times m"] \arrow[r,"\varphi^{in} \times id"] & A\times S \arrow[d,"id \times m"] \arrow[r,"in"] & S \arrow[d,"m"]\\ A' \times S' \arrow[r,"id \times \Delta"'] & A' \times S' \times S' \arrow[r,"id \times out \times id"'] &[15pt] A' \times B \times S' \arrow[r,"\varphi^{in} \times id"'] & A \times S' \arrow[r, "in"'] & S' \end{tikzcd} & \begin{tikzcd} S \arrow[d,"m"'] \arrow[r,""] & B \arrow[d,"id"] \arrow[r,"\varphi^{out}"] & B' \arrow[d,"id"]\\ S' \arrow[r,""] & B \arrow[r,"\varphi^{out}"'] & B' \end{tikzcd} \end{tabular} \end{center} Since these diagrams respect the composition of morphisms in $\CS(A,B)$, $\CS(\varphi)$ is a functor. We can also see that $\CS(\varphi) \circ \CS(\psi) = \CS(\varphi \circ \psi)$ because both sides send change the inputs and outputs of a dynamical system in the same way.
We then define $\CS_1: \mathscr{W}_1 \to \Prof_1$ to be the map given by sending \begin{tikzcd} (A,B) \arrow[r,"{(f,g)}"] & (C,D) \in \mathscr{W}_1 \end{tikzcd} to the profunctor $\CS(A,B)^{op} \times \CS(C,D) \to \Set$ where \[ (\begin{tikzcd} A \times S \arrow[r] & S \arrow[r] & B \end{tikzcd}, \begin{tikzcd} C \times T \arrow[r] & T \arrow[r] & D \end{tikzcd}) \]
is mapped to the set of linear maps $m$ such that the following diagram commutes. \begin{center} \begin{tikzcd} A \times S \arrow[d,"f \times m"'] \arrow[r] & S \arrow[d,"m"] \arrow[r] & B\arrow[d,"g"]\\ C \times T \arrow[r] & T \arrow[r] & D \end{tikzcd} \end{center} We send the morphism \[
\begin{tikzcd} (A,B) \arrow[d,"\varphi"'] \arrow[r,tick,"{(f,g)}"] & (C,D) \arrow[d,"\psi"]\\ (A',B') \arrow[r,tick,"{(f',g')}"'] & (C',D') \end{tikzcd} \in \mathscr{W}_1 \] to the natural transformation sending \begin{center} \begin{tikzcd} A \times S \arrow[d,"f \times m"'] \arrow[r] & S \arrow[d,"m"] \arrow[r] & B\arrow[d,"g"]\\ C \times T \arrow[r] & T \arrow[r] & D \end{tikzcd} \end{center} to \begin{center} \begin{tikzcd} A' \times S \arrow[d,"f' \times m"'] \arrow[r] & S \arrow[d,"m"] \arrow[r] & B\arrow[d,"g'"]\\ C' \times T \arrow[r] & T \arrow[r] & D \end{tikzcd} \end{center} by applying $\varphi$ and $\psi$ to the corresponding inputs and outputs. This is a valid transformation because of how the squares in $\mathscr{W}_1$ commute.
Since the chosen profunctors for $\CS(U(x))$ align with the hom functor of $\CS(x)$ for any $x \in \mathscr{W}_0$, we can take $\CS_U$ to be the identity natural transformation. We define $\CS_{\odot}$ by assigning to each element $(f,g) \times (f',g') \in \mathscr{W}_1 \times_{\mathscr{W}_0} \mathscr{W}_1$ the morphism in $\Prof_1$ sending \[ \begin{tikzcd} A \times S \arrow[d] \arrow[r] & S \arrow[d,"f"] \arrow[r] & B \arrow[d]\\ A' \times S' \arrow[d] \arrow[r] & S' \arrow[d,"g"] \arrow[r] & B'\arrow[d]\\ A'' \times S'' \arrow[r] & S'' \arrow[r] & B'' \end{tikzcd} \] to \[ \begin{tikzcd} A \times S \arrow[d] \arrow[r] & S \arrow[d, "g \circ f"]\arrow[r] & B \arrow[d]\\ A'' \times S'' \arrow[r] & S'' \arrow[r] & B' \end{tikzcd} \] where each of the dynamical systems above are the representatives of their isomorphism classes. This gives us a transformation because the equivalence classes generated by profunctor composition give the same result under our dynamical system morphism composition. Associativity follows from the fact that function composition is associative and unitality follows from the fact that our profunctors and composition transformation work the same way as unit profunctors and their composition.
Now we show that $\CS$ is a lax monoidal double functor. First, we show that $\CS_0$ and $\CS_1$ are lax monoidal functors. We define $\epsilon$ to be the untial morphism of $\CS_0$ sending the trivial category to the trivial dynamical system $\begin{tikzcd} * \times * \arrow[r] & * \arrow[r] & *\end{tikzcd}$ in $\CS(*,*)$ and we define $\delta$ be the unital morphism of $\CS_1$ sending the trivial profunctor to the trivial hom from the trivial dynamical system to itself. For monoidal natural transformations, we define $\mu$ to be the monoidal natural transformation of $\CS_0$ which sends \[ (\begin{tikzcd} A_1 \times S_1 \arrow[r] & S_1 \arrow[r] & B_1 \end{tikzcd},\begin{tikzcd}A_2 \times S_2 \arrow[r] & S_2 \arrow[r] & B_2 \end{tikzcd}) \]
to \[ \begin{tikzcd} (A_1 \times A_2) \times (S_1 \times S_2) \arrow[r] & S_1 \times S_2 \arrow[r] & B_1 \times B_2\end{tikzcd}. \] and define $\nu$ to be the monoidal natural transformation of $\CS_1$ which sends an ordered pair of dynamical system morphisms to the product of the morphisms. Since everything works componentwise, the associativity and unitality constraints for our lax monoidal functors hold. We already know that $L\CS_1 = \CS_0 L$ and $R\CS_1 = \CS_0 R$ as functors so we just need to check the lax monoidal properties. All of these functors send ordered pairs of categories of dynamical systems to the category of product dynamical systems in the same way, so the equalities hold. Finally, we need to check that the composition constraints are monoidal natural transformations. The unital transformation is indeed monoidal since it is the identity transformation. For the composition transformation, we need to show that the following diagram commutes. \[ \begin{tikzcd} (\CS(M_1) \odot \CS(N_1)) \otimes (\CS(M_2) \odot \CS(N_2)) \arrow[r] \arrow[d] & \CS(M_1 \odot N_1) \otimes \CS(M_2 \odot N_2) \\ \CS(M_1 \otimes M_2) \odot \CS(N_1 \otimes N_2) \arrow[r] & \CS((M_1 \otimes M_2) \odot (N_1 \otimes N_2)) \end{tikzcd} \] This can be seen from the fact that both paths send the pair of composed dynamical system morphisms \[ \left(\begin{tikzcd} A \times S \arrow[d] \arrow[r] & S \arrow[d] \arrow[r] & B \arrow[d]\\ A' \times S' \arrow[d] \arrow[r] & S' \arrow[d] \arrow[r] & B'\arrow[d]\\ A'' \times S'' \arrow[r] & S'' \arrow[r] & B'' \end{tikzcd},\begin{tikzcd} C \times T \arrow[d] \arrow[r] & T \arrow[d] \arrow[r] & D \arrow[d]\\ C' \times T' \arrow[d] \arrow[r] & T' \arrow[d] \arrow[r] & D'\arrow[d]\\ C'' \times T'' \arrow[r] & T'' \arrow[r] & D'' \end{tikzcd}\right) \] to the morphism \[ \begin{tikzcd} (A \times C) \times (S \times T) \arrow[d] \arrow[r] & S \times T \arrow[d] \arrow[r] & B \times D \arrow[d]\\ (A'' \times C'') \times (S'' \times T'') \arrow[r] & S'' \times T'' \arrow[r] & B'' \times D'' \end{tikzcd} \] One path does this by taking the product morphism first, then composing, while the other path composes the morphisms and then takes the product. Since both paths give the same result, we see that $\CS$ is a lax monoidal double functor.
We also define the category of four-step dynamical systems. A four-step dynamical system is a discrete dynamical system along with a commuting diagram \[ \begin{tikzcd} A \times S \arrow[d] \arrow[r] & S \arrow[d] \arrow[r] & B \arrow[d]\\ * \times \{1,2,3,4\} \arrow[r] & \{1,2,3,4\} \arrow[r] & * \\ \end{tikzcd} \] where the bottom dynamical system behaves by cyclically iterating through all four elements. We will denote this bottom dynamical system by $C_4$. Another way we can portray this dynamical system is \[ \begin{tikzcd} A \times (S_1 \sqcup S_2 \sqcup S_3 \sqcup S_4) \arrow[r] & (S_1 \sqcup S_2 \sqcup S_3 \sqcup S_4) \arrow[r] & B \end{tikzcd} \] where each $S_i$ is mapped to $i$. A morphism between two four-step dynamical systems is a map $f: S \to T$ such that $S_i$ maps into $T_i$ and the diagram \[ \begin{tikzcd} A \times S \arrow[d,"f \times m"'] \arrow[r] & S \arrow[d,"m"] \arrow[r] & B\arrow[d,"g"]\\ C \times T \arrow[r] & T \arrow[r] & D \end{tikzcd} \] commutes. As in the case for $\CS$, we describe $(\DS/ C_4)(A,B)$ to be the category of isomorphism classes of four-step dynamical systems with input set $A$ and output set $B$. Then we define $(\DS/ C_4)_0$ to be the category with objects of the form $(\DS / C_4)(A,B)$ and morphisms being wiring diagrams, and we define $(\DS/ C_4)_1$ to be the category of profunctors describing the morphisms of dynamical system classes with different inputs or outputs. Like we did for $\CS$, we take $(\DS / C_4)_U$ to be the identity and $(\DS / C_4)_\odot$ to be the composition natural transformation. The proof that these give a lax monoidal double functor is very similar to the proof for $\CS$.
\section{Runge-Kutta as a Double Transformation} \label{runge-kutta}
\begin{definition} The Runge-Kutta map $\RK_h(A,B): \CS(A,B) \to \DS/ C_4(A,B)$ is defined by sending a dynamical system \begin{center} \begin{tikzcd} A \times S \arrow[r] & S \arrow[r] & B \end{tikzcd} \end{center} to the dynamical system \begin{center} \begin{tikzcd} A \times (S \sqcup S^2 \sqcup S^3 \sqcup S^4) \arrow[d] \arrow[r] & S \sqcup S^2 \sqcup S^3 \sqcup S^4 \arrow[d] \arrow[r] & B\arrow[d]\\ * \times \{1,2,3,4\} \arrow[r] & \{1,2,3,4\} \arrow[r] & * \end{tikzcd} \end{center} where \begin{align*} {f'}^{upd}(a,s) &= (a,s,f^upd(s))\\ {f'}^upd(a,s,k_1) &= (a,s,k_1,f^upd(s + \frac{h}{2} k_1))\\ {f'}^upd(a,s,k_1,k_2) &= (a,s,k_1,k_2,f^upd(s + \frac{h}{2} k_2))\\ {f'}^upd(a,s,k_1,k_2,k_3) &= (a,s + \frac{h}{6} k_1 + \frac{h}{3} k_2 + \frac{h}{3} k_3 + \frac{h}{6} f^upd(a,s + hk_3)) \end{align*} and \begin{align*} f'^rdt(a,s) &= s\\ f'^rdt(a,s,k_1) &= s + \frac{h}{2} k_1\\ f'^rdt(a,s,k_1,k_2) &= s + \frac{h}{2}k_2\\ f'^rdt(a,s,k_1,k_2,k_3) &= s + hk_3\\ \end{align*} \end{definition}
\begin{lemma} Let $X \in \CS(A,B)$ be a dynamical system. Then for any wiring diagram $\varphi: (A,B) \to (C,D)$, we have \[ \RK_h(\varphi(X)) = \varphi(\RK_h(X)) \] \end{lemma}
\begin{proof} Let $X$ be the dynamical system \[ \begin{tikzcd} A \times S \arrow[r,"f^{upd}"] & S \arrow[r,"f^{out}"] & B \end{tikzcd} \] Then $\varphi(X)$ is the dynamical system \begin{tikzcd} C \times S \arrow[r,"g^{upd}"] & S \arrow[r,"g^{out}"] & B \end{tikzcd} where $g^{upd}(c,s) = f^{upd}(\varphi^{in}(c,f^{out}(s)),s)$ and $g^{out} = \varphi^{out}(f^{out}(s))$. We then compute that $\RK_h(\varphi(X))$ is the dynamical system \[ \begin{tikzcd} C \times S \sqcup S^2 \sqcup S^3 \sqcup S^4 \arrow[r,"h_1^{upd}"] & S \sqcup S^2 \sqcup S^3 \sqcup S^4 \arrow[r,"h_1^{out}"] & D \end{tikzcd} \] with \begin{align*} h_1^{upd}(c,s) &= (c,s,g^{upd}(c,s)) = (c,s,f^{upd}(\varphi^{in}(c,f^{out}(s)),s))\\ h_1^{upd}(c,s,k_1) &= (c,s,k_1,g^{upd}(c,s + \frac{h}{2} k_1)) = (c,s,k_1,f^{upd}(\varphi^{in}(c,f^{out}(s + \frac{h}{2} k_1)),s + \frac{h}{2} k_1))\\ h_1^{upd}(c,s,k_1,k_2) &= (c,s,k_1,k_2,g^{upd}(c,s + \frac{h}{2} k_2)) = (c,s,k_1,k_2,f^{upd}(\varphi^{in}(c,f^{out}(s + \frac{h}{2} k_2)),s + \frac{h}{2} k_2))\\ h_1^{upd}(c,s,k_1,k_2,k_3) &= (c,s + \frac{h}{6} k_1 + \frac{h}{3} k_2 + \frac{h}{3} k_3 + \frac{h}{6} g^{upd}(c,s + hk_3)) \\ &= (c,s + \frac{h}{6} k_1 + \frac{h}{3} k_2 + \frac{h}{3} k_3 + \frac{h}{6} f^{upd}(\varphi^{in}(c,f^{out}(s + hk_3)),s + hk_3) \end{align*} \end{proof}
Finally, we restate Theorem 1 and prove it.
\rkthm*
\begin{proof}
We first show that $(RK_h)_0$ defines a natural transformation $\CS_0 \to (\DS/ C_4)_0$. For this, we need to show that $RK_h$ gives a functor from $\CS_0(A,B)$ to $(\DS/ C_4)(A,B)$. This fact follows from the above lemma and the fact that we are sending the $\CS$-morphism \[ \begin{tikzcd} A \times S \arrow[d,"\varphi \times m"'] \arrow[r,"f^{upd}"] & S \arrow[d,"m"]\arrow[r,"f^{out}"] & B \arrow[d,"\varphi"]\\ A \times S' \arrow[r,"f'^{upd}"'] & S' \arrow[r,"f'^{out}"'] & B\\ \end{tikzcd} \] to the $\DS/ C_4$-morphism \[ \begin{tikzcd} A \times (S \sqcup S^2 \sqcup S^3 \sqcup S^4) \arrow[d,"{\varphi \times (m \sqcup m^2 \sqcup m^3 \sqcup m^4)}"'] \arrow[r] & S \sqcup S^2 \sqcup S^3 \sqcup S^4 \arrow[d,"m \sqcup m^2 \sqcup m^3 \sqcup m^4"] \arrow[r] & B\arrow[d,"\varphi"]\\ A \times (S' \sqcup S'^2 \sqcup S'^3 \sqcup S'^4) \arrow[r] & S' \sqcup S'^2 \sqcup S'^3 \sqcup S'^4 \arrow[r] & B \end{tikzcd} \] Since the morphisms are component-wise, we have that the functoral properties hold. Now we just need to show that these morphisms in $\Prof$ behave naturally with $\CS$ and $\DS / C_4$, but this follows from our above Lemma, so we can see that $(RK_h)_0$ gives us a natural transformation between $\CS_0$ and $(\DS / C_4)_0$.
We then show that $(RK_h)_1$ gives a natural transformation $\CS_1$ to $(\DS / C_4)_1$. This is not hard to check since we are sending \[ \begin{tikzcd} A \times S \arrow[d,"\varphi \times m"'] \arrow[r,"f^{upd}"] & S \arrow[d,"m"]\arrow[r,"f^{out}"] & B \arrow[d,"\varphi"]\\ A' \times S' \arrow[r,"f'^{upd}"] & S' \arrow[r,"f'^{out}"] & B'\\ \end{tikzcd} \] to \[ \begin{tikzcd} A \times (S \sqcup S^2 \sqcup S^3 \sqcup S^4) \arrow[d,"{\varphi \times (m \sqcup m^2 \sqcup m^3 \sqcup m^4)}"'] \arrow[r] & S \sqcup S^2 \sqcup S^3 \sqcup S^4 \arrow[d,"m \sqcup m^2 \sqcup m^3 \sqcup m^4"] \arrow[r] & B\arrow[d,"\varphi"]\\ A' \times (S' \sqcup S'^2 \sqcup S'^3 \sqcup S'^4) \arrow[r] & S' \sqcup S'^2 \sqcup S'^3 \sqcup S'^4 \arrow[r] & B' \end{tikzcd} \] and all the morphisms are component-wise. One can check the diagrams in the definition commute since our functions act component-wise and also compose component-wise.
Finally, we just need to show that these natural transformations are monoidal so that we have a lax monoidal double transformation. For this, we need the natural transformation to commute with our monoidal transformations. This is true for both natural transformations, since multiplying the dynamical systems and then applying Runge-Kutta gives an isomorphic dynamical system as applying Runge-Kutta to two dynamical systems and then merging the two systems. The only difference is that the order of the products in the discrete sum may be different, but since our maps work component-wise, we still get isomorphic results, so our natural transformation indeed commutes with our monoidal transformations. Our last requirement, the unital requirement for the natural transformations, is easy to check, so Runge-Kutta indeed gives a monoidal double transformation. \end{proof}
\end{document} | arXiv |
\begin{document}
\title{Non-uniform dependence on initial data\\ for
the CH equation on the line
}
\author{{\it A. Alexandrou Himonas \& Carlos Kenig}}
\date{September 23, 2010 [Corrected version of: A. Himonas and C. Kenig, {\it Non-uniform dependence on initial data for the CH equation on the line,} Differential and Integral Equations, Vol. 22, No. 3-4 (2009) pp. 201-224.]}
\keywords{CH equation, integrable, non-periodic, Cauchy problem, Sobolev spaces, well-posedness, non-uniform dependence on initial data,
approximate solutions.}
\subjclass[2000]{Primary: 35Q53}
\begin{abstract} For $s>3/2$ two sequences of CH solutions living in a bounded set of the Sobolev space $H^s(\mathbb{R})$ are constructed, whose distance at the initial time is converging to zero while at any later time is bounded below by a positive constant. This implies that the solution map of the CH equation is not uniformly continuous in $H^s(\mathbb{R})$. \end{abstract}
\maketitle \markboth{ Non-uniform dependence for CH equation on the line}
{Alex Himonas and Carlos Kenig}
\parindent0in \parskip0.1in
\section{Introduction}
\setcounter{equation}{0}
We consider the Cauchy problem for the Camassa-Holm equation (CH) \begin{equation} \label{CH} \partial_t u + u\partial_x u + \partial_x \Big( 1 - \partial_x^2 \Big)^{-1}
\Big[ u^2 + \frac{1}{2}(\partial_x u)^2 \Big] = 0, \end{equation} \begin{equation} \label{CH-data} u(x, 0) = u_0 (x),\,\, \ x \in \mathbb{R}, \;\; \ t \in \mathbb{R}. \end{equation} This equation appeared initially in the context of hereditary symmetries studied by Fuchssteiner and Fokas \cite{ff}. However, it was written explicitly as a water wave equation by Camassa and Holm \cite{ch}, who showed that CH is biHamiltonian and studied its ``peakon" solutions. Since then CH has been rederived in various ways by Misio\l ek \cite{mi}, Johnson \cite{j}, Constantin and Lannes \cite{cl}, and Ionescu-Kruse \cite{i}.
Well-posedness on the line was first established by Li and Olver. In \cite{lo} they showed that if $s>3/2$ then CH is locally well-posed in $H^s(\mathbb{R})$ with solutions depending {\it continuously} on initial data. The proof was based on a regularization technique similar to the one used by Bona and Smith for the KdV equation \cite{bs}. A similar result has also been proved by Rodriguez-Blanco \cite{rb} by using Kato's theory for quasilinear equations \cite{k}. Moreover, global well-posedness in $H^1(\mathbb{R})$ for the CH equation has been studied by Bressan and Constantin in \cite{bc}. However, well-posednes of CH in $H^s(\mathbb{R})$ for $s\in (1, 3/2]$ remains an open question.
In this paper, we show that dependence of CH solutions on initial data in Sobolev spaces can not be better than continuous. More precisely, we prove the following result.
\begin{theorem} \label{CH-non-unif-dependence}
If $s>3/2$ then the flow map $u_0 \to u(t)$ for the CH equation is not uniformly continuous from any bounded set of $H^s(\mathbb{R})$ into $C([-T, T]; H^s(\mathbb{R}))$. More precisely, there exist two sequences of CH solutions $u_n(t)$ and $v_n(t)$ in $C([-T, T]; H^s(\mathbb{R}))$ such that
\begin{equation} \label{H-s-bdd}
\| u_n(t) \|_{H^s(\mathbb{R})}
+
\| v_n(t) \|_{H^s(\mathbb{R})} \lesssim
1, \end{equation}
\begin{equation}
\label{zero-limit-at-0}
\lim_{n\to\infty}
\| u_n(0) - v_n(0)
\|_{H^s(\mathbb{R})} = 0, \end{equation}
and
\begin{equation}
\label{bdd-away-from-0}
\liminf_{n\to\infty}
\| u_n(t) - v_n(t)
\|_{H^s(\mathbb{R})} \gtrsim
\sin t,
\quad
|t|<T\le 1. \end{equation}
\end{theorem}
For $s=1$ Theorem \ref{CH-non-unif-dependence} has been already proved by Himonas, Misio\l ek and Ponce in \cite{hmp} by using traveling wave solutions that are smooth except at finitely many points at which the slope is $\pm \infty$ (cuspons). Also, in \cite{hmp} the analogues result for the periodic CH was proved.
For $s\ge 2$ non-uniform continuity of the CH solution map in the periodic case was established in \cite{hm} using high frequency traveling wave solutions and following an approach similar to the one used in \cite{kpv} by Kenig, Ponce and Vega. We mention that this method does not work in the non-periodic case because the traveling wave solutions do not live in $H^s(\mathbb{R})$.
Also, it is worth mentioning the following implication of Theorem \ref{CH-non-unif-dependence} concerning ways for proving local well-posedness for CH.
The fact that the data-to-solution map is not uniformly continuous from any bounded set of $H^s(\mathbb{R})$ into $C([-T, T]; H^s(\mathbb{R}))$
tells us that local well-posedness of CH in $H^s$ cannot be established by a solely contraction principle argument.
The proof of Theorem \ref{CH-non-unif-dependence} is based on the method of approximate solutions used by Koch and Tzvetkov in \cite{kt} and Christ, Colliander and Tao in \cite{cct}. The idea is to choose approximate solutions consisting of a low-frequency part and a high-frequency part, which satisfy the three conclusions of Theorem \ref{CH-non-unif-dependence}. Furthermore, solving the Cauchy problem with initial data given by evaluating the approximate solutions at $t = 0$ must yield actual solutions whose difference from the approximate solutions is negligible.
The literature about CH is extensive. For some other results about this equation we refer the reader to McKean \cite{mc}, Constantin and Strauss \cite{cs}, Himonas, Misio\l ek, Ponce and Zhou \cite{hmpz}, and Molinet's survey article \cite{mo}.
The paper is structured as follows. In section 2 we recall the well-posedness result of Li and Olver and use it to prove the basic energy estimate (see \eqref{CH-diff-ineq}) from which we derive a lower bound for the lifespan of the solution as well an estimate of the $H^s$ norm of the solution $u(t)$ in terms of the the $H^s$ norm of the initial data $u_0$ (see Proposition \ref{Lifespan-u-size}).
In section 3 we construct approximate solutions consisting of a low-frequance part and a high-frequency part,
and compute the error.
In section 4 we estimate the $H^1$-norm of this error.
In section 5 we solve the Cauchy problem for the CH equation with initial data given by the approximate solutions evaluated at time zero, and estimate the $H^1$-norm of the difference beween actual and approximate solutions (see Lemma \ref{CH-differ-H1-est-lem}).
Finally, in section 6 we conclude with the proof of Theorem \ref{CH-non-unif-dependence}.
\section{Local well-posedness} \setcounter{equation}{0}
We shall need the following well-posedness result,
proved in \cite{lo} using a regularization technique.
\begin{theorem} \label{CH-wp} [Li-Olver] Suppose that the function $u_0(x)$ belongs to the Sobolev space $H^s(\mathbb{R})$ for some $s >3/2$. Then there is a $T>0$, which depends
only on $\|u_0\|_{H^s}$, such that there exists a unique function $u(x, t)$ solving the Cauchy problem \eqref{CH}--\eqref{CH-data} in the sense of distributions with $u \in C([0, T]; H^s)$. When $s\ge 3$, $u$ is also a classical solution to \eqref{CH}--\eqref{CH-data}. Moreover, the solution $u$ depends continuously on the initial data $u_0$ in the sense that the mapping of the initial data to the solution is continuous from the Sobolev space $H^s$ to the space $C([0, T]; H^s)$. \end{theorem}
Using the information provided by Theorem \ref{CH-wp}, next we shall prove an explicit estimate for the time of existence $T$ of the solution $u(t)$. Also, we will show that at any time $t$ in the time interval $[0, T]$ the $H^s$ norm of the solution $u(t)$ is dominated by the $H^s$ norm of the initial data $u_0$.
\begin{proposition} \label{Lifespan-u-size} Let $s>3/2$. If $u$ is the solution of the Cauchy problem \eqref{CH}--\eqref{CH-data} described in Theorem \ref{CH-wp} then its lifespan (the maximal existence time)
is greater than
\begin{equation}
\label{Lifespan-size}
T
\doteq
\frac{1}{2c_s}
\frac{1}{ \|u_0 \|_{H^s(\mathbb{R})}},
\end{equation}
where $c_s$ is a constant depending only on $s$.
Also, we have that
\begin{equation}
\label{u-u0-Hs-bound}
\|u(t)\|_{H^s(\mathbb{R}))}
\le
2
\|u_0 \|_{H^s(\mathbb{R})},
\quad
0\le t \le T.
\end{equation}
\end{proposition}
\noindent {\bf Proof.}
The derivation of the lower bound for the lifespan \eqref{Lifespan-size} and the solution size estimate \eqref{u-u0-Hs-bound} is based on the following differential inequality for the solution $u$
\begin{equation} \label{CH-diff-ineq} \frac 12 \frac{d}{dt}
\|u(t)\|_{H^{s}(\mathbb{R})}^2 \le c_s
\|u(t)\|_{H^{s}(\mathbb{R})}^3,
\quad
0\le t \le T. \end{equation}
This inequality can be extracted from the proof of Theorem \ref{CH-wp} in \cite{lo} using the energy estimate (3.6) proved for the following regularization $$ \partial_tu - \partial_x^2\partial_tu +\varepsilon\partial_x^4\partial_tu+3u\partial_xu -\partial_xu\partial_x^2 u-u\partial_x^3u=0 $$ of the CH equation and letting $\varepsilon$ go to zero. Here, we shall prove inequality \eqref{CH-diff-ineq} by following the approach used for quasilinear symmetric hyperbolic systems in Taylor \cite{t1}.
For any $s\in \mathbb{R}$ let $D^s=(1-\partial_x^2)^{s/2}$ be the operator defined by
$$ \widehat{D^s f}(\xi) \doteq (1 + \xi^2)^{s/2} \widehat{f}(\xi), $$
where $ \widehat{f}$ is the Fourier transform
$$ \widehat{f}(\xi) = \int_\mathbb{R} e^{-ix\xi}\,\, f(x) dx. $$
Then for $f \in H^s(\mathbb{R})$ we have $$
\|f\|_{H^s(\mathbb{R})} =
\int_\mathbb{R} \big(1 + \xi^2\big)^{s}| \widehat{f}(\xi)|^2 \frac{d\xi}{2\pi}
=
\|D^sf\|_{L^2(\mathbb{R})}. $$ Now let $u$ be the solution to the Cauchy problem \eqref{CH}--\eqref{CH-data}, which according to Theorem \ref{CH-wp} belongs in $ C([0, T]; H^s)$. Solving \eqref{CH} for $\partial_t u$ we obtain
\begin{equation} \label{CH-ut-form} \partial_t u = -u\partial_x u - D^{-2} \partial_x
\Big[ u^2 + \frac{1}{2}(\partial_x u)^2 \Big]. \end{equation}
Starting with \eqref{CH-ut-form} we want to derive the energy estimate in $H^s$
expressed by inequality \eqref{CH-diff-ineq}. We can form $\frac{d}{dt}\|u\|_{H^s(\mathbb{R})}^2$ by applying formally the operator $D^s$
to both sides of \eqref{CH-ut-form}, then multiply the resulting equation by $D^su$ and integrate it with respect to $x$. Note that since $u\in H^s$ the second term in the right-hand side of \eqref{CH-ut-form} is in $H^s$ too. However, the first term, that is the product $u\partial_x u$ is only in $H^{s-1}$. To deal with this problem we will replace \eqref{CH-ut-form} by its molified smooth version
\begin{equation} \label{CH-ut-form-moli} \partial_t J_\varepsilon u = -J_\varepsilon(u\partial_x u) - D^{-2} \partial_x
\Big[ J_\varepsilon(u^2) + \frac{1}{2}J_\varepsilon[(\partial_x u)^2] \Big], \end{equation}
where for each $\varepsilon\in (0, 1]$ the operator $J_\varepsilon$ is the Friedrichs mollifier defined by
\begin{equation} \label{Fried-moli} J_\varepsilon f(x)= j_\varepsilon \ast f (x). \end{equation}
Here $j(x)$ is a $C^\infty$ function supported in the interval $[-1, 1]$ such that $j(x)\ge 0$, $\int_\mathbb{R} j(x)dx =1$ and
$$ j_\varepsilon(x)= \frac{1}{\varepsilon} \, j\big(\frac{x}{\varepsilon}\big). $$
Applying the operator $D^s$ to both sides of \eqref{CH-ut-form-moli}, then multiplying the resulting equation by $D^s J_\varepsilon u$ and integrating it for $x\in\mathbb{R}$ gives
\begin{equation} \label{CH-moli-int} \begin{split} \frac 12
\frac{d}{dt} \|J_\varepsilon u \|_{H^s}^2 &= -
\int_\mathbb{R}
D^sJ_\varepsilon(u\partial_x u) \cdot D^sJ_\varepsilon u \, dx -
\int_\mathbb{R}
D^{s-2} \partial_x J_\varepsilon(u^2) \cdot D^sJ_\varepsilon u \, dx
\\ &-
\frac 12
\int_\mathbb{R}
D^{s-2} \partial_x J_\varepsilon[(\partial_x u)^2] \cdot D^sJ_\varepsilon u \, dx.
\end{split} \end{equation}
In what follows next we use the fact that $D^s$ and $J_\varepsilon$ commute and that $J_\varepsilon$ satisfies the properties
\begin{equation} \label{J-e-inner-prod-property} (J_\varepsilon f, g)_{L^2}=( f, J_\varepsilon g)_{L^2}, \end{equation}
and
\begin{equation} \label{Je-u-Hs}
\| J_\varepsilon u \|_{H^s}
\le
\| u \|_{H^s}. \end{equation}
\noindent {\bf Estimating the Burgers term.} To estimate the first integral in the right-hand side of \eqref{CH-moli-int} we write it as follows
\begin{equation} \label{int1-est-calc1} \begin{split}
\int_\mathbb{R}
D^sJ_\varepsilon(u\partial_x u) \cdot D^sJ_\varepsilon u \, dx
&=
\int_\mathbb{R}
D^s(u\partial_x u) \cdot J_\varepsilon D^sJ_\varepsilon u \, dx
\\
&=
\int_\mathbb{R} \big[ D^s(u\partial_x u) - u D^s (\partial_xu) \big]
J_\varepsilon D^sJ_\varepsilon u \, dx
\\
&+
\int_\mathbb{R}
u D^s (\partial_xu) \cdot J_\varepsilon D^sJ_\varepsilon u \, dx. \end{split} \end{equation}
Now, we estimate the first term in the right-hand side of \eqref{int1-est-calc1}. Applying the Cauchy-Schwarz inequality gives
\begin{equation} \label{int1-est-calc2} \begin{split}
\Big|
\int_\mathbb{R} \big[ D^s(u\partial_x u) - u D^s (\partial_xu) \big]
J_\varepsilon D^sJ_\varepsilon u \, dx
\Big| &\le
\| D^s(u\partial_x u) - u D^s (\partial_xu)
\|_{L^2}
\| J_\varepsilon D^sJ_\varepsilon u
\|_{L^2} \\ &\le
\| D^s(u\partial_x u) - u D^s (\partial_xu)
\|_{L^2}
\|
u
\|_{H^s} \\ &\le
2c_s \| \partial_x u \|_{L^\infty}
\| u \|_{H^s}^2, \end{split} \end{equation}
where the last step follows from the estimate
\begin{equation} \label{int1-est-calc3}
\| D^s(u\partial_x u) - u D^s (\partial_xu) \|_{L^2}
\le
2c_s \| \partial_x u \|_{L^\infty}
\| u \|_{H^s}, \end{equation} which we prove below by using the following Kato-Ponce commutator estimate \cite{kp} (see also Ionescu and Kenig \cite{ik}).
\begin{lemma}
\label{KP-lemma}
[Kato-Ponce]
If $s>0$ then there is $c_s>0$ such that for any $f, g \in H^s(\mathbb{R})$
\begin{equation} \label{KP-com-est}
\| D^{s} \big(fg) - f D^s g\|_{L^2}
\le
c_s\big(
\| D^{s}f \|_{L^2} \| g \|_{L^\infty}
+
\| \partial_xf \|_{L^\infty} \| D^{s-1}g \|_{L^2}
\big).
\end{equation}
\end{lemma}
In fact, applying this estimate with $f=u$ and $g=\partial_xu$ gives
\begin{equation} \label{int1-est-calc4} \begin{split}
\| D^s(u\partial_x u) - u D^s (\partial_xu) \|_{L^2} &
\le
c_s\big(
\| D^{s}u \|_{L^2} \| \partial_x u \|_{L^\infty}
+
\| \partial_xu \|_{L^\infty} \| D^{s-1}\partial_x u \|_{L^2}
\big)
\\
& \le
c_s \| \partial_x u \|_{L^\infty}
\big(
\| D^{s}u \|_{L^2}
+
\| D^{s}u \|_{L^2}
\big)
\\
& \le
2c_s \| \partial_x u \|_{L^\infty}
\| u \|_{H^s}, \end{split}
\end{equation}
which is the desired estimate \eqref{int1-est-calc3}.
Next, we estimate the second integral in the right-hand side of \eqref{int1-est-calc1}.
Note if there were no $J_\varepsilon$'s involved then this would have been done in
a straightforward manner as follows
\begin{equation}
\label{int1-est-calc5}
\begin{split}
\Big|
\int_\mathbb{R}
u D^s (\partial_xu) \cdot D^su \, dx
\Big|
&=
\Big|
\frac 12 \ \int_\mathbb{R}
u \partial_x\big[(D^s u)^2\big] \, dx
\Big|
\\
& =
\Big|
- \frac 12 \ \int_\mathbb{R}
\partial_xu \, (D^s u)^2 \, dx
\Big|
\\
&\le
\frac 12 \| \partial_x u \|_{L^\infty}
\| u \|_{H^s}^2.
\end{split}
\end{equation}
When the $J_\varepsilon$'s are involved the idea is the same. However, the implementation is
more technical since we need to commute $J_\varepsilon$ so that is
grouped correctly. We accomplish this as follows
\begin{equation} \label{int1-est-calc6} \begin{split}
\int_\mathbb{R}
u D^s (\partial_xu) \cdot J_\varepsilon D^sJ_\varepsilon u \, dx
&=
\int_\mathbb{R}
J_\varepsilon u D^s (\partial_xu) \cdot D^sJ_\varepsilon u \, dx
\\
&=
\int_\mathbb{R}
\Big(
[J_\varepsilon, u ] D^s (\partial_xu) + u J_\varepsilon D^s (\partial_xu)
\Big) \cdot D^sJ_\varepsilon u \, dx
\\
&=
\int_\mathbb{R}
[J_\varepsilon, u ]\partial_x D^s u
\cdot D^sJ_\varepsilon u \, dx
\\
&+
\int_\mathbb{R} u \partial_xD^s J_\varepsilon u \cdot D^sJ_\varepsilon u \, dx. \end{split} \end{equation}
Estimating the second integral of the right-hand side of \eqref{int1-est-calc6}
like we have done in \eqref{int1-est-calc5} we get
\begin{equation}
\label{int1-est-calc7}
\begin{split}
\Big|
\int_\mathbb{R} u \partial_xD^s J_\varepsilon u \cdot D^sJ_\varepsilon u \, dx
\Big|
&=
\Big|
\frac 12 \ \int_\mathbb{R}
u \partial_x\big[(D^s J_\varepsilon u)^2\big] \, dx
\Big|
\\
& =
\Big|
- \frac 12 \ \int_\mathbb{R}
\partial_xu \, (D^s J_\varepsilon u)^2 \, dx
\Big|
\\
&\le
\frac 12 \| \partial_x u \|_{L^\infty}
\| J_\varepsilon u \|_{H^s}^2
\\
&\le
\frac 12 \| \partial_x u \|_{L^\infty}
\| u \|_{H^s}^2.
\end{split}
\end{equation}
For estimating the first integral of the right-hand side of \eqref{int1-est-calc6}
we apply the Cauchy-Schwarz inequality and we have
\begin{equation} \label{int1-est-calc8}
\begin{split}
\Big|
\int_\mathbb{R}
[J_\varepsilon, u ]\partial_x D^s u
\cdot D^sJ_\varepsilon u \, dx
\Big|
&\le
\| [J_\varepsilon, u ]\partial_x D^s u \|_{L^2}
\| D^sJ_\varepsilon u \|_{L^2}
\\
&\le
\| [J_\varepsilon, u ]\partial_x D^s u \|_{L^2}
\| u \|_{H^s}
\\
&\le
c
\| \partial_x u \|_{L^\infty}
\| u \|_{H^s}^2,
\end{split}
\end{equation}
where the last step of the above inequality is justified by the following result.
\begin{lemma}
\label{Je-u-com} Let $u(x)$ be a function such that $\| \partial_x u \|_{L^\infty} <\infty$. Then, there is $c>0$ such that for any $f \in L^2(\mathbb{R})$ we have
\begin{equation} \label{Je-u-com-L2-est}
\| [J_\varepsilon, u ]\partial_x f \|_{L^2}
\le
c
\| \partial_x u \|_{L^\infty}
\| f \|_{L^2}.
\end{equation}
\end{lemma}
{\bf Proof.} We have
\begin{equation} \label{Je-u-com-calc1} \begin{split}
[J_\varepsilon, u ]\partial_x f (x)
&=
J_\varepsilon( u \partial_x f )(x) - u J_\varepsilon(\partial_x f )(x)
\\
&=
j_\varepsilon \ast ( u \partial_x f )(x)
-
u(x) (j_\varepsilon \ast \partial_x f) (x)
\\
&=
\int_\mathbb{R}
j_\varepsilon(x-y) u(y) f'(y) \, dy
-
u(x) \int_\mathbb{R}
j_\varepsilon(x-y)f'(y) \, dy
\\
&=
\int_\mathbb{R}
\frac{1}{\varepsilon}
j\big(\frac{x-y}{\varepsilon}\big) \big[u(y)-u(x)\big] f'(y) \, dy.
\end{split}
\end{equation}
Integrating by parts and using the mean value theorem gives
\begin{equation} \label{Je-u-com-calc2} \begin{split}
[J_\varepsilon, u ]\partial_x f (x)
&=
-
\int_{\mathbb{R}}
\frac{1}{\varepsilon}
j\big(\frac{x-y}{\varepsilon}\big) u'(y) f(y) \, dy
\\
&
+
\int_{\mathbb{R}}
\frac{1}{\varepsilon^2}
j' \big(\frac{x-y}{\varepsilon}\big) \big[u(y)-u(x)\big] f(y) \, dy
\\
&=
-
\int_{|y-x|<\varepsilon}
\frac{1}{\varepsilon}
j\big(\frac{x-y}{\varepsilon}\big) u'(y) f(y) \, dy
\\
&+
\int_{|y-x|<\varepsilon}
\frac{1}{\varepsilon^2}
j' \big(\frac{x-y}{\varepsilon}\big) u'(\xi(x, y))(y-x) f(y) \, dy.
\end{split}
\end{equation}
Above we have used our assumption that $j(x)$ is supported on the interval $[-1, 1]$. So,
using the bound $|(x-y)/\varepsilon| <1$ and taking absolute values we obtain that
\begin{equation} \label{Je-u-com-calc3} \begin{split}
\big| [J_\varepsilon, u ]\partial_x f (x) \big|
&\le
\| \partial_x u \|_{L^\infty}
\Big(
\int_{\mathbb{R}}
\frac{1}{\varepsilon}
j\big(\frac{x-y}{\varepsilon}\big) |f(y)| \, dy
\\
&
+
\int_{\mathbb{R}}
\frac{1}{\varepsilon}
\big| j' \big(\frac{x-y}{\varepsilon}\big) \big| |f(y)| \, dy
\Big)
\\
&=
\| \partial_x u \|_{L^\infty}
\Big(
j_\varepsilon \ast |f| (x)
+
|j_\varepsilon' |\ast |f| (x)
\Big).
\end{split}
\end{equation}
Finally, applying Young's inequality we get
\begin{equation} \label{Je-u-com-calc4} \begin{split}
\| [J_\varepsilon, u ]\partial_x f \|_{L^2}
&\le
\| \partial_x u \|_{L^\infty}
\big(
\| j_\varepsilon \|_{L^1} \| f \|_{L^2}
+
\| j_\varepsilon' \|_{L^1} \| f \|_{L^2}
\big)
\\
& =
\big(
\| j \|_{L^1} + \| j' \|_{L^1}
\big)
\| \partial_x u \|_{L^\infty}
\| f \|_{L^2},
\end{split}
\end{equation}
which gives the desired inequality \eqref{Je-u-com-L2-est}
with constant $c= \| j \|_{L^1} + \| j' \|_{L^1}$.
\hfil $\square$
Combining the inequalities \eqref{int1-est-calc1}, \eqref{int1-est-calc2}, \eqref{int1-est-calc7} and \eqref{int1-est-calc7} we obtain the following estimate for the Burgers term of the CH equation
\begin{equation} \label{Burgers-energy-est}
\Big|
\int_\mathbb{R}
D^sJ_\varepsilon(u\partial_x u) \cdot D^sJ_\varepsilon u \, dx
\Big|
\le
c_s
\| \partial_x u \|_{L^\infty}
\| u \|_{H^s}^2. \end{equation}
\noindent {\bf Estimating the nonlocal $D^{s-2} \partial_x J_\varepsilon(u^2)$.} To estimate the second integral in the right-hand side of \eqref{CH-moli-int}
we apply the Cauchy-Schwarz inequality and we get
\begin{equation} \label{int2-est-calc1} \begin{split}
\Big|
\int_\mathbb{R}
D^{s-2} \partial_x J_\varepsilon(u^2) \cdot D^sJ_\varepsilon u \, dx
\Big|
&\le
\| D^{s-2} \partial_x J_\varepsilon(u^2) \|_{L^2}
\| D^sJ_\varepsilon u \|_{L^2}
\\
&\le
\| u^2 \|_{H^{s-1}} \| u \|_{H^s}
\\
&\le
\| u^2 \|_{H^s} \| u \|_{H^s}.
\end{split} \end{equation}
Now, we use the following estimate for the Sobolev norm of a product, which can be found in Taylor \cite{t2} (see Corollary 10.6). For any $s>0$ and $1<p<\infty$ there is $C=C_{s,p}>0$ such that
\begin{equation} \label{best-Sob-product-est}
\| f g \|_{H^{s,p}}
\le
C \Big[
\| f \|_{H^{s,p}} \| g\|_{L^\infty} + \| f \|_{L^\infty} \| g\|_{H^{s,p}}
\Big].
\end{equation}
Using this result with $s=2$ and $f=g=u$ from \eqref{int2-est-calc1} we obtain that
\begin{equation} \label{nonloc-u2-energy-Sob-best}
\Big|
\int_\mathbb{R}
D^{s-2} \partial_x J_\varepsilon(u^2) \cdot D^sJ_\varepsilon u \, dx
\Big| \le 2 c_s
\| u \|_{L^\infty} \| u\|_{H^s}^2. \end{equation}
\noindent {\bf Estimating the nonlocal term $ D^{s-2} \partial_x J_\varepsilon[(\partial_x u)^2] $.}
As before, applying the Cauchy-Schwarz inequality we have
\begin{equation} \label{nonloc-ux2-energy-Sob-best} \begin{split}
\Big|
\int_\mathbb{R}
D^{s-2} \partial_x J_\varepsilon[(\partial_x u)^2] \cdot D^sJ_\varepsilon u \, dx
\Big|
&\le
\| D^{s-2} \partial_x J_\varepsilon[(\partial_x u)^2] \|_{L^2}
\| D^sJ_\varepsilon u \|_{L^2}
\\
&\le
\| (\partial_x u)^2 \|_{H^{s-1}} \| u \|_{H^s}
\\
&\le
c_s
\| \partial_x u \|_{H^{s-1}}^2 \| u \|_{H^s}
\\
& \le
2 c_s \| \partial_xu \|_{L^\infty} \| u \|_{H^s}^2,
\end{split} \end{equation}
where in the last step we used estimate \eqref{best-Sob-product-est} applied with $s$ replace by $s-1>0$ and $f=g=\partial_xu$.
Now, combining equation
\eqref{Fried-moli} and estimates
\eqref{Burgers-energy-est},
\eqref{nonloc-u2-energy-Sob-best},
\eqref{nonloc-ux2-energy-Sob-best}
we obtain the differential inequality
\begin{equation} \label{CH-moly-ineq} \frac 12 \frac{d}{dt}
\|J_\varepsilon u(t)\|_{H^s}^2 \le c_s
\| u(t) \|_{C^1}
\|u(t)\|_{H^s}^2,
\quad
0\le t \le T. \end{equation}
Next, integrating \eqref{CH-moly-ineq} from 0 to $t$, $t<T$, gives
\begin{equation} \label{CH-moly-ineq-int1} \frac 12
\|J_\varepsilon u(t)\|_{H^s}^2
-
\frac 12
\|J_\varepsilon u(0)\|_{H^s}^2 \le c_s \int_0^t
\| u(\tau) \|_{C^1}
\|u(\tau)\|_{H^s}^2\, d\tau. \end{equation}
Then, letting $\varepsilon$ go to $0$ \eqref{CH-moly-ineq-int1} gives
\begin{equation} \label{CH-moly-ineq-int2}
\frac 12
\|u(t)\|_{H^s}^2
-
\frac 12
\|u(0)\|_{H^s}^2 \le c_s \int_0^t
\| u(\tau) \|_{C^1}
\|u(\tau)\|_{H^s}^2\, d\tau. \end{equation}
Finally, from \eqref{CH-moly-ineq-int2} using Gronwall's inequality we obtain the following lemma, which summarizes our estimates thus far.
\begin{lemma} \label{CH-energy-inequality}
Let $s>3/2$ and $u \in C([0, T]; H^s)$ be the solution of the Cauchy problem \eqref{CH}--\eqref{CH-data}. Then
\begin{equation} \label{CH-energy-ineq} \frac 12 \frac{d}{dt}
\|u(t)\|_{H^s}^2 \le c_s
\| u(t) \|_{C^1}
\|u(t)\|_{H^s}^2,
\quad
0\le t \le T. \end{equation}
\end{lemma}
Since $s>3/2$ using Sobolev's inequality \begin{equation} \label{Sob-C1-ineq}
\| u(t) \|_{C^1}
\le c_s
\|u(t)\|_{H^s}, \end{equation} from \eqref{CH-energy-ineq} we obtain the desired inequality \eqref{CH-diff-ineq}.
\vskip0.1in \noindent {\bf Lifespan estimate.}
To derive an explicit formula for $T=T( \|v(0)\|_{H^{s}})$ we proceed as follows. Letting $y(t)= \|u(t)\|_{H^{s}}^2$ inequality \eqref{CH-diff-ineq} takes the form
\begin{equation} \label{energy-y-ineq} \frac 12
y^{-3/2}\frac{dy}{dt} \le c_s,
\qquad
y(0)=y_0= \|u_0\|_{H^{s}}^2. \end{equation}
Integrating \eqref{energy-y-ineq} from 0 to $t$ gives
\begin{equation} \label{energy-y-ineq-calc1} \frac{1}{\sqrt{y_0}} - \frac{1}{\sqrt{y(t)}} \le c_s t. \end{equation}
Replacing $y(t)$ with $\|u(t)\|_{H^{s}}^2$ and solving for $\|u(t)\|_{H^s}$ we obtain the formula
\begin{equation} \label{norm-u(t)-formula}
\|u(t)\|_{H^s} \le
\frac{ \|u_0\|_{H^s}}{1-c_s\|u_0\|_{H^s} t}. \end{equation}
Now, from \eqref{norm-u(t)-formula} we see that $\|u(t)\|_{H^s}^2$ is finite if
\begin{equation*} \label{Lifespan-calc1}
c_s \|u_0\|_{H^s} t<1, \end{equation*}
or
\begin{equation} \label{Lifespan-calc1} t <
\frac{1}{ c_s \|u_0\|_{H^s}}. \end{equation}
Therefore, the solution $u(t)$ to the CH Cauchy problem certainly exists for $0\le t <T_0$, where
\begin{equation} \label{CH-Lifespan} T_0 =
\frac{1}{ c_s \|u_0\|_{H^s}}. \end{equation}
\noindent {\bf Size of the solution estimate.} If we choose $T=1/2 T_0$, that is
\begin{equation} \label{T-def} T =
\frac{1}{2 c_s \|u_0\|_{H^s}}, \end{equation}
then for $0\le t \le T$ inequality \eqref{norm-u(t)-formula} gives
\begin{equation*} \label{u(t)-u(0)-bound}
\|u(t)\|_{H^{s}} \le
\frac{ \|u_0\|_{H^s}}{1-(c_s\|u_0\|_{H^s})/(2 c_s \|u_0\|_{H^s})}, \end{equation*}
or
\begin{equation} \label{u(t)-u(0)-bound}
\|u(t)\|_{H^{s}}
\le 2 \|u_0\|_{H^s}, \quad 0\le t \le T. \end{equation} This completes the proof of Proposition \ref{Lifespan-u-size}. \hfil $\square$
\vskip0.1in \noindent {\bf Remark.} Inequality \eqref{CH-energy-ineq} can be used to show that
if $u \in C([0, T]; H^s)$, $s>3/2$, is a solution of Cauchy problem \eqref{CH}--\eqref{CH-data}
such that
$ \sup_{0\le t<T}
\| u(t) \|_{C^1}
<\infty $
then $u(t)$ persists to be a solution beyond the time $T$. In particular, we can show that if the lifespan $T$ of $u$ is finite then
$ \sup_{0\le t<T}
\| u(t) \|_{C^1} = \infty $
(see Theorem 6.2 in \cite{lo}).
\section{Construction of approximate solutions }
\setcounter{equation}{0}
Here we shall construct a two-parameter family of approximate solutions
$u^{\omega, \lambda}=u^{\omega, \lambda}(x, t)$,
each member of which consists of two parts, that is
\begin{equation} \label{approx-sln} u^{\omega, \lambda} = u_\ell + u^h. \end{equation}
The high frequency part $u^h$ is given by
\begin{equation} \label{high-frequency-approx-sln} u^h=u^{h,\omega, \lambda}(x, t)
=
\lambda^{-\delta/2 - s}
\varphi (\frac{x}{\lambda^{\delta}})
\cos (\lambda x-\omega t),
\end{equation}
and is not a solution of CH.
Here $\varphi$ is a $C^{\infty}$ function such that
\begin{equation} \label{phi} \varphi(x)= \begin{cases}
&1, \text { if } |x|<1,\\
&0, \text { if } |x|\ge 2. \end{cases}
\end{equation}
The low frequency part
$
u_\ell =u_{\ell, \omega, \lambda}(x, t)
$
is the solution to the following Cauchy problem for CH
\begin{equation} \label{low-frequency-sln} \partial_t u_\ell + u_\ell\partial_x u_\ell + \Lambda^{-1}
\Big[ u_\ell^2 + \frac{1}{2}(\partial_x u_\ell)^2 \Big] = 0, \end{equation}
\begin{equation} \label{low-frequency-data} u_\ell(x, 0) =
\omega \lambda^{-1} \tilde{\varphi} (\frac{x}{\lambda^{\delta}}), \ x \in \mathbb{R}, \;\; \ t \in \mathbb{R}, \end{equation}
where $\tilde{\varphi}$ is a $C_0^\infty(\mathbb{R})$ function
such that
\begin{equation} \label{phi-relation} \tilde{\varphi}(x)=1,
\,\, \text{if} \,\,
x\in \text{supp }\varphi.
\end{equation}
Furthermore,
$\Lambda^{-1}$ denotes the order $-1$ pseudodifferential operator
\begin{equation} \label{Lambda-1-def} \Lambda^{-1} = \partial_x \Big( 1 - \partial_x^2 \Big)^{-1}. \end{equation}
As it is explained in Lemma \ref{u-low-cp-info} below, the initial value problem \eqref{low-frequency-sln}--\eqref{low-frequency-data} has a unique smooth solution $u_\ell$ belonging in $H^s(\mathbb{R})$ for all $s$. Thus, the approximate solutions $u^{\omega, \lambda}$ belong in every Sobolev space.
Substituting the approximate solution $ u^{\omega, \lambda} = u_\ell + u^h $
into CH equation we obtain the following expression
\begin{equation*} \label{approx-sln-error} \begin{split} F &= \partial_t
u^h + u_\ell \partial_x u^h + u^h \partial_x u_\ell + u^h \partial_x u^h + \Lambda^{-1} \Big[ 2u_\ell u^h + (u^h)^2 + \partial_xu_\ell \partial_xu^h + \frac 12 (\partial_x u^h)^2 \Big] \\ &+ \partial_t u_\ell + u_\ell\partial_x u_\ell + \partial_x \Big( 1 - \partial_x^2 \Big)^{-1}
\Big[ u_\ell^2 + \frac{1}{2}(\partial_x u_\ell)^2 \Big]. \end{split} \end{equation*}
Now, taking into consideration that $u_\ell$ solves CH we obtain the following error for the approximate solution
\begin{equation} \label{approx-sln-error} \begin{split} F &= \partial_t
u^h + u_\ell \partial_x u^h + u^h \partial_x u_\ell + u^h \partial_x u^h \\ & + \Lambda^{-1} \Big[ 2u_\ell u^h + (u^h)^2 + \partial_xu_\ell \partial_xu^h + \frac 12 (\partial_x u^h)^2 \Big]. \end{split} \end{equation}
Computing $\partial_t u^h$ gives
\begin{equation} \label{t-der-of-u-high-1} \partial_tu^h(x, t) =
\omega
\lambda^{-\delta/2 - s}
\varphi (\frac{x}{\lambda^{\delta}})
\sin (\lambda x-\omega t ).
\end{equation}
Furthermore, since $\tilde{\varphi}$ is equal to 1 on the support of $\varphi$ we see that we can write $\partial_t u^h$ in the following form
\begin{equation} \label{t-der-of-u-high} \begin{split} \partial_tu^h(x, t)
&=
\omega
\tilde{\varphi} (\frac{x}{\lambda^{\delta}})
\lambda^{-\delta/2 - s}
\varphi (\frac{x}{\lambda^{\delta}})
\sin (\lambda x-\omega t ) \\ &= \lambda u_\ell(x, 0) \cdot
\lambda^{-\delta/2 - s}
\varphi (\frac{x}{\lambda^{\delta}})
\sin (\lambda x-\omega t ). \end{split}
\end{equation}
Computing the spacial derivative of $u^h$ gives
\begin{equation} \label{x-der-of-u-high} \begin{split} \partial_xu^h(x, t)
&= -
\lambda \cdot
\lambda^{-\delta/2 - s}
\varphi (\frac{x}{\lambda^{\delta}})
\sin (\lambda x-\omega t), \\ &+
\lambda^{-\frac 32 \delta - s}
\partial_x\varphi (\frac{x}{\lambda^{\delta}})
\cos (\lambda x-\omega t ). \end{split}
\end{equation}
Then, using \eqref{t-der-of-u-high} and \eqref{x-der-of-u-high} we find that
\begin{equation} \label{dt-uh-calc} \begin{split} \partial_t
u^h + u_\ell \partial_x u^h &= \lambda \Big[ u_\ell(x, 0) - u_\ell(x, t) \Big]
\lambda^{-\delta/2 - s}
\varphi (\frac{x}{\lambda^{\delta}})
\sin (\lambda x-\omega t )
\\
&+
u_\ell(x, t) \cdot
\lambda^{-\frac 32 \delta - s}
\partial_x\varphi (\frac{x}{\lambda^{\delta}})
\cos (\lambda x-\omega t ). \end{split} \end{equation}
Therefore, the error \eqref{approx-sln-error} of the approximate solution $u^{\omega, \lambda}$ is given by
\begin{equation} \label{F-sum-of Fj} F = F_1 + F_2+ \cdots +F_8, \end{equation}
where
\begin{equation} \label{F-j-def} \begin{split} &F_1 = \lambda \Big[ u_\ell(x, 0) - u_\ell(x, t) \Big]
\lambda^{-\delta/2 - s}
\varphi (\frac{x}{\lambda^{\delta}})
\sin (\lambda x-\omega t )
\\
&
F_2=
u_\ell(x, t) \cdot
\lambda^{-\frac 32 \delta - s}
\partial_x\varphi (\frac{x}{\lambda^{\delta}})
\cos (\lambda x-\omega t )
\\
&
F_3 = u^h \partial_x u_\ell
\\
&
F_4= u^h \partial_x u^h
\\
&
F_5= \Lambda^{-1} \big[ 2u_\ell u^h \big]
\\
&
F_6= \Lambda^{-1} \big[ (u^h)^2 \big]
\\
&
F_7= \Lambda^{-1} \big[ \partial_xu_\ell \partial_xu^h \big]
\\
&
F_8= \Lambda^{-1} \big[ \frac 12 (\partial_x u^h)^2 \big]. \end{split} \end{equation}
Next we shall estimate the size of the error $F$.
\section{ Estimating the $H^1$ norm of the error } \setcounter{equation}{0}
To estimate the $H^1$ norm of the error $F$ it suffices to estimate the $H^1$ norm of each term $F_j$. Observe that each $F_j$ is expressed in terms of $u_\ell$ and $u^h$. The high frequency part $u^h$ is defined by formula \eqref{high-frequency-approx-sln} and
\begin{equation}
\label{Hs-norm-of-u-h}
\|
u^h(t)
\|_{H^s(\mathbb{R})} \approx 1, \quad \text{for } \, \lambda>>1, \end{equation}
because of the following result.
\begin{lemma}
\label{lem:Hs-norm-of-ap-sl}
Let
$\psi \in \mathcal{S}(\mathbb{R})$, $1<\delta<2$ and $\alpha \in \mathbb{R}$. Then for any $s\ge 0$ we have that
\begin{equation}
\label{Hs-norm-of-ap-sl-2}
\lim_{\lambda\to \infty}
\lambda^{-\frac{1}{2}\delta-s}
\|
\psi (\frac{x}{\lambda^\delta}) \cos (\lambda x - \alpha)
\|_{H^s(\mathbb{R})} = \frac{1}{\sqrt{2}}
\|
\psi
\|_{L^2(\mathbb{R})}.
\end{equation}
Relation \eqref{Hs-norm-of-ap-sl-2} is also true if $\cos$ is replaced by $\sin$. \end{lemma}
Although this lemma can be found in \cite{kt}, we include its proof here for the convenience of the reader.
\vskip0.1in
\noindent {\bf Proof.} Since
\begin{equation*}
\begin{split} \Big( \psi (\frac{x}{\lambda^\delta}) \cos (\lambda x - \alpha) \widehat {\Big)}(\xi) = \frac 12 \lambda^\delta \big[ e^{-i\alpha} \widehat{ \psi} (\lambda^\delta (\xi -\lambda)) + e^{i\alpha} \widehat{ \psi} (\lambda^\delta (\xi + \lambda)) \big],
\end{split}
\end{equation*}
we have that
\begin{equation*}
\label{Lambda-1-trig-calc}
\begin{split}
\lambda^{-\delta-2s}
\| \psi(\frac{x}{\lambda^{\delta}})
\cos (\lambda x-\alpha)
\|_{H^s(\mathbb{R})}^2 & = \frac{ \lambda^{-2s+\delta}}{8\pi} \int_\mathbb{R} \big(1+\xi^2)^s
\big| e^{-i\alpha} \widehat{ \psi} (\lambda^\delta (\xi -\lambda)) + e^{i\alpha} \widehat{ \psi} (\lambda^\delta (\xi + \lambda))
\big|^2 d\xi \\ &=
\frac{ \lambda^{-2s+\delta}}{8\pi}
\Big[ \int_\mathbb{R} \big(1+\xi^2)^s
\big| \widehat{ \psi} (\lambda^\delta (\xi -\lambda))
\big|^2 d\xi \\ &+ \int_\mathbb{R} \big(1+\xi^2)^s
\big| \widehat{ \psi} (\lambda^\delta (\xi +\lambda))
\big|^2 d\xi \\ &+ 2 \int_\mathbb{R} \big(1+\xi^2)^s \text{Re}
\big[ e^{-2i\alpha} \widehat{ \psi} (\lambda^\delta (\xi - \lambda)) \bar{ \widehat{\psi} } (\lambda^\delta (\xi + \lambda)) \big] d\xi \Big].
\end{split}
\end{equation*}
Now, in the first and third integral we make the change of variables $\eta=\lambda^\delta (\xi-\lambda)$, while in the second we let $\eta=\lambda^\delta (\xi+\lambda)$. Thus, we have
\begin{equation*}
\label{Lambda-1-trig-calc}
\begin{split}
\lambda^{-\delta-2s}
\| \psi(\frac{x}{\lambda^{\delta}})
\cos (\lambda x-\alpha)
\|_{H^s(\mathbb{R})}^2 & = \frac{ \lambda^{-2s}}{8\pi}
\Big[ \int_\mathbb{R} \Big(1+\big(\frac{\eta}{\lambda^\delta}+\lambda)^2 \Big)^s
\big| \widehat{ \psi} (\eta)
\big|^2 d\eta \\ &+ \int_\mathbb{R} \Big(1+\big(\frac{\eta}{\lambda^\delta}- \lambda)^2 \Big)^s
\big| \widehat{ \psi} (\eta)
\big|^2 d\eta \\ &+ 2 \int_\mathbb{R} \Big(1+\big(\frac{\eta}{\lambda^\delta}+\lambda)^2 \Big)^s \text{Re}
\big[ e^{-2i\alpha} \widehat{ \psi} (\eta) \bar{ \widehat{\psi} } (\eta+2\lambda^{\delta+1}) \big] d\xi \Big].
\end{split}
\end{equation*}
Moving the factor $\lambda^{-2s}$ inside the integrals gives
\begin{equation*}
\label{Lambda-1-trig-calc}
\begin{split}
\lambda^{-\delta-2s}
\| \psi(\frac{x}{\lambda^{\delta}})
\cos (\lambda x-\alpha)
\|_{H^s(\mathbb{R})}^2 & = \frac{1}{8\pi}
\Big[ \int_\mathbb{R} \Big( \frac{1}{\lambda^2}+\big(\frac{\eta}{\lambda^{\delta+1}}+ 1)^2 \Big)^s
\big| \widehat{ \psi} (\eta)
\big|^2 d\eta \\ &+ \int_\mathbb{R} \Big( \frac{1}{\lambda^2}+\big(\frac{\eta}{\lambda^{\delta+1}} - 1)^2 \Big)^s
\big| \widehat{ \psi} (\eta)
\big|^2 d\eta \\ &+ 2 \int_\mathbb{R} \Big( \frac{1}{\lambda^2}+\big(\frac{\eta}{\lambda^{\delta+1}}+ 1)^2 \Big)^s \text{Re}
\big[ e^{-2i\alpha} \widehat{ \psi} (\eta) \bar{ \widehat{\psi} } (\eta+2\lambda^{\delta+1}) \big] d\xi \Big].
\end{split}
\end{equation*}
Since $\psi \in \mathcal{S}(\mathbb{R})$ we have that
$\widehat{\psi} (\eta+2\lambda^{\delta+1}) \to 0$ as $\lambda \to \infty$.
Therefore, applying the dominated convergence theorem we see that
the third integral goes to zero while each of the other two goes to $\|\widehat{\psi}\|_{L^2}^2$.
Therefore, we obtain that
\begin{equation*}
\label{Lambda-1-trig-calc}
\lim_{\lambda \to \infty}
\lambda^{-\delta-2s}
\| \psi(\frac{x}{\lambda^{\delta}})
\cos (\lambda x-\alpha)
\|_{H^s(\mathbb{R})}^2 = \frac{1}{4\pi}
\|\widehat{\psi}\|_{L^2}^2 = \frac{1}{2}
\|\psi\|_{L^2}^2,
\end{equation*}
which proves the lemma. \,\, $\square$
As we have stated earlier, the low frequency part $u_\ell$ is the solution of the Cauchy problem \eqref{low-frequency-sln}--\eqref{low-frequency-data}. Next lemma summarizes the basic information about $u_\ell$.
\begin{lemma} \label{u-low-cp-info} Let $\omega$ be bounded, $0<\delta <2$
and $\lambda>>1$. Then, the initial value problem
\eqref{low-frequency-sln}--\eqref{low-frequency-data}
has a unique smooth solution $u_\ell
\in C([0, 1]; H^s(\mathbb{R}))$, for all $s > 3/2$, and satisfying the
estimate
\begin{equation}
\label{Hs-norm-of u-ell-t-est}
\|
u_\ell(t)
\|_{H^s(\mathbb{R})} \le c_s
\lambda^{-1+\delta/2},
\quad
0\le t \le 1.
\end{equation}
\end{lemma}
{\bf Proof.} Let $s\ge 0$.
For any function $\psi\in \mathcal{S}(\mathbb{R})$ we have
\begin{equation}
\label{psi-delta-estmate}
\| \psi(\frac{x}{\lambda^{\delta}})
\|_{H^s(\mathbb{R})} \le
\lambda^{\delta/2}
\,\,
\| \psi
\|_{H^s(\mathbb{R})}.
\end{equation}
In fact, using the relation
$\widehat{\psi(x/\rho)} (\xi)
=
\rho
\widehat{\psi}(\rho \xi)
$
and making the change of variables $\eta=\lambda^\delta \xi$
we obtain
\begin{equation*}
\label{psi-delta-est}
\begin{split}
\| \psi(\frac{x}{\lambda^{\delta}})
\|_{H^s}^2 &= \frac{1}{2\pi}
\int_{\mathbb{R}}
(1+\xi^2)^s
| \lambda^\delta \, \widehat{ \psi} (\lambda^\delta \xi)
|^2
d\xi \\
&
= \frac{1}{2\pi}
\int_{\mathbb{R}} \Big(1+\frac{\eta^2}{\lambda^{2\delta}}\Big)^s
\cdot \lambda^{2\delta}
| \widehat{ \psi} (\eta)
|^2 \,
\frac{d\eta}{\lambda^\delta}
\\
&
= \lambda^{\delta} \cdot
\frac{1}{2\pi}
\int_{\mathbb{R}} \Big(1+\frac{\eta^2}{\lambda^{2\delta}}\Big)^s
| \widehat{ \psi} (\eta)
|^2 \,
d\eta
\\
&
\le \lambda^{\delta} \cdot
\frac{1}{2\pi}
\int_{\mathbb{R}}
\big(1+ \eta^2 \big)^s
| \widehat{ \psi} (\eta)
|^2 \,
d\eta
\\
&
=
\lambda^{\delta}
\,\,
\| \psi
\|^2_{H^s(\mathbb{R})}.
\end{split}
\end{equation*}
Now, using inequality \eqref{psi-delta-estmate} we have that the initial data $u_\ell (0)$ satisfy the estimate
\begin{equation}
\label{Hs-norm-of u-ell-0-est}
\|
u_\ell(0)
\|_{H^s(\mathbb{R})} \le
|\omega|
\lambda^{-1+\delta/2}
\,\,
\| \tilde{\varphi}
\|_{H^s(\mathbb{R})},
\end{equation}
which for $\omega$ bounded decays if
\begin{equation}
\label{delta-condition}
\delta < 2.
\end{equation}
Next, using estimate \eqref{Lifespan-size} from Proposition \ref{Lifespan-u-size} we have that the lifespan $T$ of the solution $u_\ell(t)$ satisfies
\begin{equation*}
\label{Lifespan-est}
T
\ge
\frac{1}{ 2c_s
\|
u_\ell(0)
\|_{H^s(\mathbb{R})}}
\ge
\frac{c_s'}{
\lambda^{-1+\delta/2}} \ge 1,
\quad
\text{ for }
\lambda >>1,
\end{equation*}
since $\delta<2$.
Finally, if $s \ge 0$ then from estimate \eqref{u-u0-Hs-bound}
of Proposition \ref{Lifespan-u-size} we have
\begin{equation*}
\label{u-ell-t-est}
\|
u_\ell(t)
\|_{H^s(\mathbb{R})}
\le
\|
u_\ell(t)
\|_{H^{s+2}(\mathbb{R})}
\le
c_s
\|
u_\ell(0)
\|_{H^{s+2}(\mathbb{R})} \le
c_s
\lambda^{-1+\delta/2}.
\quad \square
\end{equation*}
\vskip0.1in
Now we are ready to estimate the $H^1$ norm of each error $F_j$.
\noindent
{\bf Estimating the $H^1$-norm of $F_1$.} We have
\begin{equation}
\label{H1-est-F1-calc1}
\begin{split}
\|F_1\|_{H^1(\mathbb{R})} &=
\|
\lambda \Big[ u_\ell(x, 0) - u_\ell(x, t) \Big]
\lambda^{-\delta/2 - s}
\varphi (\frac{x}{\lambda^{\delta}})
\sin (\lambda x-\omega t )\|_{H^1(\mathbb{R})} \\ &=
\lambda^{1-\delta/2 - s}
\|
\varphi (\frac{x}{\lambda^{\delta}})
\sin (\lambda x-\omega t )
\big[ u_\ell(x, 0) - u_\ell(x, t) \big]
\|_{H^1(\mathbb{R})}. \end{split}
\end{equation}
Using the inequality
\begin{equation}
\label{H1-product}
\| f g
\|_{H^1(\mathbb{R})} \le \sqrt{2} \,
\|
f
\|_{C^1(\mathbb{R})}
\|
g
\|_{H^1(\mathbb{R})},
\end{equation}
from \eqref{H1-est-F1-calc1} we get
\begin{equation*}
\label{H1-est-F1-calc2}
\begin{split}
\|F_1\|_{H^1(\mathbb{R})} &\lesssim
\lambda^{1-\delta/2 - s}
\|
\varphi (\frac{x}{\lambda^{\delta}})
\sin (\lambda x-\omega t )
\|_{C^1(\mathbb{R})}
\| u_\ell(x, 0) - u_\ell(x, t)
\|_{H^1(\mathbb{R})} \end{split}
\end{equation*}
And, since
$\|
\varphi (\frac{x}{\lambda^{\delta}})
\sin (\lambda x-\omega t )
\|_{C^1(\mathbb{R})}
=
\| \varphi\|_{L^\infty} \lambda
$
the last inequality gives
\begin{equation}
\label{H1-est-F1-calc2}
\|F_1\|_{H^1(\mathbb{R})} \lesssim
\lambda^{2-\delta/2 - s}
\| u_\ell(x, 0) - u_\ell(x, t)
\|_{H^1(\mathbb{R})}.
\end{equation}
To estimate the $H^1$ norm of the difference $u_\ell(t)-u_\ell(0)$ we apply
the fundamental theorem of calculus in the time variable to obtain
\begin{equation}
\label{FTC-in-t}
u_\ell(x, t) - u_\ell(x, 0)=\int_0^t \partial_t u_\ell (x, \tau) d\tau.
\end{equation}
Then, taking the $H^1$ norm of the space variable to both sides
of \eqref{FTC-in-t}
and passing the norm inside the integral gives
\begin{equation}
\label{FTC-in-t-H1}
\| u_\ell(x, 0) - u_\ell(x, t)
\|_{H^1(\mathbb{R})} \le \int_0^t
\| \partial_t u_\ell (x, \tau)
\|_{H^1(\mathbb{R})}
d\tau,
\qquad
t\in [0, 1].
\end{equation}
Next we estimate
$
\| \partial_t u_\ell (x, \tau)
\|_{H^1(\mathbb{R})}. $ For this we solve equation \eqref{low-frequency-sln} for $\partial_t u_\ell$
to get
\begin{equation} \label{t-der-u-low-relation} \partial_t u_\ell (x, \tau) = -
u_\ell\partial_x u_\ell - \Lambda^{-1}
\big[ u_\ell^2 + \frac{1}{2}(\partial_x u_\ell)^2 \big]. \end{equation}
Thus, at any time in $[0, T]$ we have
\begin{equation} \label{t-der-u-low-H1}
\| \partial_t u_\ell (x, \tau)
\|_{H^1(\mathbb{R})} \le
\|
u_\ell\partial_x u_\ell
\|_{H^1(\mathbb{R})} +
\| \Lambda^{-1}
\big[ u_\ell^2 + \frac{1}{2}(\partial_x u_\ell)^2 \big]
\|_{H^1(\mathbb{R})} \end{equation}
Now, using the inequality
\begin{equation} \label{H1-algebra}
\| fg
\|_{H^1 (\mathbb{R})} \le c
\|
f
\|_{H^1(\mathbb{R})}
\| g
\|_{H^1(\mathbb{R})} \end{equation}
and the estimate
\begin{equation} \label{Lambda-1-est}
\|
\Lambda^{-1} f
\|_{H^1 (\mathbb{R})} \le
\| f
\|_{L^2(\mathbb{R})} \end{equation}
from \eqref{t-der-u-low-H1} we obtain that
\begin{equation}
\begin{split} \label{t-der-u-low-H1-2}
\| \partial_t u_\ell (x, \tau)
\|_{H^1(\mathbb{R})}
&\lesssim
\|
u_\ell
\|_{H^1 (\mathbb{R})}
\|
\partial_xu_\ell
\|_{H^1 (\mathbb{R})} +
\| u_\ell^2 + \frac{1}{2}(\partial_x u_\ell)^2
\|_{L^2 (\mathbb{R})}
\\
&\lesssim
\|
u_\ell
\|_{H^1 (\mathbb{R})}
\|
u_\ell
\|_{H^2 (\mathbb{R})} +
\| u_\ell^2
\|_{L^2 (\mathbb{R})}
+
\|
(\partial_x u_\ell)^2
\|_{L^2 (\mathbb{R})}
\\
&\lesssim
\|
u_\ell
\|_{H^2 (\mathbb{R})}^2 +
\| u_\ell
\|_{L^\infty (\mathbb{R})}
\| u_\ell
\|_{L^2 (\mathbb{R})}
+
\|
\partial_x u_\ell
\|_{L^\infty(\mathbb{R})}
\|
\partial_x u_\ell
\|_{L^2 (\mathbb{R})}
\\
&\lesssim
\|
u_\ell
\|_{H^2 (\mathbb{R})}^2 +
\|
u_\ell
\|_{H^1 (\mathbb{R})}^2
+
\|
u_\ell
\|_{H^2 (\mathbb{R})}^2
\\
&\lesssim
\|
u_\ell
\|_{H^2 (\mathbb{R})}^2.
\end{split} \end{equation}
Using estimate \eqref{Hs-norm-of u-ell-t-est}, from the last inequality we get
\begin{equation} \label{t-der-u-low-H1-fin}
\| \partial_t u_\ell (x, \tau)
\|_{H^1(\mathbb{R})} \lesssim
\lambda^{-2+\delta}.
\end{equation}
Substituting \eqref{t-der-u-low-H1-fin} into \eqref{FTC-in-t-H1} we obtain
\begin{equation}
\label{H1-of-u-ell-dif-est}
\| u_\ell(x, 0) - u_\ell(x, t)
\|_{H^1(\mathbb{R})} \lesssim
\lambda^{-2+\delta}.
\end{equation}
Finally, combining \eqref{H1-of-u-ell-dif-est} and \eqref{H1-est-F1-calc2}
gives
\begin{equation}
\label{H1-est-F1-calc4}
\|F_1\|_{H^1(\mathbb{R})} \lesssim
\lambda^{2-\delta/2 - s}
\cdot
\lambda^{-2+\delta},
\end{equation}
which gives
\begin{equation}
\label{H1-est-F1}
\|F_1\|_{H^1(\mathbb{R})} \lesssim
\lambda^{- s + \delta/2},
\qquad
\lambda>>1.
\end{equation}
\noindent
{\bf Estimating the $H^1$-norm of $F_2$.} Reading $F_2$ from \eqref {F-j-def} we have
\begin{equation}
\label{H1-est-F2-calc1}
\begin{split}
\|F_2\|_{H^1(\mathbb{R})} &=
\|
u_\ell(x, t) \cdot
\lambda^{-\frac 32 \delta - s}
\partial_x\varphi (\frac{x}{\lambda^{\delta}})
\cos (\lambda x-\omega t )
\|_{H^1(\mathbb{R})} \\ &\lesssim
\lambda^{-\frac 32 \delta - s}
\|
\partial_x\varphi (\frac{x}{\lambda^{\delta}})
\cos (\lambda x-\omega t )
\|_{C^1(\mathbb{R})}
\|
u_\ell(x, t)
\|_{H^1(\mathbb{R})} \\ &\lesssim
\lambda^{-\frac 32 \delta - s}
\|
\partial_x\varphi (\frac{x}{\lambda^{\delta}})
\cos (\lambda x-\omega t )
\|_{C^1(\mathbb{R})}
\|
u_\ell(x, t)
\|_{H^2(\mathbb{R})} \\ &\lesssim
\lambda^{-\frac 32 \delta - s} \cdot \lambda \cdot
\lambda^{-1+ \frac 12 \delta},
\end{split}
\end{equation}
which gives
\begin{equation}
\label{H1-est-F2}
\|F_2\|_{H^1(\mathbb{R})} \lesssim
\lambda^{ - s - \delta}.
\end{equation}
\noindent
{\bf Estimating the $H^1$-norm of $F_3$.} From \eqref {F-j-def} we have
\begin{equation*}
\label{H1-est-F3-calc1}
\begin{split}
\|F_3(t)\|_{H^1(\mathbb{R})} &=
\|
u^h(t)
\partial_xu_\ell(t)
\|_{H^1(\mathbb{R})} \\ &\lesssim
\|
u^h(t)
\|_{C^1(\mathbb{R})}
\|
\partial_xu_\ell(t)
\|_{H^1(\mathbb{R})} \\ &\lesssim
\|
u^h(t)
\|_{C^1(\mathbb{R})}
\|
u_\ell(t)
\|_{H^2(\mathbb{R})}.
\end{split}
\end{equation*}
Using formula \eqref{high-frequency-approx-sln} for $u^h$ and estimate \eqref{Hs-norm-of u-ell-t-est} for $u_\ell$, from the last inequality we obtain that
\begin{equation*}
\label{H1-est-F3-calc2}
\|F_3(t)\|_{H^1(\mathbb{R})} \lesssim
\lambda^{-\frac 12 \delta - s +1} \cdot
\lambda^{-1+ \frac 12 \delta},
\end{equation*}
which gives
\begin{equation}
\label{H1-est-F3}
\|F_3(t)\|_{H^1(\mathbb{R})} \lesssim
\lambda^{ - s},
\qquad
\lambda>>1.
\end{equation}
\noindent
{\bf Estimating the $H^1$-norm of $F_4$.} Reading $F_4$ from \eqref {F-j-def}
and using \eqref{best-Sob-product-est} we have
\begin{equation}
\label{H1-est-F4-calc1}
\begin{split}
\|F_4(t)\|_{H^1(\mathbb{R})} &=
\|
u^h(t)
\partial_xu^h(t)
\|_{H^1(\mathbb{R})} \\ &\lesssim
\|
u^h(t)
\|_{H^1(\mathbb{R})}
\|
\partial_xu^h(t)
\|_{L^\infty(\mathbb{R})} +
\|
u^h(t)
\|_{L^\infty(\mathbb{R})}
\|
\partial_xu^h(t)
\|_{H^1(\mathbb{R})} \\ &\lesssim
\|
u^h(t)
\|_{H^1(\mathbb{R})}
\|
\partial_xu^h(t)
\|_{L^\infty(\mathbb{R})} +
\|
u^h(t)
\|_{L^\infty(\mathbb{R})}
\|
u^h(t)
\|_{H^2(\mathbb{R})}.
\end{split}
\end{equation}
Since
$$
\|
u^h(t)
\|_{L^\infty(\mathbb{R})}
\lesssim
\lambda^{ -\frac 12 \delta - s},
\quad
\|
\partial_xu^h(t)
\|_{L^\infty(\mathbb{R})}
\lesssim
\lambda^{ -\frac 12 \delta - s+1},
$$
and since, by Lemma \ref{lem:Hs-norm-of-ap-sl}, we have
\begin{equation*}
\label{Hk-u-h}
\begin{split}
\|
u^h(t)
\|_{H^k(\mathbb{R})} &=
\lambda^{-\delta/2 - s}
\|
\varphi (\frac{x}{\lambda^{\delta}})
\cos (\lambda x-\omega t )
\|_{H^k(\mathbb{R})} \\ &=
\lambda^{-s + k}
\cdot
\lambda^{-\delta/2 - k}
\|
\varphi (\frac{x}{\lambda^{\delta}})
\cos (\lambda x-\omega t )
\|_{H^k(\mathbb{R})} \\ &\lesssim
\lambda^{-s + k},
\end{split}
\end{equation*}
estimate \eqref{H1-est-F4-calc1} gives
\begin{equation*}
\label{H1-est-F4-calc2}
\|F_4(t)\|_{H^1(\mathbb{R})} \lesssim
\lambda^{-s + 1}
\cdot
\lambda^{ -\frac 12 \delta - s+1} +
\lambda^{ -\frac 12 \delta - s} \cdot
\lambda^{-s + 2}.
\end{equation*}
Thus,
\begin{equation}
\label{H1-est-F4}
\|F_4(t)\|_{H^1(\mathbb{R})} \lesssim
\lambda^{ -2 s -\frac 12 \delta +2},
\qquad
\lambda>>1.
\end{equation}
\noindent
{\bf Estimating the $H^1$-norm of $F_5$.} We have
\begin{equation*}
\label{H1-est-F5-calc1}
\begin{split}
\|F_5\|_{H^1(\mathbb{R})} &=
\|
\Lambda^{-1} \big[ 2u_\ell u^h \big]
\|_{H^1(\mathbb{R})} \\ &\le 2
\| u_\ell u^h
\|_{L^2(\mathbb{R})} \\ &\lesssim
\|
u^h
\|_{L^\infty(\mathbb{R})}
\|
u_\ell
\|_{L^2(\mathbb{R})} \\ &\lesssim
\|
u^h
\|_{L^\infty(\mathbb{R})}
\|
u_\ell
\|_{H^2(\mathbb{R})} \\ &\lesssim
\lambda^{ -\frac 12 \delta - s} \cdot
\lambda^{-1 + \frac 12 \delta },
\end{split}
\end{equation*}
which gives
\begin{equation}
\label{H1-est-F5}
\|F_5(t)\|_{H^1(\mathbb{R})} \lesssim
\lambda^{ -s -1},
\qquad
\lambda>>1.
\end{equation}
\noindent
{\bf Estimating the $H^1$-norm of $F_6$.} From \eqref {F-j-def}
and Lemma \ref{lem:Hs-norm-of-ap-sl} we have
\begin{equation*}
\label{H1-est-F6-calc1}
\begin{split}
\|F_5\|_{H^1(\mathbb{R})} &=
\|
\Lambda^{-1} \big[ (u^h)^2 \big]
\|_{H^1(\mathbb{R})} \\ &\le
\| (u^h)^2
\|_{L^2(\mathbb{R})} \\ &\lesssim
\|
u^h
\|_{L^\infty(\mathbb{R})}
\|
u^h
\|_{L^2(\mathbb{R})} \\ &\lesssim
\lambda^{ -\frac 12 \delta - s} \cdot
\lambda^{-s },
\end{split}
\end{equation*}
which gives
\begin{equation}
\label{H1-est-F6}
\|F_6(t)\|_{H^1(\mathbb{R})} \lesssim
\lambda^{ -2s - \frac 12 \delta},
\qquad
\lambda>>1.
\end{equation}
\noindent
{\bf Estimating the $H^1$-norm of $F_7$.} Also, we have
\begin{equation*}
\label{H1-est-F7-calc1}
\begin{split}
\|F_7\|_{H^1(\mathbb{R})} &=
\| \Lambda^{-1} \big[ \partial_xu_\ell \partial_xu^h \big]
\|_{H^1(\mathbb{R})} \\ &\le
\| \partial_xu_\ell \partial_xu^h
\|_{L^2(\mathbb{R})} \\ &\lesssim
\| \partial_xu^h
\|_{L^\infty(\mathbb{R})}
\|
\partial_xu_\ell
\|_{L^2(\mathbb{R})} \\ &\lesssim
\| \partial_xu^h
\|_{L^\infty(\mathbb{R})}
\|
u_\ell
\|_{H^2(\mathbb{R})} \\ &\lesssim
\lambda^{ -\frac 12 \delta - s +1} \cdot
\lambda^{-1 +\frac 12 \delta },
\end{split}
\end{equation*}
which gives
\begin{equation}
\label{H1-est-F6}
\|F_6(t)\|_{H^1(\mathbb{R})} \lesssim
\lambda^{ -s },
\qquad
\lambda>>1.
\end{equation}
\noindent
{\bf Estimating the $H^1$-norm of $F_8$.} Finally, we have
\begin{equation*}
\label{H1-est-F8-calc1}
\begin{split}
\|F_8\|_{H^1(\mathbb{R})} &=
\| \Lambda^{-1} \big[ \frac 12 (\partial_x u^h)^2 \big] \\ &\le \frac 12
\| (\partial_x u^h)^2
\|_{L^2(\mathbb{R})} \\ &\lesssim
\| \partial_xu^h
\|_{L^\infty(\mathbb{R})}
\| \partial_xu^h
\|_{L^2(\mathbb{R})} \\ &\lesssim
\| \partial_xu^h
\|_{L^\infty(\mathbb{R})}
\| u^h
\|_{H^1(\mathbb{R})} \\ &\lesssim
\lambda^{ -\frac 12 \delta - s +1} \cdot
\lambda^{-s +1},
\end{split}
\end{equation*}
which gives
\begin{equation}
\label{H1-est-F6}
\|F_6(t)\|_{H^1(\mathbb{R})} \lesssim
\lambda^{ -2s - \frac 12 \delta +2},
\qquad
\lambda>>1.
\end{equation}
Collecting all error estimates together gives the following proposition.
\begin{proposition}
\label{high-s-error-estimate-prop}
Let $s>1$ and $1<\delta<2$. Then,
for $\omega$ bounded
and $ \lambda>>1$ we have that
\begin{equation}
\label{F-error-estimate-2}
\|
F(t)
\|_{H^1(\mathbb{R})}
\lesssim
\lambda^{-r_s},
\qquad
\text{ for }
\,
\lambda >>1,
\end{equation}
with
\begin{equation} \label{decay-exponent}
r_s
\doteq \big(s - \frac 12\delta\big)>0, \qquad \text{ if } \, s> \frac 12\delta. \end{equation}
\end{proposition}
\section{ Estimating the difference between approximate and actual solutions} \setcounter{equation}{0}
Let $u_{\omega, \lambda}(x, t)$ be the solution to CH equation with initial data the value of the approximate solution $u^{\omega, \lambda}(x, t)$ at time zero. That is, $u_{\omega, \lambda}(x, t)$ solves the Cauchy problem
\begin{equation} \label{CH-with-appox-data} \partial_t u_{\omega, \lambda} +
u_{\omega, \lambda}
\partial_xu_{\omega, \lambda}
+ \Lambda^{-1}
\Big[
u_{\omega, \lambda}^2 + \frac{1}{2}(\partial_x u_{\omega, \lambda})^2 \Big] = 0, \ x \in \mathbb{R}, \;\; \ t \in \mathbb{R}, \end{equation} \begin{equation} \label{CH-appox-data} u_{\omega, \lambda}(x, 0) = u^{\omega, \lambda}(x, 0) = \omega \lambda^{-1} \tilde{\varphi}(\frac{x}{\lambda^{\delta}}) +
\lambda^{-\delta/2 - s}
\varphi (\frac{x}{\lambda^{\delta}})
\cos (\lambda x). \end{equation}
Note that $u^{\omega, \lambda}(0)$ is in $H^s(\mathbb{R})$, $s\ge 0$, and
\begin{equation} \label{approx-sln-at-0-est}
\| u^{\omega, \lambda}(0)
\|_{H^s} \le
\| u_\ell(0)
\|_{H^{s}} +
\|u^h(0)
\|_{H^s} \lesssim \lambda^{-1+\frac 12 \delta} +1. \end{equation}
Therefore, if $s>3/2$ then
using Theorem \ref{CH-wp} and Proposition \ref{Lifespan-u-size}
we see that for any $\omega$ in a bounded set and $\lambda>>1$
the Cauchy problem \eqref{CH-with-appox-data}--\eqref{CH-appox-data}
has a unique solution $u_{\omega, \lambda}$ in $C([0, T]; H^s(\mathbb{R}))$ with
\begin{equation} \label{T-indep-of-lambda}
T
\gtrsim
\frac{1}{
\| u^{\omega, \lambda}(0)
\|_{H^s(\mathbb{R})}}
\gtrsim
\frac{1}{
\lambda^{-1+\delta/2} +1}
\gtrsim 1.
\end{equation}
In fact, $u_{\omega, \lambda}(t)$ is in $C^\infty$ for each $t\in [0, T]$.
To estimate the difference between approximate and actual solutions we form the differential equation which it satisfies. So, if we let
\begin{equation}
\label{CH-difference} v = u^{\omega, \lambda} - u_{\omega, \lambda},
\end{equation}
then a straightforward computation shows that $v$ satisfies the Cauchy problem
\begin{equation} \label{CH-difference-eqn} \partial_t v -
v\partial_xv
+ u^{\omega, \lambda} \partial_xv
+
\partial_xu^{\omega, \lambda} v -
\Lambda^{-1}
\Big[
v^2
+
\frac{1}{2}(\partial_x v)^2
-
2u^{\omega, \lambda} v
-
\partial_xu^{\omega, \lambda} \partial_xv \Big] = F(x, t), \end{equation} \begin{equation} \label{CH-difference-data} v(x, 0) = 0, \ x \in \mathbb{R}, \;\; \ t \in \mathbb{R}. \end{equation}
where $F$ is defined by
\begin{equation} \label{Burgers-F} F \doteq
\partial_tu^{\omega, \lambda} + u^{\omega, \lambda} \partial_xu^{\omega, \lambda} + \Lambda^{-1}
[(u^{\omega, \lambda})^2 + \frac 12 (\partial_xu^{\omega, \lambda})^2], \end{equation}
and which it has been shown to satisfy the $H^1$-estimate \eqref{F-error-estimate-2}.
\begin{lemma} \label{CH-differ-H1-est-lem}
Let $1<\delta<2$. If $s>3/2$ then
\begin{equation}
\| v(t)
\|_{H^1(\mathbb{R})} \doteq \label{differ-H1-est}
\| u^{\omega, \lambda}(t) -
u_{\omega, \lambda}(t)
\|_{H^1(\mathbb{R})} \lesssim
\lambda^{-r_s},
\quad
0 \le t \le T, \end{equation}
where $r_s=s- \delta/2>0$ (see \eqref{decay-exponent}). \end{lemma}
{\bf Proof.} We have
\begin{equation} \label{t-deriv-of-H1-v} \frac 12 \frac{d}{dt}
\| v(t)
\|_{H^1(\mathbb{R})}^2 = \int_{\mathbb{R}} \big[ v\partial_tv + \partial_xv \partial_x\partial_tv \big] dx \end{equation}
Applying to both sides of \eqref{CH-difference-eqn} the operator $(1-\partial_x^2)$ and solving for $\partial_t v$ we obtain
\begin{equation} \label{t-deriv-of-v} \begin{split} \partial_t v &= (1-\partial_x^2) F \\ & -
(1-\partial_x^2) \big[ u^{\omega, \lambda} \partial_xv
+
\partial_xu^{\omega, \lambda} v
\big]
\\ &
-
\partial_x
\big[ 2u^{\omega, \lambda} v
+
\partial_xu^{\omega, \lambda} \partial_xv
\big]
\\
&
+
3v\partial_x v - 2
\partial_xv\partial_x^2 v
-
v\partial_x^3 v
+
\partial_t \partial_x^2v \end{split} \end{equation}
Substituting $\partial_tv$ from \eqref{t-deriv-of-v} to \eqref{t-deriv-of-H1-v} we get
\begin{equation} \label{deriv-H1-norm-v-sq-1} \begin{split} \frac 12 \frac{d}{dt}
\| v(t)
\|_{H^1(\mathbb{R})}^2 & = \int_{\mathbb{R}} v (1-\partial_x^2) F dx \\ & - \int_{\mathbb{R}} v (1-\partial_x^2) \big[ u^{\omega, \lambda} \partial_xv
+
\partial_xu^{\omega, \lambda} v
\big]
dx
\\ &
- \int_{\mathbb{R}} v
\partial_x
\big[ 2u^{\omega, \lambda} v
+
\partial_xu^{\omega, \lambda} \partial_xv \big] dx \\ &
+ \int_{\mathbb{R}}
\big[ v
(3v\partial_x v -
2\partial_xv\partial_x^2 v
-
v\partial_x^3 v
+
\partial_t \partial_x^2v)
+
\partial_xv \partial_x\partial_tv \big] dx. \end{split} \end{equation}
Noting that the last integral can be rewritten as
$$ \int_{\mathbb{R}} \Big[ \partial_x\big(v^3\big) - \partial_x\big(v^2 \partial_x^2v\big) + \partial_x\big(v\partial_t\partial_xv\big)
\Big] \, dx
= 0, $$ which is a property special to CH, we see that equation \eqref{deriv-H1-norm-v-sq-1} takes the form
\begin{equation} \label{deriv-H1-norm-v-sq} \begin{split} \frac 12 \frac{d}{dt}
\| v(t)
\|_{H^1(\mathbb{R})}^2 & = \int_{\mathbb{R}} v (1-\partial_x^2) F dx \\ & - \int_{\mathbb{R}} v (1-\partial_x^2) \big[ u^{\omega, \lambda} \partial_xv
+
\partial_xu^{\omega, \lambda} v
\big]
dx
\\ &
- \int_{\mathbb{R}} v
\partial_x
\big[ 2u^{\omega, \lambda} v
+
\partial_xu^{\omega, \lambda} \partial_xv \big] dx. \end{split} \end{equation}
Integrating by parts and applying the Cauchy-Schwarz inequality,
we estimate the three integrals in the right-hand side
of \eqref{deriv-H1-norm-v-sq} as follows.
For the first integral we have
\begin{equation} \label{RHS-1}
\Big|
\int_{\mathbb{R}} v (1-\partial_x^2) F dx
\Big| =
\Big|
\int_{\mathbb{R}}
[
v F
+ \partial_xv\partial_xF ] dx
\Big| \le
\| F(t)
\|_{H^1(\mathbb{R})}
\| v(t)
\|_{H^1(\mathbb{R})}. \end{equation}
Also, for the third integral we have
\begin{equation} \label{RHS-3} \begin{split}
\Big| \int_{\mathbb{R}} v
\partial_x
\big[ 2u^{\omega, \lambda} v
&+
\partial_xu^{\omega, \lambda} \partial_xv \big] dx
\Big| =
\Big| \int_{\mathbb{R}} \partial_xv
\big[ 2u^{\omega, \lambda} v
+
\partial_xu^{\omega, \lambda} \partial_xv \big] dx
\Big| \\ & \le 2 \Big(
\| u^{\omega, \lambda}(t)
\|_{L^\infty(\mathbb{R})} +
\| \partial_xu^{\omega, \lambda}(t)
\|_{L^\infty(\mathbb{R})} \Big)
\| v(t)
\|_{H^1(\mathbb{R})}^2. \end{split} \end{equation}
Integrating by parts, we write the second integral in the form
\begin{equation} \label{RHS-3} \begin{split} \int_{\mathbb{R}} v (1-\partial_x^2) \big[ u^{\omega, \lambda} \partial_xv
&+
\partial_xu^{\omega, \lambda} v
\big]
dx = \int_{\mathbb{R}} v \big[ u^{\omega, \lambda} \partial_xv
+
\partial_xu^{\omega, \lambda} v
\big] dx \\ &+
\int_{\mathbb{R}} \partial_x v \partial_x \big[ u^{\omega, \lambda} \partial_xv \big] dx +
\int_{\mathbb{R}}
\partial_x v \partial_x \big[
\partial_xu^{\omega, \lambda} v
\big]
dx \end{split} \end{equation}
and estimate its first part by
\begin{equation} \label{RHS-2.1}
\Big| \int_{\mathbb{R}} v \big[ u^{\omega, \lambda} \partial_xv
+
\partial_xu^{\omega, \lambda} v
\big]
dx
\Big| \le \Big(
\| u^{\omega, \lambda}(t)
\|_{L^\infty(\mathbb{R})} +
\| \partial_xu^{\omega, \lambda}(t)
\|_{L^\infty(\mathbb{R})} \Big)
\| v(t)
\|_{H^1(\mathbb{R})}^2. \end{equation}
Its second part we can be written as
\begin{equation*}
\label{RHS-2.2}
\int_{\mathbb{R}} \partial_x v \partial_x \big[ u^{\omega, \lambda} \partial_xv \big] dx = \int_{\mathbb{R}}
\big[
\frac 12 u^{\omega, \lambda} \partial_x(\partial_xv)^2
+ \partial_x u^{\omega, \lambda} (\partial_xv)^2
\big] dx
=
\frac 12
\int_{\mathbb{R}}
\partial_xu^{\omega, \lambda} (\partial_xv)
\big] dx, \end{equation*}
which gives that
\begin{equation}
\label{RHS-2.2}
\Big|
\int_{\mathbb{R}} \partial_x v \partial_x \big[ u^{\omega, \lambda} \partial_xv \big] dx
\Big| \le
\| \partial_xu^{\omega, \lambda}(t)
\|_{L^\infty(\mathbb{R})}
\| v(t)
\|_{H^1(\mathbb{R})}^2. \end{equation}
Finally, writing the last part as follows
\begin{equation*}
\label{RHS-2.3}
\int_{\mathbb{R}}
\partial_x v \partial_x \big[
\partial_xu^{\omega, \lambda} v
\big]
dx = \int_{\mathbb{R}}
\big[ \partial_xu^{\omega, \lambda} (\partial_xv)^2
+ \partial_x^2 u^{\omega, \lambda} v \partial_xv
\big] dx \end{equation*}
we see that it can be estimated as follow
\begin{equation}
\label{RHS-2.3}
\Big|
\int_{\mathbb{R}}
\partial_x v \partial_x \big[
\partial_xu^{\omega, \lambda} v
\big]
dx
\Big| \le \Big(
\| \partial_xu^{\omega, \lambda}(t)
\|_{L^\infty(\mathbb{R})} +
\| \partial_x^2u^{\omega, \lambda}(t)
\|_{L^\infty(\mathbb{R})} \Big)
\| v(t)
\|_{H^1(\mathbb{R})}^2. \end{equation}
Combining the above estimates gives
\begin{equation} \label{deriv-H1-norm-v-sq-2} \begin{split} \frac 12 \frac{d}{dt}
&\| v(t)
\|_{H^1(\mathbb{R})}^2 \lesssim
\| F(t)
\|_{H^1(\mathbb{R})}
\| v(t)
\|_{H^1(\mathbb{R})} + \\ & + \Big(
\| u^{\omega, \lambda}(t)
\|_{L^\infty(\mathbb{R})} +
\| \partial_xu^{\omega, \lambda}(t)
\|_{L^\infty(\mathbb{R})} +
\| \partial_x^2u^{\omega, \lambda}(t)
\|_{L^\infty(\mathbb{R})} \Big)
\| v(t)
\|_{H^1(\mathbb{R})}^2. \end{split} \end{equation}
From \eqref{high-frequency-approx-sln} we have
\begin{equation} \label{CH-xx-deriv} \begin{split} \partial_x^2u^h &=
\lambda^{-\frac 52 \delta - s}
\partial_x^2\varphi (\frac{x}{\lambda^{\delta}})
\cos (\lambda x-\omega t)
\\
&
- 2
\lambda^{-\frac 32 \delta - s+1}
\partial_x\varphi (\frac{x}{\lambda^{\delta}})
\sin (\lambda x-\omega t)
- 2
\lambda^{-\frac 12 \delta - s+2}
\varphi (\frac{x}{\lambda^{\delta}})
\cos (\lambda x-\omega t).
\end{split}
\end{equation}
so that
\begin{equation} \label{u-h-ap-x-sup}
\| u^h(t)
\|_{L^\infty(\mathbb{R})} +
\| \partial_xu^h(t)
\|_{L^\infty(\mathbb{R})} +
\| \partial_x^2u^h(t)
\|_{L^\infty(\mathbb{R})} \lesssim \lambda^{-({\frac 12 \delta + s-2})}. \end{equation}
For $u_\ell$ we have
\begin{equation} \label{u-ell-x-sup}
\| u_\ell(t)
\|_{L^\infty(\mathbb{R})} +
\| \partial_xu_\ell(t)
\|_{L^\infty(\mathbb{R})} +
\| \partial_x^2u_\ell(t)
\|_{L^\infty(\mathbb{R})} \lesssim
\| u_\ell(t)
\|_{H^3(\mathbb{R})} \lesssim \lambda^{-(1-\frac 12 \delta)}. \end{equation}
Therefore
\begin{equation} \label{u-ap-x-sup}
\| u^{\omega, \lambda}(t)
\|_{L^\infty(\mathbb{R})} +
\| \partial_xu^{\omega, \lambda}(t)
\|_{L^\infty(\mathbb{R})} +
\| \partial_x^2u^{\omega, \lambda}(t)
\|_{L^\infty(\mathbb{R})} \lesssim \lambda^{-\rho_s}, \end{equation}
where
\begin{equation} \label{rho-def} \rho_s \doteq \min\{1-\frac 12 \delta, \frac 12 \delta + s-2 \}>0, \end{equation} for any any \textcolor{Red}{ $s>3/2$} if $\delta$ is chosen appropriately in the interval $(1, 2)$.
Using \eqref{u-ap-x-sup} and the $H^1$- estimate \eqref{F-error-estimate-2}
for the error $F$, from \eqref{deriv-H1-norm-v-sq-2} we get
\begin{equation*} \label{deriv-H1-norm-v-sq-3} \frac 12 \frac{d}{dt}
\| v(t)
\|_{H^1(\mathbb{R})}^2 \lesssim
\lambda^{-\rho_s}
\| v(t)
\|_{H^1(\mathbb{R})}^2 +
\lambda^{-r_s}
\| v(t)
\|_{H^1(\mathbb{R})}, \end{equation*}
which gives the differential inequality
\begin{equation} \label{deriv-H1-norm-v-sq-3} \frac{d}{dt}
\| v(t)
\|_{H^1(\mathbb{R})} \lesssim
\lambda^{-\rho_s}
\| v(t)
\|_{H^1(\mathbb{R})} +
\lambda^{-r_s}. \end{equation}
Since $ \|v(0)\|_{H^1(\mathbb{R})}=0$ and for
$s>1$ we can choose $\delta$ such that $\rho_s \ge 0$ from \eqref{deriv-H1-norm-v-sq-3} and Gronwall's inequality we obtain that
\begin{equation} \label{v-differ-est}
\| v(t)
\|_{H^1(\mathbb{R})} \lesssim
\lambda^{-r_s},
\quad
0 \le t \le T, \end{equation}
which concludes the proof of the lemma. \, $\square$
\section{ Non-uniform dependence in $H^s(\mathbb{R})$ for $s>3/2$} \setcounter{equation}{0}
Next we shall prove non-uniform dependence for CH
by taking advantage of the information provided
by Theorem \ref{CH-wp} and Proposition \ref{Lifespan-u-size}, and the $H^1$- estimate \eqref{differ-H1-est} on
the difference between approximate solutions and solutions
with same initial data.
For this, let $u_{1, \lambda}(x, t)$ and $u_{-1, \lambda}(x, t)$ be the unique solutions
to the the Cauchy problem
\eqref{CH-with-appox-data}--\eqref{CH-appox-data}
with initial data
$u^{1, \lambda}(x, 0)$ and $u^{-1, \lambda}(x, 0)$ correspondingly.
By Theorem \ref{CH-wp} these solutions belong in $C([0, T]; H^s(\mathbb{R}))$.
Recall, using Proposition \ref{Lifespan-u-size} we proved
estimate \eqref{T-indep-of-lambda} which says that
$T$ is independent of $\lambda>>1$. Also, for $s>3/2$,
using estimate \eqref{u-u0-Hs-bound}, we have
\begin{equation}
\label{CH-slns-Hs-norm}
\| u_{\pm 1, \lambda}(t)
\|_{H^s(\mathbb{R})}
\lesssim
\| u^{\pm 1, \lambda}(0)
\|_{H^s(\mathbb{R})},
\quad 0\le t\le T.
\end{equation}
Furthermore, since our $s$-dependent initial data $u^{\pm, \lambda}(0)$ belong to every Sobolev space they do belong to $H^{[s]+2}(\mathbb{R})$. Since $s>3/2$ by the argument in the last remark of section 2 we obtain a companion estimate
to \eqref{CH-slns-Hs-norm}
\begin{equation}
\label{CH-slns-Hk-norm}
\| u_{\pm 1, \lambda}(t)
\|_{H^{[s]+2}(\mathbb{R})}
\lesssim
\| u^{\pm 1, \lambda}(0)
\|_{H^{[s]+2}(\mathbb{R})}, \quad 0\le t\le T.
\end{equation}
Now let $k=[s]+2$.
If $\lambda$ is large enough then from
\eqref{Hs-norm-of-ap-sl-2} and \eqref{Hs-norm-of u-ell-t-est}
we have
\begin{equation*}
\label{ap-data-size}
\begin{split}
\| u^{\pm 1, \lambda}(t)
\|_{H^k(\mathbb{R})}
& \le
\| u_{\ell, \pm 1, \lambda}(t)
\|_{H^k(\mathbb{R})}
+
\lambda^{-\frac{1}{2}\delta-s}
\|
\varphi(\frac{x}{\lambda^\delta}) \cos (\lambda x-\lambda t)
\|_{H^k(\mathbb{R})}
\\
& \lesssim
\lambda^{-1+\frac 12 \delta}
+
\lambda^{k-s} \cdot \lambda^{-\frac{1}{2}\delta-k}
\|
\varphi(\frac{x}{\lambda^\delta}) \cos (\lambda x-\lambda t)
\|_{H^k(\mathbb{R})} \\
& \lesssim
\lambda^{-1+\frac 12 \delta}
+
\lambda^{k-s}
\|
\varphi
\|_{L^2(\mathbb{R})},
\end{split}
\end{equation*}
which gives
\begin{equation}
\label{ap-data-Hk-size}
\| u^{\pm 1, \lambda}(t)
\|_{H^k(\mathbb{R})}
\lesssim
\lambda^{k-s},
\quad
\text{hence by \eqref{CH-slns-Hk-norm} }
\quad
\| u_{\pm 1, \lambda}(t)
\|_{H^k(\mathbb{R})}
\lesssim
\lambda^{k-s}.
\quad
\end{equation}
Therefore, from \eqref{ap-data-Hk-size} we
obtain the following estimate for the $H^k$-norm of the difference of
$u_{\pm 1, \lambda}$ and $u_{\pm 1, \lambda}$
\begin{equation}
\label{CH-differ-Hk-norm}
\| u^{\pm 1, \lambda}(t) - u_{\pm 1, \lambda}(t)
\|_{H^k(\mathbb{R})}
\lesssim
\lambda^{k-s},
\quad
0 \le t \le T.
\end{equation}
Applying \eqref{differ-H1-est} with our particular choice of $\omega=\pm 1$ we have
\begin{equation} \label{CH-differ-H1-est}
\| u^{\pm 1, \lambda}(t) - u_{\pm 1, \lambda}(t)
\|_{H^1(\mathbb{R})} \lesssim
\lambda^{-r_s },
\quad
0 \le t \le T. \end{equation}
Now, applying the interpolation inequality
\begin{equation*} \label{Hs-interpolation}
\| \psi
\|_{H^s(\mathbb{R})} \le
\| \psi
\|_{H^{s_1}(\mathbb{R})}^{(s_2-s)/(s_2-s_1)}
\| \psi
\|_{H^{s_2}(\mathbb{R})}^{(s-s_1)/(s_2-s_1)} \end{equation*}
with $s_1=1$ and $s_2=[s] +2=k$ and
using estimates \eqref{CH-differ-H1-est} and \eqref{CH-differ-Hk-norm} gives
\begin{equation} \label{CH-differ-Hs-est-interpo} \begin{split}
\| u^{\pm 1, \lambda}(t) - u_{\pm 1, \lambda}(t)
\|_{H^s(\mathbb{R})} &\le
\| u^{\pm 1, \lambda}(t) - u_{\pm 1, \lambda}(t)
\|_{H^1(\mathbb{R})}^{(k-s)/(k-1)}
\\
&\cdot
\|
u^{\pm 1, \lambda}(t) - u_{\pm 1, \lambda}(t)
\|_{H^k(\mathbb{R})}^{(s-1)/(k-1)}
\\
&
\lesssim
\lambda^{(-r_s)[(k-s)/(k-1)]}
\lambda^{(k-s)[(s-1)/(k-1)]}
\\
&
\lesssim
\lambda^{-(r_s-s+1)[(k-s)/(k-1)]}.
\end{split} \end{equation}
From the last inequality we obtain that
\begin{equation} \label{CH-differ-Hs-dacay-est}
\| u^{\pm 1, \lambda}(t) - u_{\pm 1, \lambda}(t)
\|_{H^s(\mathbb{R})}
\lesssim
\lambda^{-\varepsilon_s},
\,\,\,
0 \le t \le T,
\end{equation}
where $\varepsilon_s$ is given by
\begin{equation} \label{epsilon-decay-exponent}
\varepsilon_s
= (1 - \frac 12 \delta)/(s+2).
\end{equation}
Note that
\begin{equation} \label{epsilon-s-positive}
\varepsilon_s>0,
\quad
\text{for}
\quad s>1. \end{equation}
Next, we shall use estimate \eqref{CH-differ-Hs-dacay-est} to prove non-uniform dependence when $s>3/2$.
\noindent {\bf Behavior at time zero.} Since $\delta<2$, at $t=0$ we have
\begin{equation} \label{CH-slns-differ-t-0} \begin{split}
\| u_{1, \lambda}(0) - u_{-1, \lambda}(0)
\|_{H^s(\mathbb{R})} &=
\| 2 \lambda^{-1} \tilde{\varphi}(\frac{x}{\lambda^{\delta}})
\|_{H^s(\mathbb{R})}
\\ &\le 2
\lambda^{-1+\frac 12 \delta}
\| \tilde{\varphi}
\|_{H^s(\mathbb{R})}
\longrightarrow
0
\,\,
\text{as}
\,\,
\lambda \to \infty.
\end{split}
\end{equation}
\noindent {\bf Behavior at time $t>0$.} Then, we write
\begin{equation}
\label{CH-slns-differ-t-pos} \begin{split}
\| u_{1, \lambda}(t) - u_{- 1, \lambda}(t)
\|_{H^s(\mathbb{R})}
& \ge
\| u^{1, \lambda}(t) - u^{- 1, \lambda}(t)
\|_{H^s(\mathbb{R})}
\\
&
-
\| u^{1, \lambda}(t) - u_{1, \lambda}(t)
\|_{H^s(\mathbb{R})}
\\
&
-
\| u^{-1, \lambda}(t) - u_{-1, \lambda}(t)
\|_{H^s(\mathbb{R})}
\end{split} \end{equation}
Using estimate \eqref{CH-differ-Hs-dacay-est} for the last two terms in \eqref{CH-slns-differ-t-pos} we obtain
\begin{equation}
\label{CH-slns-differ-t-pos-est}
\| u_{1, \lambda}(t) - u_{- 1, \lambda}(t)
\|_{H^s(\mathbb{R})} \ge
\| u^{1, \lambda}(t) - u^{- 1, \lambda}(t)
\|_{H^s(\mathbb{R})} - c
\lambda^{-\varepsilon_s}. \end{equation}
In \eqref{CH-slns-differ-t-pos-est} letting $\lambda$ go to $\infty$ gives
\begin{equation}
\label{CH-slns-to-ap-est}
\liminf_{\lambda\to\infty}
\| u_{1, \lambda}(t) - u_{- 1, \lambda}(t)
\|_{H^s(\mathbb{R})} \ge
\liminf_{\lambda\to\infty}
\| u^{1, \lambda}(t) - u^{- 1, \lambda}(t)
\|_{H^s(\mathbb{R})}. \end{equation}
Inequality \eqref{CH-slns-to-ap-est} is a key estimate since it reduces finding a lower positive bound for the difference of the {\bf uknown} solution sequences to
finding a lower positive bound for the difference of the known approximate solution sequences. Using the identity
$$ \cos \alpha -\cos \beta = -2 \sin(\frac{\alpha + \beta}{2}) \sin(\frac{\alpha - \beta}{2}) $$ gives $$ u^{1, \lambda}(t) - u^{- 1, \lambda}(t) = u_{\ell, 1, \lambda}(t) - u_{\ell, -1, \lambda}(t) + 2
\lambda^{-\frac 12 \delta - s}
\varphi (\frac{x}{\lambda^{\delta}})
\sin (\lambda x) \sin t.
$$
Therefore
\begin{equation}
\label{B--ap-below-est-1}
\begin{split}
\| u^{1, \lambda}(t) - u^{- 1, \lambda}(t)
\|_{H^s(\mathbb{R})}
&
\ge
2
\lambda^{-\frac 12 \delta - s}
\|
\varphi (\frac{x}{\lambda^{\delta}})
\sin (\lambda x)
\|_{H^s(\mathbb{R})}
|\sin t|
\\
& -
\|
u_{\ell, 1, \lambda}(t)
\|_{H^s(\mathbb{R})}
-
\|
u_{\ell, -1, \lambda}(t)
\|_{H^s(\mathbb{R})}
\\
&
\gtrsim
2
\lambda^{-\frac 12 \delta - s}
\|
\varphi (\frac{x}{\lambda^{\delta}})
\sin (\lambda x)
\|_{H^s(\mathbb{R})}
|\sin t|
-
\lambda^{-1+\frac 12 \delta}.
\end{split} \end{equation}
Now letting $\lambda$ go to $\infty$, \eqref{B--ap-below-est-1} gives
\begin{equation}
\label{CH-ap-below-est}
\liminf_{\lambda\to\infty}
\| u^{1, \lambda}(t) - u^{- 1, \lambda}(t)
\|_{H^s(\mathbb{R})}
\gtrsim
\| \varphi
\|_{L^2(\mathbb{R})}
|\sin t|. \end{equation}
Combining \eqref{CH-slns-to-ap-est} and \eqref{CH-ap-below-est} gives
\begin{equation}
\label{CH-slns-below-est-fin}
\liminf_{\lambda\to\infty}
\| u_{1, \lambda}(t) - u_{- 1, \lambda}(t)
\|_{H^s(\mathbb{R})}
\gtrsim
\| \varphi
\|_{L^2(\mathbb{R})}
|\sin t |, \end{equation}
which proves Theorem \ref{CH-non-unif-dependence}.
\vskip0.1in \noindent {\em{ \bf ACKNOWLEDGEMENTS}}. The first author thanks the Department of Mathematics of the University of Chicago for the hospitality during his stay there in the Fall of 2007. The second author acknowledges partial support from the NSF.
\vskip0.1in
\begin{minipage}[b]{6 cm} {\bf A. Alexandrou Himonas}\\ Department of Mathematics \\
University of Notre Dame\\ Notre Dame, IN 46556\\
E-mail: {\it himonas.1$@$nd.edu} \end{minipage}
\begin{minipage}[b]{7 cm} {\bf Carlos Kenig}\\ Department of Mathematics\\ The University of Chicago\\ 5734 S. University Avenue\\ Chicago, Illinois 60637\\
E-mail: {\it cek$@$math.uchicago.edu} \end{minipage}
\end{document} | arXiv |
Vectorised implementation of splines
All you need is numpy.einsum.
Estimated reading time: ~ 11 min.
This exposes a way to efficiently sample points from spline curves and 2-sphere homeomorphic splines using linear algebra operations and trivial yet useful caching operations.
First, a definition of spline curves as well as two ways of sampling it is given. Secondly, a definition of 2-sphere homeomorphic splines as well as a matricial scheme to sample several of them efficiently are given.
Finally, some tricks for the implementations are also covered briefly. I think those schemes benefit from a nearly optimal arithmetic intensity.
In the following, we clarify the following notations:
\([\![n]\!] \triangleq \{0, 1, \cdots, n - 1\}\)
As to clarify the manipulation of vectors as rows, for \(n \in \mathbb{N}\), \(\mathbb{R}^{n}\) is identified to \(\mathbb{R}^{1 \times n}\):
$$ \mathbb{R}^n \approxeq \mathbb{R}^{1 \times n} $$
Open or closed spline curves
Expression of the curve \(\gamma\)
A 2D open or closed Spline is \(\gamma\) is defined as follows:
$$ \gamma: t \in [0, 1] \longrightarrow \mathbb{R}^2 $$
$$ \gamma: t \longmapsto \sum_{k=0}^{M-1} \mathbf{c}[k]\ \varphi_{\text{per}}(Mt - k) $$
where \(t\) is the parameter, where \(\mathbf{c} \in \mathbb{R}^{M \times 2}\) are the spline coefficients also referred to in this 2D case as control points, and where \(\varphi_{\text{per}}\) is the periodisation of \(\varphi\), a given basis function.
Periodisation using wrapping
Periodisation is needed in the case of closed spline to meet boundary conditions.
One can define a wrapping function \(\text{wrap}_{M}\) to easily make \(\varphi\) periodic, that is:
$$ \forall k \in [\![M]\!],\quad \forall t \in [0,1] \quad \quad \varphi_{\text{per}}(M t - k) = \varphi_{} \circ \text{wrap}_{M}(t,k) $$
Wrapping will be used in all the coming details.
Sampling using matrices
One can efficiently, for \(t \in [0, 1]\) sample a point \(\gamma(t) \in \mathbb{R}^2 \approxeq \mathbb{R}^{1\times 2}\) using matrices multiplication.
One can compute the vector:
$$ \phi_M(t) \triangleq \left[\varphi_{\text{per}}(Mt - k)\right]_{k=0}^{M-1} \in \mathbb{R}^M \approxeq \mathbb{R}^{1\times M} $$
and get:
$$ \gamma(t) = \phi_M(t)\ \mathbf{c} $$
Thus, for a given number of points \(N\), one can easily extend this scheme to sample a set \(\mathbf{X}\) of points:
$$ \mathbf X \triangleq [\gamma(t_i)]_{i=0}^{N-1} \in \mathbb{R}^{N\times 2} $$
By computing the matrix:
$$ \mathbf{\Phi} \triangleq [\phi_M(t_i)]_{i=0}^{N-1} \in \mathbb{R}^{N\times M} $$
one gets:
$$ \mathbf X = \mathbf{\Phi}\ \mathbf{c} $$
Sampling evenly on the curve \(\gamma\)
A naive sampling consider a range of \(N\) evenly spread parameters \((t_i)_{i=0}^{N-1} \in [0, 1]^N\), that is:
$$ t_i \triangleq \frac{i}{N-1}, \quad \forall i \in [\![N]\!] $$
This generally does not give evenly sampled points \((\gamma(t_i))_{i=0}^{N-1}\) on the curve, in the sense that:
$$ || \gamma(t_{i+1}) - \gamma(t_{i})||_2 \approxeq d, \quad \forall i \in [\![N-1]\!] $$
$$ d\triangleq \frac{L}{N} $$
$$ L \triangleq \text{length}_\gamma(1) $$
$$ \text{length}_\gamma: t \longmapsto \int_0^t ||\gamma'(t) ||_2 \; \text d t $$
To sample \((\gamma(t_i))_{i=0}^{N-1}\) evenly, one can define \((t_i)_{i=0}^{N-1} \in [0, 1]^N\) as follows:
$$ t_i \triangleq \text{length}_\gamma^{-1}\left(\frac{i}{N-1} L \right), \quad \forall i \in [\![N]\!] $$
In practice, one does not compute \(\text{length}_\gamma^{-1}\) but binary-searches for preimages of \(\text{length}_\gamma\) after having it computed on a discretisation of \([0, 1]\) using \(N_{\text{oversample}} = c \ N\) points, for a given integer \(c\), for instance \(c \triangleq 5\).
This can be done efficiently for all the points using using numerical integration and a schema with the exact same matricial structure as presented above that considers \(\varphi'\) instead of \(\varphi\) for sampling \(\gamma'\).
2-Sphere homeomorphic Spline
Expression of the surface \(\eta\)
A 2-Sphere homeomorphic Spline parametrisation is given by:
$$ \eta:[0, 1]^2 \longrightarrow \mathbb{R}^3 $$
$$ \mathbf{\eta}: (s, t) \longmapsto \sum_{l=-1}^{Ms} \sum_{k=0}^{M_t -1} \mathbf{c}[l, k] \varphi_{\frac{\pi}{M_s - 1}}((M_s - 1)s - l)\ \varphi_{\frac{2\pi}{M_t}, \text{per}}(M_t t - k) $$
where the following parameters are considered:
\(s\in [0,1]\), for the longitude,
\(t\in [0,1]\), for the latitude,
where the following spline coefficients are considered:
$$ \mathbf{c} \in \mathbb{R}^{(M_s + 2)\times M_t \times 3} $$
and where the exponential spline \(\phi_a\) for \(\alpha \in \left\{\frac{2\pi}{M_t}, \frac{\pi}{M_s - 1}\right\}\) is chosen
\begin{align} \varphi_\alpha:x \longmapsto \begin{cases} \frac{\cos(\alpha|x|)\cos(\frac{\alpha}{2})-\cos(\alpha)}{\left(1-\cos(\alpha)\right)} & 0 \leq |x| < \frac{1}{2}\\ \frac{\left(1-\cos(\alpha(3/2-|x|))\right)}{2\left(1-\cos(\alpha)\right)} & \frac{1}{2} \leq |x| < \frac{3}{2} \\ 0 & \frac{3}{2} \leq |x| \end{cases}. \end{align}
On coefficients and control points
The coefficients shown above contain more than the actual control points.
Indeed, the parametrisation is fully determined by \(M_t(M_s-2)+6\) points:
the north pole \(\mathbf{c}_\mathrm{N}\)
the south pole \(\mathbf{c}_\mathrm{S}\)
the tangents at the north pole \(\mathbf{T}_{1,\mathrm{N}}\), \(\mathbf{T}_{2,\mathrm{N}}\)
the tangents at the south pole \(\mathbf{T}_{1,\mathrm{S}}\), \(\mathbf{T}_{2,\mathrm{S}}\)
\((M_s - 2)\times M_t\) actual control points (excluding \(\mathbf{c_N}\) and \(\mathbf{c_S}\), the north and south pole):
$$ \mathbf{c}[l, k],\quad (l,k) \in [\![1, M_s - 2 ]\!] \times [\![M_t]\!] $$
Moreover, \(4\times M_t\) additional control points at the extremities of each open longitude are considered to ensure correct behaviour at the boundaries are used for continuity and smoothness conditions due to the support size of the exponential spline basis:
$$ \mathbf{c}[l, k],\quad (l,k) \in \{-1, 0, M_s - 1, M_s \} \times [\![M_t]\!] $$
Their expression is given, for \(k \in [\![M_t]\!]\), by:
$$ \left\{ \begin{array} \mathbf{c}[-1,k] &=& \mathbf{c}[1,k] - \frac{\mathbf{T}_{1,\mathrm{N}}c_{M_t}[k] + \mathbf{T}_{2,\mathrm{N}}s_{M_t}[k]}{(M_s-1)\varphi'_{\frac{\pi}{M_s-1}}(1)} \\ \mathbf{c}[0,k] &=& \frac{\mathbf{c}_\mathrm{N} - \varphi_{\frac{\pi}{M_s-1}}(1)(\mathbf{c}[-1,k]+\mathbf{c}[1,k])}{\varphi_{\frac{\pi}{M_s-1}}(0)} \\ \mathbf{c}[M_s,k] &=& \mathbf{c}[M_s-2,k] - \frac{\mathbf{T}_{1,\mathrm{S}}c_{M_t}[k] + \mathbf{T}_{2,\mathrm{S}}s_{M_t}[k]}{(M_s-1)\varphi'_{\frac{\pi}{M_s-1}}(1)}\\ \mathbf{c}[M_s-1,k] &=& \frac{\mathbf{c}_\mathrm{S} - \varphi_{\frac{\pi}{M_s-1}}(1)(\mathbf{c}[M_s-2,k]+\mathbf{c}[M_s,k])}{\varphi_{\frac{\pi}{M_s-1}}(0)} \end{array} \right. $$
Periodising latitudes using argument wrapping
As in the 2D case, one can define a wrapping function \(\text{wrap}_{M_t}\) to make \(\varphi_{\frac{2\pi}{M_t}}\) periodic, that is
$$ \forall k \in [\![M_t]\!], \forall t \in [0,1] \quad \varphi_{\frac{2\pi}{M_t}, \text{per}}(M_t t - k) = \varphi_{\frac{2\pi}{M_t}} \circ \text{wrap}_{M_t}(t,k) $$
Wrapping is used in this implementation.
Matricial expression of the surface
We can rewrite the expression
$$ \mathbf{\eta} (s, t) = \sum_{l=-1}^{Ms} \sum_{k=0}^{M_t -1} \mathbf{c}[l, k] \varphi_{\frac{\pi}{M_s - 1}}((M_s - 1)s - l)\ \varphi_{\frac{2\pi}{M_t}, \text{per}}(M_t t - k) $$
as a matricial product.
To do so, we first define the following matrix:
$$ \begin{array} \ \Phi(s, t) &\triangleq& \left[\varphi_{\frac{\pi}{M_s - 1}}((M_s - 1)s - l) \ \varphi_{\frac{2\pi}{M_t}, \text{per}}(M_t t - k)\right]_{(l, k) \in [\![M_s + 2]\!]\times [\![M_t]\!]} &\in& \mathbb{R}^{(M_s + 2)\times M_t} \end{array} $$
This way we have:
$$ \mathbf{\eta} (s, t) = \sum_{l=-1}^{Ms} \sum_{k=0}^{M_t -1} \Phi(s, t)[l, k] \ \mathbf{c}[l, k] $$
To combine both sums in a sole one, we define \(\tilde\Phi(s, t)\) and \(\mathbf{\tilde c}\), which are flattened version of \(\Phi(s, t)\) and \(\mathbf{c}\):
$$ \begin{array} \ \tilde\Phi(s, t) &\triangleq& \text{flatten}(\Phi(s, t)) &\in & \mathbb{R}^{(M_s + 2)M_t} \approxeq \mathbb{R}^{1 \times (M_s + 2)M_t} \\ &=& \left[\varphi_{\frac{2\pi}{M_t}, \text{per}}(M_t t - k) \varphi_{\frac{\pi}{M_s - 1}}((M_s - 1)s - l)\right]_{l\in [\![-1, M_s]\!], k \in [\![0, M_t ]\!]} \end{array} $$
$$ \begin{array} \quad \mathbf{\tilde c} &\triangleq& \text{flatten}(\mathbf{c}) &\in& \mathbb{R}^{(M_s + 2)M_t \times 3} \\ &=& \left[\mathbf{c}[k,l]\right]_{i\triangleq (l\ M_t + k) \in [\![(M_s + 2) M_t ]\!]} \end{array} $$
So that we get the nice matricial expression:
$$ \mathbf{\eta}(s, t) = \tilde\Phi(s, t)\ \mathbf{\tilde c} \in \mathbb{R}^{1\times 3} $$
Efficiently sampling using the matricial expression
Now, for some sampling parameters \(N_s, N_t\), how to obtain:
$$ \ \mathbf{X} \triangleq \left(\eta(s_i, t_j)\right)_{(i, j) \in {[\![N_s]\!] \times [\![N_t]\!]}} \in \mathbb{R}^{N_sN_t\times 3} $$
And how to do it efficiently for several splines when \(N_s, N_t\) and \(\varphi\) are fixed? One can proceed as follows:
Compute the tensor resulting from the concatenation of the points \(\Phi\) matrices:
$$ \mathbf{\Phi} \triangleq \left[\Phi(s_i, t_j) \right]_{(i, j) \in {[\![N_s]\!] \times [\![N_t]\!]}} \in \mathbb{R}^{N_s\times N_t\times (M_s + 2)\times M_t} $$
Flatten it on its two first axis and on its two lasts, that is:
$$ \begin{array} \mathbf{\tilde\Phi} &\triangleq& \text{flatten}(\mathbf{\Phi}) \in \mathbb{R}^{N_s N_t\times (M_s + 2)M_t} \\ &=& \left[\tilde\Phi(s_i, t_j) \right]_{p\triangleq(i N_t+ j) \in {[\![N_s N_t]\!]}} \end{array} $$
Cache \(\mathbf{\tilde\Phi}\)
Use \(\mathbf{\tilde\Phi}\) on several splines' flattened coefficients \(\mathbf{\tilde c}\):
$$ \ \mathbf{X} = \mathbf{\tilde \Phi} \mathbf{\tilde c} \in \mathbb{R}^{N_sN_t \times 3} $$
Computing \(\Phi\) efficiently: some tensor wizardry
First note that \(\Phi(s, t) \in \mathbb{R}^{(M_s + 2) \times M_t}\) can be computed as the outer product of two vectors
$$ \Phi(s, t) = \phi_{M_s}^{\text{long}}(s) \otimes \phi_{M_t}^{\text{lat}}(t) $$
$$ \left\{ \begin{array} \ \phi_{M_s}^{\text{long}}(s) &\triangleq& \left[\varphi_{\frac{2\pi}{M_s -1}}((M_s - 1)s - l)\right]_{l \in [\![M_s + 2]\!]} &\in& \mathbb{R}^{M_s + 2} \\ \phi_{M_t}^{\text{lat}}(t) &\triangleq& \left[\varphi_{\frac{2\pi}{M_t}, \text{per}}(M_t t - k)\right]_{k \in [\![M_t]\!]} &\in& \mathbb{R}^{M_t} \end{array} \right. $$
This can be generalised on all the points \((i, j) \in {[\![N_s]\!] \times [\![N_t]\!]}\). Indeed suppose that we have the two following matrices which concatenates the vectors for each point:
$$ \left\{ \begin{array} \ \phi_{(N_s, M_s)}^{\text{long}} &\triangleq& \left[\phi_{M_s}^{\text{long}}(s_i)\right]_{i \in [\![N_s]\!]} &\in& \mathbb{R}^{N_s \times (M_s + 2)} \\ \phi_{(N_t, M_t)}^{\text{lat}} &\triangleq& \left[\phi_{M_t}^{\text{lat}}(t_j)\right]_{j \in [\![N_t]\!]} &\in& \mathbb{R}^{N_t \times M_t} \end{array} \right. $$
\(\mathbf{\Phi}\) can be computed by performing the outer product with respect to the two last axis of those:
$$ \mathbf{\Phi} = \left[\phi_{(N_s, M_s)}^{\text{long}}[i,:] \otimes \phi_{(N_t, M_t)}^{\text{lat}}[j,:]\right]_{(i, j) \in {[\![N_s]\!] \times [\![N_t]\!]}} \in \mathbb{R}^{N_s\times N_t\times (M_s + 2)\times M_t} $$
This can be computed in a call of numpy.einsum with the proper subscripts. See this small numpy snippet to understand the subscripts used for Einstein notations.
About the implementation of \(\text{flatten}\)
\(\text{flatten}\) is implemented using reshaping (e.g np.reshape) which generally is a constant time and memory operation as solely the tensors strides are being modified when reshaping. For more information, see the this comment.
Further work
This Python implementation allows sampling 10 million surface splines' points in a few second which should be fine for most application.
Still, the slow part of constructing \(\mathbf{\Phi}\) can be further speeded up, especially using parallelization because np.einsum is single-threaded. Might Python transpilers like Numba or Pythran help in this regards as they can translate numpy instructions into efficient machine code?
Finally and more importantly, some time is needed to reorder the implementation, release it and port it to use GPUs.
R. Delgado-Gonzalo, P. Thévenaz, M. Unser, Exponential splines and minimal-support bases for curve representation, Computer Aided Geometric Design, Volume 29, Issue 2, 2012, Pages 109-128, ISSN 0167-8396, https://doi.org/10.1016/j.cagd.2011.10.005
Understanding Multilayer Perceptron in Depth Academic Curriculum | CommonCrawl |
\begin{definition}[Definition:Kronecker Delta/Number]
Let $\Gamma$ be a set.
Then $\delta_{\alpha \beta}: \Gamma \times \Gamma \to \set {0, 1}$ is the mapping on the cartesian square of $\Gamma$ defined as:
:$\forall \tuple {\alpha, \beta} \in \Gamma \times \Gamma: \delta_{\alpha \beta} :=
\begin {cases}
1 & : \alpha = \beta \\
0 & : \alpha \ne \beta
\end {cases}$
This use of $\delta$ is known as the '''Kronecker delta notation''' or '''Kronecker delta convention'''.
\end{definition} | ProofWiki |
\begin{document}
\title{Continuous Variable Quantum Repeaters}
\author{Josephine Dias} \email{[email protected]} \author{Tim Ralph}
\affiliation{Centre for Quantum Computation and Communication Technology$,$ School of Mathematics and Physics$,$ University of Queensland$,$ St. Lucia$,$ Queensland 4072$,$ Australia}
\date{\today}
\begin{abstract} We propose a quantum repeater for continuous variable (CV) quantum optical states. Our repeater relies on an error correction protocol for loss on CV states based on CV teleportation and entanglement distillation via noiseless linear amplification. The error correction protocol is concatenated to preserve the same effective transmission coefficient for the quantum channel over increasing distance. The probability of successful operation of the repeater scales polynomially with distance. However, the protocol is limited by a trade-off between fidelity and probability of success.
\end{abstract}
\maketitle
\section{Introduction} Quantum communication enables various cryptographic protocols that outperform their classical counterparts including Quantum Key Distribution (QKD), with its promise of absolutely secure transmission of information \cite{gisin2002quantum}. The use of quantum optical systems as information carriers is currently the only practical approach to quantum communication \cite{bachor2004guide}. Never-the-less, one of the biggest challenges facing the realisation of long distance quantum communication is optical loss due to fibre or free-space attenuation. One proposed method to enable long distance transmission of quantum states is the quantum repeater \cite{briegel1998quantum}. In this model, a lossy quantum channel is segmented into smaller, more manageable attenuation lengths along which entanglement is distributed and then purified. Entanglement swapping operations are then performed resulting in entanglement being held between both ends of the quantum channel.
There have been a number of proposals for quantum repeaters that work on discrete variable quantum systems such as the polarization of single photons \cite{sangouard2011quantum}, and some elements of these have been implemented experimentally. However, quantum communication protocols can also be implemented using quantum continuous variables \cite{weedbrook2012gaussian}. To date, a complete quantum repeater protocol for continuous variables has not been described, although evidence that CV quantum repeaters can increase transmission distances has been presented \cite{campbell2013continuous} and hybrid protocols combining continuous and discrete states have been proposed \cite{van2008quantum}. It is known that regenerative stations containing only Gaussian elements cannot act as CV quantum repeaters \cite{namiki2014gaussian}.
In this paper, we outline an architecture for a quantum repeater that may be used with continuous variable quantum optical systems. Our model relies on concatenated error correction protocols consisting of continuous variable teleportation \cite{braunstein1998teleportation} and entanglement distillation via noiseless linear amplification \cite{ralph2009nondeterministic}. The paper is arranged in the following way. In the next section we review the continuous variable error correction protocol that lies at the heart of our repeater. In section III we will describe how the error correction can be concatenated in such a way that the same effective transmission coefficient is maintained even though the physical channel is growing in length. We show that the overhead for this concatenation is polynomial in the length of the channel. We also derive a lower bound for the fidelity of the channel as a function of the channel length. In section IV we evaluate the performance of the continuous variable quantum repeater assuming noiseless linear amplification is implemented via the generalized quantum scissor approach. In section V we consider alternative approaches to implementing the noiseless linear amplification before concluding in the final section.
\section{The Error Correction Protocol} We use the error correction protocol for continuous-variable states described in Ref~\cite{ralph2011quantum}. This technique for quantum error correction is effective against Gaussian noise induced by loss and proceeds by distilling entanglement and using this entanglement for teleportation. \begin{figure}
\caption{\subref{fig:loss-channel} Lossy channel \subref{fig:error-correction1} Protocol for quantum error correction from Ref~\cite{ralph2011quantum}. Here EPR entanglement is distributed through a lossy channel of transmission \(\eta\). Noiseless linear amplification is performed to distill the entanglement which is then used for teleportation.}
\label{fig:loss-channel}
\label{fig:error-correction1}
\label{fig:error-correction}
\end{figure}
The aim of the protocol is to improve the effective transmission of a quantum state through a lossy channel (Fig.1(a)). The protocol is pictured in Fig.\ref{fig:error-correction1} where an Einstein, Podolsky, Rosen (EPR or two mode squeezed) state is distributed through the lossy channel. Distillation is achieved via noiseless linear amplification (NLA) \cite{ralph2009nondeterministic} which is non-deterministic but heralded. When successful, the effect of the NLA on the entanglement is to produce an EPR state of higher purity (for a given entanglement strength) than achievable via direct transmission through the channel. After successful operation of the NLA, the distilled entanglement is used for teleportation: the input signal and the arm of the entangled state that did not pass through the loss are mixed on a 50:50 beamsplitter and conjugate quadratures are detected on each output mode via homodyne detection (also known as dual homodyne detection); the results of the measurement are sent via a classical channel to the receiver; and amplitude and phase modulation proportional to the measurement result are performed to displace the arm of the entanglement that passed through the loss and the NLA, producing the output mode.
For an input coherent state \(\ket{\alpha}\), the action of the lossy channel causes the transformation: \begin{equation} \ket{\alpha}\to\ket{\sqrt{\eta}\alpha} \label{eq:transform loss} \end{equation} where \(\eta\) is the transmission of the channel. In contrast, if the input coherent state is instead teleported using the distilled EPR state and applying gain tuning \cite{polkinghorne1999continuous} we obtain the transformation: \begin{equation} \ket{\alpha}\to\ket{g\sqrt{\eta}\chi\alpha} \label{eq:transform} \end{equation} where \(g\) is the gain of the NLA, and \(\chi\) is the strength of the entanglement. By controlling the gain of the NLA, the effective transmission of the channel can be controlled. In particular, we will be interested in the case where \(g\) is chosen to be \(\frac{1}{\eta^{1/4}\chi}\) and the output \eqref{eq:transform} of the protocol is \(\ket{\eta^{1/4}\alpha}\). That is, the channel of transmission \(\eta\) has been error corrected to an effective transmission of \(\eta_{eff}=\sqrt{\eta}\).
However, it is important to note that the transformation \eqref{eq:transform} is only achieved when the NLA operates in an unphysical asymptotic limit. When implemented with linear optics, the NLA can be constructed from an array of \(N\) modified quantum scissors devices \cite{pegg1998optical}. The input state is split evenly among the \(N\) quantum scissors devices and the state is truncated in the photon number basis to order \(N\). This inevitably limits the fidelity between the input and output states. Additionally, the operation of the NLA is probabilistic and the success probability decreases exponentially with the number of quantum scissors.
Never-the-less, as is shown in Ref~\cite{ralph2011quantum}, this protocol can still be effective at correcting errors induced by loss on field states in the high loss regime.
\section{Concatenation of the Error Correction Protocol} We now present a way to concatenate these error correction protocols in such a way that the effective transmission of the quantum channel is constant with distance and with a probability of success that scales polynomially with distance. In this section we will place bounds on the fidelity of the output state from the repeater based on an assumed fidelity for each of the basic error correction modules. In the subsequent section we will calculate the value for this fidelity under various conditions, and hence estimate the performance of the entire repeater protocol.
\begin{figure}
\caption{Structure of the quantum repeater for continuous variable states. \subref{fig:Concatenate1} Two links of the repeater. Each box labelled ``error correction" corresponds to the error correction protocol depicted in Fig.~\ref{fig:error-correction1}. \subref{fig:Concatenate2} Four links of the repeater. Nesting two error correction boxes inside a larger error correction box represents the replacement of the physical lossy channel within Fig.~\ref{fig:error-correction1} with the error corrected channel depicted in Fig.~\ref{fig:Concatenate1}. \subref{fig:Concatenate3} Eight links of the repeater. The nesting of error correction modules is concatenated again.}
\label{fig:Concatenate1}
\label{fig:Concatenate2}
\label{fig:Concatenate3}
\label{fig:Concatenation}
\end{figure}
The repeater is depicted in Figure~\ref{fig:Concatenation} where each error correction segment represents the protocol in Fig.~\ref{fig:error-correction1}. Each individual error correction segment takes the initial transmission of the channel \(\eta\) to an effective transmission \(\sqrt{\eta}\). When two error correction protocols are run in series as in Fig.~\ref{fig:Concatenate1}, a quantum channel of overall transmission \(\eta^2\) now has effective transmission \(\eta\).
To preserve this transmission \(\eta\) over double the distance, another two links of the repeater are necessary as in Fig.~\ref{fig:Concatenate2}. Furthermore, the four base level error correction protocols are nested within two higher level error correction protocols allowing the transmission \(\eta^4\) to appear as effective transmission \(\eta\). If the distance is to be doubled again, another level of concatenation is required as shown in Fig.~\ref{fig:Concatenate3}. Concatenation proceeds in this way for increasing distance where a channel of transmission \(\eta^{2^k}\) requires \(k-1\) levels of concatenation.
When run in series, two error correction protocols may operate their NLAs independently and simultaneously. Throughout this paper we implicitly assume that high quality quantum memories are available that can store quantum states without loss of fidelity till the synchronising signals arrive from the various NLAs. Therefore, if \(P\) is the success probability for one iteration of the error correction protocol, then the entire protocol in Fig.~\ref{fig:Concatenate1} also operates with success probability \(P\). However, at the first level of concatenation (Fig.~\ref{fig:Concatenate2}) the four individual error correction procedures need to herald successful operation before error correction at the next level of concatenation can proceed. The probability of success for the repeater protocol in Fig.~\ref{fig:Concatenate2} is therefore \(P^2\). Similarly, the success probability for the protocol in Fig.~\ref{fig:Concatenate3} is \(P^3\). Whilst the probability of success is dropping exponentially with the number of concatenations, the distance doubles with each concatenation. Thus, in general we have: \begin{equation} P_M = P^{\log_2 M} = M^{\log_2 P} \end{equation} where \(M\) is the number of links of the quantum repeater, and thus a polynomial scaling of success probability with distance.
To quantify how the quality of the transmitted state decays with distance we use the fidelity, $F$, between input and output states. As stated earlier, the transformation \(\ket{\alpha}\to\ket{g\sqrt{\eta}\chi\alpha}\) is only achieved for fidelity \(F<1\) due to state truncation from the NLA. Formally, the output after one segment of error correction is: \begin{equation} \hat{\rho} = F \ket{g\sqrt{\eta}\chi\alpha}\bra{g\sqrt{\eta}\chi\alpha}+(1-F)\hat{\rho}_{\tilde{T}} \end{equation} where \(\hat{\rho}_{\tilde{T}}\) is orthogonal to the target state \(\ket{g\sqrt{\eta}\chi\alpha}\). Another iteration of the error correction protocol performs the transformation: \begin{equation} \hat{\rho}^\prime = F(F \ket{g^2\eta\chi^2\alpha^2}\bra{g^2\eta\chi^2\alpha^2}+(1-F)\hat{\rho}_{\tilde{T}^2}) + (1-F) \hat{\tilde{\rho}} \end{equation} where \(\hat{\rho}_{\tilde{T}^2}\) is orthogonal to the new target state \(\ket{g^2\eta\chi^2\alpha^2}\) and \(\hat{\tilde{\rho}}\) is orthogonal to \(\hat{\rho}\). In this way, two error correction protocols in series (Fig.~\ref{fig:Concatenate1}) produce the required target state \(\ket{g^2\eta\chi^2\alpha^2}\) with fidelity of at least \(F^2\). With two error correction protocols nested within another error correction protocol, the fidelity would be at least \(F^3\) and therefore, the entire protocol in Fig.~\ref{fig:Concatenate2} would have fidelity of at least \(F^6\). In this way, we may say that for \(M\) links of the quantum repeater, the fidelity \(F_M\) between input and output states is bounded below by: \begin{equation} F_M \geq F^{2(M-1)} \label{eq:FM} \end{equation}
\section{Results} In the previous section we described the design of the CV quantum repeater and the scaling properties with distance of its probabilty of success and fidelity, in terms of the probability of success and fidelity of a single error correction module. This fundamental fidelity, \(F\), and success probability, \(P\), are dependent on the entanglement strength of the two mode squeezed state, \(\chi\), and the transmission of the channel between nodes, \(\eta\). Additionally, a higher number of quantum scissors devices employed in the NLA would increase the fidelity, but unfortunately decrease the success probability of the protocol. The task therefore becomes optimising \(F\) and \(P\) to produce the best performance for this quantum repeater. \begin{figure}
\caption{Maximum achievable fidelity as a function of effective channel transmission for two links of the quantum repeater (\(\eta^2\to \eta\)). Plotted curves in blue, red and yellow are for the protocol when the NLA is operating with one, two and three quantum scissors respectively. The green line refers to the NLA operation discussed in Section \ref{sec:opt}.}
\label{fig:FidelityM2}
\end{figure}
\begin{figure}
\caption{Log plot of probability of success of the error corrected channel as a function of effective channel transmission for two links of the quantum repeater (\(\eta^2\to \eta\)). All curves achieve constant fidelity of \(F=0.99\). The solid line shows the success probability of the protocol operating with a single quantum scissor in the linear optics implementation of the NLA, the dashed line is for two quantum scissors and the dot-dashed line is for three quantum scissors.}
\label{fig:LogPSuccess}
\end{figure}
In the Appendix we detail the calculation of \(F\) and \(P\) assuming that the NLA is implemented using the generalised quantum scissor protocol. We further assume ideal detectors, and single photon and EPR sources. These results are used in the following to examine the performance limits of the CV quantum repeater.
The results contained in Fig.~\ref{fig:FidelityM2} show the maximum achievable fidelity for the two links of the quantum repeater protocol pictured in Fig.~\ref{fig:Concatenate1}. This protocol preserves the effective transmission of a channel \(\eta\) over double the actual distance \(\eta^2\) and the plot shows the fidelities that can be achieved when the error correction protocol uses an NLA that consists of one, two or three quantum scissors. As is evidenced by this plot, using more quantum scissors enables you to achieve higher fidelities that may be impossible with fewer quantum scissors devices. The cost is an exponential decrease in probability of success with increasing numbers of quantum scissors.
The plot in Fig.~\ref{fig:LogPSuccess} compares the probability of success for two links when the NLA is operating with one, two or three quantum scissors. All plotted curves in Fig.~\ref{fig:LogPSuccess} achieve constant fidelity of \(F = 0.99\). As is shown, with more quantum scissors devices, achieving this high fidelity becomes possible at higher transmissions. However, the success probability decreases significantly in these cases.
\begin{figure}
\caption{Fidelity of the quantum repeater as a function of the effective channel transmission with the error correction protocol using a single quantum scissor and \(\chi = 0.1\). The solid line shows the fidelity for two links of the repeater (\(\eta^2\to\eta\)), the dashed line is for four links of the repeater (\(\eta^4\to\eta\)) and the dot-dashed line is for eight links (\(\eta^8\to\eta\)).}
\label{fig:Fidelity1QS}
\end{figure} \begin{table}
\caption{Fidelity and probability of success estimates for varying distances with the quantum repeater. Points correspond to plot in Fig.~\ref{fig:Fidelity1QS}. Also shown in this table are the corresponding fidelities and success probabilities for the protocol when operating with two quantum scissors.} \begin{center}
\begin{tabular}{| c | c | c | c | c | c |}
\hline
& & \multicolumn{2}{|c|}{One QS} & \multicolumn{2}{|c|}{Two QS} \\ \hline
& Distance & \(F_M\) & \(P_M\) & \(F_M\) & \(P_M\) \\ \hline
\textcolor[rgb]{0.24,0.6,0.33692}{\ding{108}} & \( \sim 200 \mathrm{km}\)& 0.98 & 0.001 & 0.99& \(1.1\times 10^{-6}\) \\ \hline
\textcolor[rgb]{0.24,0.353173,0.6}{\ding{110}} & \(\sim 400\mathrm{km}\)& 0.94 & \(1.2\times10^{-6}\) & 0.98 &\(1.2 \times 10^{-12} \) \\ \hline
\textcolor[rgb]{0.6,0.24,0.563266}{\ding{117}} & \(\sim 800\mathrm{km}\)& 0.87 & \(1.3\times10^{-9}\) & 0.97 &\(1.3 \times 10^{-18}\) \\
\hline
\end{tabular} \end{center} \label{tab:Distances} \end{table}
We now examine the performance of our quantum repeater over varying distances using the fidelity and probability of success. Using reasonable parameters, the plot in Fig.~\ref{fig:Fidelity1QS} shows an example of the fidelity that can be expected from the repeater protocol (where the NLA consists of only a single quantum scissor). The two-mode squeezed state has fixed entanglement strength of \(\chi = 0.1\) and the plot gives fidelity as a function of transmission between repeater nodes. The different lines on this plot give the fidelity for the different number of nodes used in the repeater.
With the assumption of a loss rate of 0.02dB per kilometre, we note that transmission of \(0.01\) corresponds to loss after approximately 100km of optic fibre. The points highlighted in the plot in Fig.~\ref{fig:Fidelity1QS} correspond to the fidelities with which you can preserve this effective transmission over longer actual distances. Specifically, these distances are 200km, 400km and 800km. Table~\ref{tab:Distances} gives numerical estimates for how the fidelity and probability of success decrease over these distances for the repeater when operating with one or two quantum scissors.
To improve the fidelity you can achieve with this repeater, there are two main options: distribute a weakly entangled EPR state (\(\chi\ll 1\)) and use a high gain of the NLA (\(g\gg 1\)) or employ more quantum scissors in the implementation of the NLA. Both of these options come with the unfortunate cost of a reduction in the probability of success. This signifies the most prominent limitation in using this repeater; that is a high fidelity comes at the expense of the probability of success.
\section{Error Correction with Optimal Amplification \label{sec:opt}} The trade-off between probability of success and fidelity can be improved by considering more general versions of the NLA than the quantum scissor implementation. For example, suppose we could implement the transformation: \begin{align} \ket{0} &\to\ket{0} \nonumber\\ \ket{1} &\to g\ket{1} \nonumber\\ \ket{2} &\to g^2 \ket{2} \label{eq:Ideal} \end{align} This is unlike the transformation of the NLA when implemented with two quantum scissors (which performs \(\ket{0}\to\ket{0}\), \(\ket{1}\to g\ket{1}\) and \(\ket{2}\to\frac{1}{2}g^2 \ket{2}\) up to a normalisation factor). The implementation of the transformation in equation \ref{eq:Ideal} would have an improvement on the fidelity, specifically shown by the green line in Fig.~\ref{fig:FidelityM2}. Here, it can be seen that the fidelity achievable using \eqref{eq:Ideal} is comparable to that of the NLA consisting of three quantum scissors. A physical implementation of eq.~\eqref{eq:Ideal} has been proposed using linear optics \cite{lund2014private}, however the probabilities of a successful transformation are orders of magnitude below its quantum scissors counterpart. The theoretical maximum for probability of success of noiseless amplification has been shown to scale as \(g^{-2N}\) where \(N\) is the order of state truncation \cite{pandey2013quantum}. For the transformation given in \eqref{eq:Ideal}, this would be \(g^{-4}\). Amplification of this type has been theoretically modelled in Ref~\cite{mcmahon2014optimal}. Hence, in principle we can achieve fidelities similar to 3 quantum scissors with probabilities similar to 2 quantum scissors in this way. However, note that the explicit construction in Ref~\cite{mcmahon2014optimal} requires non-linear optical interactions.
\section{Conclusion} In summary, we have proposed a method to concatenate error correction protocols to produce a quantum repeater that works with CV states. The error correction relies on continuous variable teleportation and entanglement distillation through noiseless linear amplification. While teleportation of CV states is advantageous because of its deterministic operation, it also limits the channel transmission improvement achievable between input and output states. Fidelity is limited by the NLA due to the state truncation. However, the use of CV teleportation also means the protocol will work on any field state and is therefore not limited to a particular optical encoding of quantum information.
The repeater protocol we present here is limited due to the inevitable trade-off between fidelity and probability of success. As such, there remains significant room for improvement with the protocol used for entanglement distillation. It remains an open question as to how the protocol may be amended to produce higher fidelities while maintaining (or improving) the probability of success.
\begin{acknowledgments} We thank Remi Blandino and Austin Lund for useful discussions. This research was funded by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (Project No. CE110001027). \end{acknowledgments}
\section{Appendix}
In this appendix, we provide details on the calculation of the fidelity \(F\), and the probability of success \(P\) of a single error correction module (pictured in Fig.~\ref{fig:error-correction1}). The continuous variable teleportation protocol uses EPR entanglement of the form. \begin{equation} \ket{EPR} = \sqrt{1-\chi^2}\sum_{n=0}^\infty \chi^n \ket{n}\ket{n} \label{eq:EPR_state} \end{equation} In the error correction protocol, one arm of this entanglement is mixed with the input signal \(\ket{\alpha}\) and then conjugate quadratures are detected. The state after detection can be described as: \begin{equation} \bra{p_a}\bra{x_b} \hat{U}_{BS} \ket{\alpha}_a \ket{EPR}_b \label{eq:coherent_epr_projection} \end{equation} In \cite{blandino2014channel}, it was shown that this dual homodyne measurement can also be expressed using the equivalence \begin{equation} \bra{p_a}\bra{x_b} \hat{U}_{BS} \hat{D}_a(\alpha)\propto \bra{p_a} \bra{x_a}\hat{U}_{BS} \hat{D}_b(-\alpha^*) \end{equation} That is, with measurements on the \(\hat{X}\) and \(\hat{P}\) quadratures after a beam splitter, a coherent state of amplitude \(\alpha\) incoming on mode \(a\) is equivalent to a displacement of \(-\alpha^*\) on mode \(b\) and the vacuum \(\ket{0}\) entering mode \(a\). We now use the result from \cite{blandino2014channel} that states \(\bra{p_a}\bra{x_b} \hat{U}_{BS} \ket{0}_a =\frac{1}{\sqrt{\pi}} \bra{\beta} \), or dual homodyne detection corresponds to a projection onto a coherent state \(\bra{\beta}\) where \(\beta = \frac{x+ip}{\sqrt{2}}\). The state can now be written: \begin{equation} \frac{1}{\sqrt{\pi}} \bra{\beta} \hat{D}_b(-\alpha^*)\ket{EPR}_{bc} \end{equation} Expanding the EPR state in the number basis gives: \begin{align} \sqrt{\frac{1-\chi^2}{\pi}}&\bra{\beta} \hat{D}_b(-\alpha^*) \sum_n \chi^n \ket{n}_b \ket{n}_c \\ =& \sqrt{\frac{1-\chi^2}{\pi}}\bra{\beta} \hat{D}^\dagger_b(\alpha^*) \sum_n \chi^n \ket{n}_b \ket{n}_c \\ =& \sqrt{\frac{1-\chi^2}{\pi}}\bra{\beta+\alpha^*} \sum_n \chi^n \ket{n}_b \ket{n}_c \\
=&\sqrt{\frac{1-\chi^2}{\pi}} \sum_n e^{-\frac{1}{2}|\beta^*+\alpha|^2}\frac{(\chi(\beta^*+\alpha))^n}{\sqrt{n!}} \ket{n} \\
=& \sqrt{\frac{1-\chi^2}{\pi}} e^{\frac{1}{2}|\beta^*+\alpha|^2(\chi^2-1)}\ket{\chi(\beta^*+\alpha)} \end{align} This pure coherent state passess through a lossy channel of transmission \(\eta\), and is transformed as: \begin{equation}
\hat{\rho}= \frac{1-\chi^2}{\pi}e^{|\beta^*+\alpha|^2(\chi^2-1)}\ket{\sqrt{\eta}\chi(\beta^*+\alpha)}\bra{\sqrt{\eta}\chi(\beta^*+\alpha)} \label{eq:eta_chi_beta_alpha} \end{equation} We then use the Noiseless Linear Amplifier to purify the entanglement. In the quantum scissors implementation of the NLA, with \(N\) quantum scissors, an input number state \(\ket{n}\) is transformed as \cite{ralph2009nondeterministic}: \begin{equation} \hat{T}_N \ket{n} = \left(\frac{1}{1+g^2}\right)^{\frac{N}{2}} \frac{N!}{(N-n)!N^n}g^n\ket{n} \label{eq:T_N} \end{equation} For a single quantum scissor, this transformation is: \begin{equation} \hat{T}_1(\alpha\ket{0} +\beta\ket{1}) = \sqrt{\frac{1}{g^2+1}}(\alpha\ket{0} + g\beta\ket{1}) \label{eq:single_qs} \end{equation} with all higher order terms truncated. Therefore, the state after action of the NLA becomes: \begin{equation}
\sqrt{\frac{1-\chi^2}{1+g^2}} \frac{1}{\sqrt{\pi}} e^{\frac{1}{2}|\beta^*+\alpha|^2(\chi^2-1-\eta\chi^2)}\left(\ket{0}+g\sqrt{\eta}\chi(\beta^*+\alpha) \ket{1}\right) \label{eq:qs_state} \end{equation} The last remaining step in this error correction protocol is a displacement depending on the result from the dual homodyne measurements, \(\beta\) and a scaling by \(g\sqrt{\eta}\chi\) to account for the entanglement, lossy channel and gain of NLA. The displacement operator \(\hat{D}(-g\sqrt{\eta}\chi\beta^*)\) is applied to \eqref{eq:qs_state} and finally we obtain the un-normalised output state of the error correction protocol: \begin{align}
\ket{\psi_{1}} = \sqrt{\frac{1-\chi^2}{1+g^2}} \frac{1}{\sqrt{\pi}} e^{\frac{1}{2} |\beta^*+\alpha|^2(\chi^2-1-\eta\chi^2)}\hat{D}(-g\sqrt{\eta}\chi\beta^*)\nonumber\\ \left(\ket{0}+g\sqrt{\eta}\chi(\beta^*+\alpha)\ket{1}\right) \label{eq:alpha_out} \end{align} We can now compute the fidelity, or overlap of this output state with the required target state \(\ket{g\sqrt{\eta}\chi\alpha}\). Fidelity is defined as: \begin{equation}
F( \ket{\varphi},\ket{\psi}) = |\braket{\varphi|\psi}|^2 \end{equation} Note that the state \(\ket{\psi_1}\) is dependent on the measurement outcome \(\beta\) from the homodyne detection. In computing the fidelity between this state and the target state \(\ket{g\sqrt{\eta}\chi\alpha}\), we need to average over all possible \(\beta\): \begin{equation}
F(\ket{g\sqrt{\eta}\chi\alpha}, \ket{\psi_{1}}) = \frac{1}{\int \braket{\psi_{1}|\psi_{1}}\mrm{d}^2\beta} \int|\braket{g\sqrt{\eta}\chi\alpha|\psi_{1}}|^2\mrm{d}^2\beta \end{equation}
where integration over the complex amplitude \(\mrm{d}^2\beta\) denotes integration over the real and imaginary components of \(\beta\). The factor of \(\frac{1}{\int \braket{\psi_{1}|\psi_{1}} \mrm{d}^2 \beta}\) is needed for normalisation, and it is also important to note that the norm of the un-normalised state gives the probability of success: \begin{equation}
P_{suc} = \int \braket{\psi_{1}|\psi_{1}}\mrm{d}^2\beta \end{equation} \begin{widetext} We now compute the probability of success. \begin{align}
\braket{\psi_{1}|\psi_{1}} = \frac{1-\chi^2}{1+g^2}\frac{1}{\pi} & e^{|\beta^*+\alpha|^2(\chi^2-1-\eta\chi^2)}\left(\bra{0} +g\sqrt{\eta}\chi(\beta+\alpha^*)\bra{1}\right) \hat{D}^\dagger(-g\sqrt{\eta}\chi\beta^*)\nonumber \\& \hat{D}(-g\sqrt{\eta}\chi\beta^*)\left(\ket{0}+g\sqrt{\eta}\chi(\beta^*+\alpha)\ket{1}\right) \nonumber \\
= \frac{1-\chi^2}{1+g^2}\frac{1}{\pi} & e^{|\beta^*+\alpha|^2(\chi^2-1-\eta\chi^2)}\left(\bra{0} +g\sqrt{\eta}\chi(\beta+\alpha^*)\bra{1}\right)\left(\ket{0}+g\sqrt{\eta}\chi(\beta^*+\alpha)\ket{1}\right) \nonumber \\
= \frac{1-\chi^2}{1+g^2}\frac{1}{\pi} & e^{|\beta^*+\alpha|^2(\chi^2-1-\eta\chi^2)}\left( 1+g^2\eta\chi^2|\beta^*+\alpha|^2 \right) \nonumber \\ \end{align} Integrating over \(\beta\): \begin{align}
\int\braket{\psi_{1}|\psi_{1}} \mrm{d}^2\beta & =\nonumber \\ \int_{-\infty}^\infty \int_{-\infty}^\infty &\frac{1-\chi^2}{1+g^2}\frac{1}{\pi} e^{|\beta^*+\alpha|^2(\chi^2-1-\eta\chi^2)}\left( 1+g^2\eta\chi^2|\beta^*+\alpha|^2 \right) \mrm{d}\,\mrm{Re}(\beta)\; \mrm{d}\,\mrm{Im}(\beta) \nonumber \\ P_{suc}& = \frac{1-\chi^2}{1+g^2}\frac{1+(-1+\eta+g^2\eta)\chi^2}{(1+(-1+\eta)\chi^2)^2} \end{align} We now compute the fidelity: \begin{align}
\braket{g\sqrt{\eta}\chi\alpha|\psi_{1}} &= \sqrt{\frac{1-\chi^2}{1+g^2}} \frac{1}{\sqrt{\pi}} e^{\frac{1}{2} |\beta^*+\alpha|^2(\chi^2-1-\eta\chi^2)}\bra{g\sqrt{\eta}\chi\alpha}\hat{D}(-g\sqrt{\eta}\chi\beta^*)\left(\ket{0}+g\sqrt{\eta}\chi(\beta^*+\alpha)\ket{1}\right) \nonumber \\
&=\sqrt{\frac{1-\chi^2}{1+g^2}} \frac{1}{\sqrt{\pi}} e^{\frac{1}{2} |\beta^*+\alpha|^2(\chi^2-1-\eta\chi^2)}\bra{g\sqrt{\eta}\chi(\alpha+\beta^*)} \left(\ket{0}+g\sqrt{\eta}\chi(\beta^*+\alpha)\ket{1}\right)\nonumber \\
& = \sqrt{\frac{1-\chi^2}{1+g^2}} \frac{1}{\sqrt{\pi}} e^{\frac{1}{2} |\beta^*+\alpha|^2(\chi^2-1-\eta\chi^2-g^2\eta\chi^2)}(\braket{0|0}+g^2\eta\chi^2|\beta^*+\alpha|^2\braket{1|1})\nonumber \\
&= \sqrt{\frac{1-\chi^2}{1+g^2}} \frac{1}{\sqrt{\pi}} e^{\frac{1}{2} |\beta^*+\alpha|^2(\chi^2-1-\eta\chi^2-g^2\eta\chi^2)}(1+g^2\eta\chi^2|\beta^*+\alpha|^2) \end{align} \begin{equation}
|\braket{g\sqrt{\eta}\chi\alpha|\psi_{1}} |^2=\frac{1-\chi^2}{1+g^2}\frac{1}{\pi}e^{|\beta^*+\alpha|^2(\chi^2-1-\eta\chi^2-g^2\eta\chi^2)}(1+g^2\eta\chi^2|\beta^*+\alpha|^2)^2 \end{equation} Integrating over \(\beta\): \begin{multline}
\int |\braket{g\sqrt{\eta}\chi\alpha|\psi_{1}} |^2\mrm{d}^2\beta = \\ \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{1-\chi^2}{1+g^2}\frac{1}{\pi}e^{|\beta^*+\alpha|^2(\chi^2-1-\eta\chi^2-g^2\eta\chi^2)}(1+g^2\eta\chi^2|\beta^*+\alpha|^2)^2 \mrm{d}\,\mrm{Re}(\beta)\; \mrm{d}\,\mrm{Im}(\beta) \nonumber \end{multline} \begin{align}
F( & \ket{g\sqrt{\eta}\chi\alpha}, \ket{\psi_{1}}) = \frac{1}{\int \braket{\psi_{1}|\psi_{1}}\mrm{d}^2\beta} \int|\braket{g\sqrt{\eta}\chi\alpha|\psi_{1}}|^2\mrm{d}^2\beta \nonumber \\ &= \frac{(1+(-1+\eta)\chi^2)^2}{1+(-1+\eta+g^2\eta)\chi^2} \frac{\left(2 \chi ^2 \left(2 \eta g^2+\eta -1\right)+\chi ^4 \left(\eta \left(5 \eta g^4+4 (\eta -1) g^2+\eta -2\right)+1\right)+1\right)}{\left(\chi ^2 \left(\eta g^2+\eta -1\right)+1\right)^3} \nonumber \\ &= \frac{\left(1+(-1+\eta ) \chi ^2\right)^2 \left(1+2 \left(-1+\eta +2 g^2 \eta \right) \chi ^2+\left(1+\eta \left(-2+4 g^2 (-1+\eta )+\eta +5 g^4 \eta \right)\right) \chi ^4\right)}{\left(1+\left(-1+\eta +g^2 \eta \right) \chi ^2\right)^4} \end{align} \end{widetext} Thus we have calculated the fidelity \(F\) and probability of success \(P\) of a single error correction module for the protocol when operating with a single quantum scissor. This fidelity is independent of coherent amplitude and is therefore valid for a coherent state of any amplitude and also an ensemble of coherent states.
Results for two and three quantum scissors are derived similarly, with the only difference being the transformation \eqref{eq:T_N} is applied with \(N=2\) and \(N=3\) respectively. Results for error correction with optimal amplification use \eqref{eq:Ideal} in place of this transformation.
\end{document} | arXiv |
\begin{document}
\title[Counting planar curves in $\mathbb{P}^3$ with degenerate singularities]{Counting planar curves in $\mathbb{P}^3$ with degenerate singularities}
\author[Nilkantha Das]{Nilkantha Das} \address{School of Mathematical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar, Odisha- 752 050, India.}
\email{[email protected] } \author[Ritwik Mukherjee]{Ritwik Mukherjee}
\address{School of Mathematical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar, Odisha- 752 050, India.} \email{[email protected]}
\subjclass[2010]{14N35, 14J45}
\date{}
\begin{abstract} In this paper, we consider the following question: how many degree $d$ curves are there in $\mathbb{P}^3$ (passing through the right number of generic lines and points), whose image lies inside a $\mathbb{P}^2$, having $\delta$ nodes and one singularity of codimension $k$. We obtain an explicit formula for this number when $\delta+k \leq 4$ (i.e. the total codimension of the singularities is not more than four). We use a topological method to compute the degenerate contribution to the Euler class; it is an extension of the method that originates in the paper by A.~Zinger (\cite{Zin}) and which is further pursued by S.~Basu and the second author in \cite{R.M}, \cite{BM13_2pt_published} and \cite{BM8}. Using this method,
we have obtained formulas when the singularities present are more degenerate than nodes (such as cusps, tacnodes and triple points). When the singularities are only nodes, we have verified that our answers are consistent with those obtained by by S.~Kleiman and R.~Piene (in \cite{KP2}) and by T.~Laarakker (in \cite{TL}). We also verify that our answer for the characteristic number of planar cubics with a cusp and the number of planar quartics with two nodes and one cusp is consistent with the answer obtained by R.~Singh and the second author (in \cite{RS}), where they compute the characteristic number of rational planar curves in $\mathbb{P}^3$ with a cusp. We also verify some of the numbers predicted by the conjecture made by Pandharipande in \cite{RPDeg}, regarding the enumerativity of BPS numbers for $\mathbb{P}^3$.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction} One of the most fundamental and studied problems in enumerative geometry is the following: what is
the number of degree $d$ curves in $\mathbb{P}^2$ that have $\delta$ distinct nodes and that pass through $\frac{d(d+3)}{2} - \delta$ generic points? A more general question is to enumerate the characteristic number of curves that have more degenerate singularities. To make this precise, let us make the following definition. \begin{defn} Let $f:\mathbb{P}^2 \longrightarrow \mathcal{O}(d)$ be a holomorphic section.
A point $q \in f^{-1}(0)$ is said to have a singularity of type $A_k$ or $D_k$ if there exists a coordinate system $(x,y) :(U,q) \longrightarrow (\mathbb{C}^2,0)$ such that $f^{-1}(0) \cap U$ is given by \begin{align*} A_{k \geq 1}: y^2 + x^{k+1} &=0 \qquad \textnormal{and} \qquad D_{k \geq 4}: y^2 x + x^{k-1} =0.
\end{align*} \end{defn} \hf \hf In more common terminology, $q$ is a
{\it simple node} (or just node) if its singularity type is $A_1$; a {\it cusp} if its type is $A_2$; a {\it tacnode} if its type is $A_3$ and an {\it ordinary triple point} if its type is $D_4$.
\begin{rem}
We will frequently use the phrase ``a singularity of codimension $k$''. Roughly speaking, this refers to the number of conditions having that singularity imposes on the space of curves. More precisely, it is the expected codimension of the equisingular strata. Hence, a singularity of type $A_k$ or $D_k$
is a singularity of codimension $k$.
\end{rem} \hf \hf A classical question in enumerative geometry is this: what is $N_d(A_1^{\delta})$, the number of degree $d$ curves in $\mathbb{P}^2$, that have $\delta$ distinct (ordered) nodes, that pass through $\frac{d(d+3)}{2}-\delta$ generic points? More generally, one can ask what is $N_d(A_1^{\delta} \mathfrak{X})$, the number of degree $d$ curves in $\mathbb{P}^2$, that have $\delta$ distinct (ordered) nodes and one singularity of type $\mathfrak{X}$, that pass through $\frac{d(d+3)}{2}-\delta-c_{\mathfrak{X}}$ generic points, where $c_{\mathfrak{X}}$ is the codimension of the singularity $\mathfrak{X}$? \\ \hf \hf The question of computing $N_d(A_1^{\delta})$ and $N_d(A_1^{\delta} \mathfrak{X})$ has been studied for a very long time starting with Zeuthen (\cite{Zu}) more than a hundred years ago. It has been studied extensively in the last thirty years from various perspectives by numerous mathematicians including amongst others, Z.~Ran (\cite{Ran1}, \cite{Ran}), I.~Vainsencher (\cite{Van}), L.~Caporaso and J.~Harris (\cite{CH}), M.~Kazarian (\cite{Kaz}), S.~Kleiman and R.~Piene (\cite{KP1}), D.~Kerner (\cite{Ker1} and \cite{Ker2}), F.~Block (\cite{FB}), Y.~Tzeng and J.~Li (\cite{Tz}, \cite{Tzeng_Li}), M.~Kool, V.~Shende and R.~Thomas (\cite{KST}), S.~Fomin and G.~Mikhalkin (\cite{FoMi}), G.~Berczi (\cite{Berczi}) and S.~Basu and R.~Mukherjee (\cite{R.M}, \cite{BM13_2pt_published} and \cite{BM8}).\\ \hf \hf This problem motivates a natural generalization considered by Kleiman and Piene in \cite{KP2}, where they study the enumerative geometry of nodal curves in a moving family of surfaces (i.e. a fiber bundle version of the earlier question). More recently, this question has been studied further by T.Laarakker in \cite{TL}. \\ \hf \hf Let us now state the question more precisely. We define a \textbf{planar curve} in $\mathbb{P}^3$ to be a curve in $\mathbb{P}^3$, whose image lies inside some $\mathbb{P}^2$. Let us define \begin{align*} N_d^{\textnormal{Planar}, \mathbb{P}^3 }(A_1^{\delta} \mathfrak{X}; r,s) \end{align*} to be the number of planar degree $d$ curves in $\mathbb{P}^3$, intersecting $r$ lines and passing through $s$ points, and having $\delta$ distinct nodes and one singularity of type $\mathfrak{X}$, where $r+2s = \frac{d(d+3)}{2}+3-(\delta +c_{\mathfrak{X}})$ and $c_{\mathfrak{X}}$ is the codimension of the singularity $\mathfrak{X}$.
The result of S. Kleiman and R. Piene (\cite{KP2}) can be used to obtain a formula for $N_d^{\textnormal{Planar}, \mathbb{P}^3 }(A_1^{\delta}; r,s)$, if $\delta \leq 8$ (see section \ref{KP_check} for details). In \cite{TL}, T.Laarakker obtains a formula for $N_d^{\textnormal{Planar}, \mathbb{P}^3 }(A_1^{\delta}; r,s)$, for all $\delta$. \\ \hf \hf The main result of this paper is as follows: \begin{thm} \label{main_thm} Let $\mathfrak{X}$ be a singularity of codimension $c_{\mathfrak{X}}$ and $\delta$ a non negative integer.
We obtain an explicit formula for $N_d^{\textnormal{Planar}, \mathbb{P}^3}(A_1^{\delta} \mathfrak{X}, r,s)$, when $\delta + c_{\mathfrak{X}} \leq 4$, provided $d \geq d_{\textnormal{min}}$, where $d_{\textnormal{min}}:= c_{\mathfrak{X}} + 2\delta.$ \end{thm} In section \ref{low_degree_checks}, we verify that when the singularities present are only nodes,
our answers agree with the answers obtained by S.~Kleiman and R.~Piene (in \cite{KP2}) and by T.~Laarakker (in \cite{TL}). We also verify some of the numbers predicted by the conjecture made by R.~Pandharipande in \cite{RPDeg}, regarding the enumerativity of the BPS numbers for $\mathbb{P}^3$. \\ \hf \hf Very recently, a stable map version of this question has been studied by the second author, A.~Paul and R.~Singh (in \cite{MPS}). In that paper, the authors find a formula for the characteristic number of planar genus zero (rational) degree $d$-curves in $\mathbb{P}^3$.
Building up on the results of that paper, the second author of this paper and R.~Singh obtain a formula for the characteristic number of planar genus zero (rational) degree $d$-curves in $\mathbb{P}^3$ having a cusp (in \cite{RS}). In section \ref{low_degree_checks}, we also verify that our formula for $N_d^{\textnormal{Planar}, \mathbb{P}^3 }(A_2; r,s)$ and $N_d^{\textnormal{Planar}, \mathbb{P}^3 }(A_1^2 A_2; r,s)$ is logically consistent with the formula obtained in \cite{RS},
when $d=3$ and $d=4$ respectively. \begin{rem} \label{KP_remark} In \cite[Theorem 1.2]{KP1}, the authors compute the corresponding numbers $N(A_1^{\delta} \mathfrak{X})$ for a fixed surface, while in \cite{KP2} an algorithm is developed to compute $N(A_1^{\delta})$ for a family of surfaces.
It ought to be possible to generalize the algorithm developed in \cite{KP2} to higher singularities and compute all the numbers obtained by \Cref{main_thm} (this has been point out to us by S. Kleiman \cite{KP3}). \end{rem}
\section{Setup and Notation} \label{notation} \noindent Let us now describe the setup develop some notation to obtain a formula for the numbers stated in \Cref{main_thm}. Our basic objects are planar degree degree $d$ curves in $\mathbb{P}^3$, i.e. degree $d$ curves in $\mathbb{P}^3$ whose image lies inside a $\mathbb{P}^2$.
Let us denote the dual of $\mathbb{P}^3$ by $\widehat{\mathbb{P}}^3$; this is the space of $\mathbb{P}^2$ inside $\mathbb{P}^3$. An element of $\widehat{\mathbb{P}}^3$ can be thought of as a nonzero linear functional $\eta: \mathbb{C}^4 \longrightarrow \mathbb{C}$ upto scaling (i.e., it is the projectivization of the dual of $\mathbb{C}^4$). Given such an $\eta$, we define the projectivization of its zero set as $\mathbb{P}^2_{\eta}$. In other words, \begin{align*} \mathbb{P}^2_{\eta} &:= \mathbb{P}(\eta^{-1}(0)). \end{align*} Note that this $\mathbb{P}^2_{\eta}$ is a subset of $\mathbb{P}^3$.\\ \hf\hf Next, given a positive integer $\delta$, let us define \begin{align*} \mathcal{S}_{\delta} &:= \{ ([\eta], q_1, \ldots, q_{\delta}) \in \widehat{\mathbb{P}}^3 \times (\mathbb{P}^3)^{\delta}: \eta(q_1) =0, \ldots, \eta(q_{\delta}) =0\}. \end{align*} Clearly $\mathcal{S}_{\delta}$ is a fiber bundle over $\widehat{\mathbb{P}}^3$ with fiber $(\P^2)^{\delta}$. This is a plane in $\mathbb{P}^3$ and a collection of $\delta$ points that lie on that plane. We will often abbreviate $\mathcal{S}_1$ as $\mathcal{S}$.
Let us consider the section of the following line bundle induced by the evaluation map, i.e. \begin{align*} \textnormal{ev}:\widehat{\mathbb{P}}^3 \times \mathbb{P}^3 \longrightarrow \gamma_{\widehat{\mathbb{P}}^3}^{*}\otimes \gamma_{\mathbb{P}^3}^*, \qquad \textnormal{given by} \qquad \{\textnormal{ev}([\eta], [q])\}(\eta \otimes q):= \eta(q), \end{align*} where $\gamma_{\widehat{\mathbb{P}}^3}^{*}$ and $\gamma_{\mathbb{P}^3}^*$ are dual of the tautological line bundles over $\widehat{\mathbb{P}}^3$ and $\mathbb{P}^3$ respectively (or equivalently $\mathcal{O}_{\widehat{\mathbb{P}}^3}(1)$ and $\mathcal{O}_{\mathbb{P}^3}(1)$ respectively). Note that \begin{align} \mathcal{S}&= \textnormal{ev}^{-1}(0). \label{s_ev} \end{align} Next, let us denote $\mathcal{D} \longrightarrow \widehat{\mathbb{P}}^3$ to be a fiber bundle over $\widehat{\mathbb{P}}^3$, such that the fiber over each $[\eta] \in \widehat{\mathbb{P}}^3$ is the space of degree $d$ curves in $\mathbb{P}^2_{\eta}$. Next, we note that $\widehat{\mathbb{P}}^3$ is naturally isomorphic to $\mathbb{G}(3,4)$. Let us denote $\gamma_{3,4} \longrightarrow \mathbb{G}(3,4)$ to be the tautological three plane bundle over the Grassmannian. Hence, via this isomorphism we note that
\begin{align*} \mathcal{D} & \approx \mathbb{P}(\textnormal{Sym}^{d} \gamma_{3,4}^{*}) \longrightarrow \widehat{\mathbb{P}}^3. \end{align*} Hence, $\mathcal{D}$ is a fiber bundle over $\widehat{\mathbb{P}}^3$, whose fibers are isomorphic to $\mathbb{P}^{\frac{d(d+3)}{2}}$. An element of $\mathcal{D}$ will be denoted by $([f], [\eta])$; this means that $f$ is a homogeneous degree $d$-polynomial defined on $\mathbb{P}^2_{\eta}$. \\ \hf \hf Next, given a positive integer $\delta$, let us define
\begin{align*} \mathcal{S}_{\mathcal{D}_{\delta}} := \{ ([f], [\eta], q_1, \ldots, q_{\delta}) \in \mathcal{D} \times (\mathbb{P}^3)^{\delta}&: ([\eta], q_1, \ldots, q_{\delta}) \in \mathcal{S}_{\delta}\}.
\end{align*} Note that $\mathcal{S}_{\mathcal{D}_{\delta}}$ can be considered as pull back bundle of $\mathcal{D}$ via the fiber bundle map $\pi: \mathcal{S}_{\delta} \rightarrow \widehat{\mathbb{P}}^3$, i.e. the following diagram \begin{displaymath} \xymatrix{ \mathcal{S}_{\mathcal{D}_{\delta}} \ar[d]_-{\pi_{\mathcal{D}}^*} \ar[r] & \mathcal{D} \ar[d]^-{\pi_{\mathcal{D}}} \\ \mathcal{S}_{\delta} \ar[r]_-{} & \widehat{\mathbb{P}}^3 } \end{displaymath} commutes. We will abbreviate $\mathcal{S}_{\mathcal{D}_{1}}$ as $\mathcal{S}_{\mathcal{D}}$. Next, let $X_1, X_2, \ldots, X_{\delta}$ be subsets of $\mathcal{S}_{\mathcal{D}}$. We define \begin{align*} X_1 \circ X_2 \circ \ldots \circ X_{\delta} := \{ ([f], [\eta], q_1, \ldots, q_{\delta}) \in \mathcal{S}_{\mathcal{D}_{\delta}}& : ([f], [\eta], q_i) \in X_i ~~\forall ~i = 1 ~\textnormal{to} ~\delta \qquad \textnormal{and} \\
& \qquad \qquad \qquad \qquad \quad q_i \neq q_j \qquad \textnormal{if} ~~i \neq j \}. \end{align*} We will make the following abbreviation \begin{align*} X_1^{\delta_1} \circ X_2^{\delta_2} \ldots X_m^{\delta_m} &:= \underbrace{X_1 \circ \ldots \circ X_1}_{\textnormal{$\delta_1$ times}} \circ \underbrace{X_2 \circ \ldots \circ X_2}_{\textnormal{$\delta_2$ times}} \circ \ldots \circ \underbrace{X_m \circ \ldots \circ X_m}_{\textnormal{$\delta_m$ times}}. \end{align*} When $\delta_i =1$, we will omit writing the superscript. For example, \begin{align*} X_1 \circ X_2^{3} \circ X_3 &= X_1^1 \circ X_2^{3} \circ X_3^1 = X_1 \circ X_2 \circ X_2 \circ X_2 \circ X_3. \end{align*} Next, let $\mathfrak{X}$ be a singularity of a given type. We will also denote $\mathfrak{X}$ to be the space of curves and a marked point such that the curve has a singularity of type $\mathfrak{X}$ at the marked point. More precisely, \begin{align*} \mathfrak{X} &:= \{ ([f], [\eta], q) \in \mathcal{S}: ~f ~~\textnormal{has a signularity of type $\mathfrak{X}$ at $q$} \}. \end{align*} For example, \begin{align*} A_2 &:= \{ ([f], [\eta], q) \in \mathcal{S}: ~f ~~\textnormal{has a signularity of type $A_2$ at $q$} \}. \end{align*}
For example, $A_1^{2} \circ A_2$ is the space of curves with three ordered points, where the curve has a simple node at the first two points and a cusp at the last point and all the three points are distinct. Similarly, $A_1^{2} \circ \overline{A}_2$ is the space of curves with three distinct ordered points, where the curve has a simple node at the first two points and a singularity at least as degenerate as a cusp at the last point; the curve could have a tacnode at the last marked point (here $\overline{X}$ indicates the closure of $X$). \\
\hf \hf Next, consider the following rank two vector bundle $\pi:W \longrightarrow \mathcal{S}$, where the fiber over each point $([\eta], q)$ is the tangent space of $\mathbb{P}^2_{\eta}$ at the point $q$, i.e. \begin{align}
\pi^{-1}([\eta], q) &:= T\mathbb{P}^2_{\eta}|_{q}. \label{W_defn} \end{align} Let $W_{\DD}\longrightarrow \mathcal{S}_{\DD}$ denote the pullback of $W$ to $\mathcal{S}_{\DD}$ and let $\mathbb{P}W_{\DD} \longrightarrow \mathcal{S}_{\DD}$ denote the projectivization of $W_{\DD}$.
We can now define the space of curves having a singularity singularity of certain type together with a direction, i.e. if $\mathfrak{X}$ be singularity of a given type, then define \begin{align*} \widehat{\mathfrak{X}} & := \{ ([f], [\eta], l_q) \in \mathbb{P} W_{\mathcal{D}} : f \text{ has a singularity of type } \mathfrak{X} \text{ at } q \}. \end{align*} We can also define the space of curves with a singularity and a specific direction along which certain directional derivatives vanish, i.e. \begin{align*}
\mathcal{P}\A_k &:= \{ ([f], [\eta], l_q) \in \mathbb{P} W_{\mathcal{D}}: ([f], [\eta], q) \in \A_k, ~~\nabla^2 f|_{q}(v, \cdot) =0 \qquad \forall v \in l_q\} \qquad \textnormal{if ~~$k \geq 2$}. \end{align*} For example, $\PP A_2$ is the space of curves with a marked point and a marked direction, such that the curve has a cusp at the marked point and the marked direction belongs to the kernel of the Hessian. Note that
the projection map $\pi : \PP A_k \longrightarrow A_k$ is one to one.
Next, let us define \begin{align*}
\mathcal{P}\A_1 &:= \{ ([f], [\eta], l_q) \in \mathbb{P} W_{\mathcal{D}}: ([f], [\eta], q) \in \A_1, ~~\nabla^2 f|_q(v, v) =0 \qquad \forall v \in l_q\} \qquad \textnormal{and} \\
\mathcal{P}\D_4 &:= \{ ([f], [\eta], l_q) \in \mathbb{P} W_{\mathcal{D}}: ([f], [\eta], q) \in \D_4, ~~\nabla^3 f|_q(v, v, v) =0 \qquad \forall v \in l_q\}. \end{align*} In other words, $\PP A_1$ is the space of curves with a marked point and a marked direction, such that the curve has a node at the marked point and the second derivative along the marked direction vanishes. Note that there are two such distinguished directions. Hence, the projection map $\pi : \PP A_1 \longrightarrow A_1$ is two to one. Similarly, the projection map $\pi : \PP D_4 \longrightarrow D_4$ is three to one.\\ \hf \hf Next, let $\mathcal{S}_{\mathcal{D}_{\delta}}\times_{\mathcal{D}} \mathbb{P} W_{\mathcal{D}}$ denote the fibered product of $\mathcal{S}_{\mathcal{D}_{\delta}}$ and $\mathbb{P} W_{\mathcal{D}}$ over $\mathcal{D}$ via the natural forgetful map. It can be considered as a fiber bundle over $\widehat{\mathbb{P}}^3$ whose fiber over each point $[\eta] \in \widehat{\mathbb{P}}^3$ is \begin{align*} \mathbb{P}(H^0(\mathcal{O}(d), \mathbb{P}^2_{\eta})) \times (\mathbb{P}^2_{\eta})^{\delta} \times \mathbb{P}(T \mathbb{P}^2_{\eta}). \end{align*} Let $\pi:\mathcal{S}_{\mathcal{D}_{\delta}}\times_{\mathcal{D}} \mathbb{P} W_{\mathcal{D}} \longrightarrow \mathcal{S}_{\mathcal{D}_{\delta+1}}$ denote the projection map. If $S$ is a subset of $\mathcal{S}_{\mathcal{D}_{\delta+1}}$, then we define
\begin{align} \widehat{S} &:= \{ ([f], [\eta], q_1, \ldots, q_{\delta}, l_{q_{\delta+1}}) \in \mathcal{S}_{\mathcal{D}_{\delta}}\times_{\mathcal{D}} \mathbb{P} W_{\mathcal{D}}: ([f], [\eta], q_1, \ldots, q_{\delta+1}) \in S \} = \pi^{-1}(S). \label{hat_S_defn} \end{align}
Finally, if $S_1, \ldots S_n$ are subsets of $\mathcal{S}_{\mathcal{D}}$ and $T$ is a subset of $\mathbb{P}W_{\mathcal{D}}$, then we define \begin{align*} S_1 \circ S_2 \circ \ldots \circ S_{\delta}\circ T &:= \{ ([f], [\eta], q_1, \ldots, q_{\delta}, l_{q_{\delta+1}}) \in \mathcal{S}_{\mathcal{D}_{\delta}}\times_{\mathcal{D}} \mathbb{P} W_{\mathcal{D}}: ([f], q_1) \in S_1, \ldots, ([f], q_{\delta}) \in S_{\delta}, \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad ([f], l_{q_{\delta+1}}) \in T \qquad \textnormal{and}\\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad q_1, \ldots, q_{\delta}, q_{\delta+1} \qquad \textnormal{are all distinct}\}. \end{align*}
As an example, $A_1^{2} \circ \PP A_2$ is the space of curves with three distinct ordered points, where the curve has a simple node at the first two points and a cusp at the last point and a distinguished direction at the last marked point, such that the Hessian vanishes along that direction.
\section{Cohomology ring structure of projective fiber bundles} \label{coh_ring} We now recapitulate some basic facts about the cohomology ring of the various spaces we will encounter. First, we recall that via the annihilation map, $\widehat{\mathbb{P}}^3$ is isomorphic to $\mathbb{G}(3,4)$. Via this isomorphism, we can think of $a$ (which is actually a generator of $H^*(\widehat{\mathbb{P}}^3)$) as a generator of $H^*(\mathbb{G}(3,4))$. We note that \begin{align*} c(\gamma_{3,4}^*) & = 1 + a + a^2 + a^3. \end{align*} Next, using the splitting principle, we conclude that \begin{align} c(\textnormal{Sym}^d \gamma_{3,4}^*) & = 1 + s_1 a + s_2 a^2 + s_3 a^3, \qquad \textnormal{where} \\ s_1 & := \frac{d(d+1)(d+2)}{6}, \nonumber \qquad s_2 := \frac{d(d+1)(d+2)(d+3)(d^2+2)}{6} \qquad \textnormal{and} \nonumber \\ s_3 &:= \frac{d(d+1)(d+2)(d+3)(d^2+2)(d^3+3d^2+12d+12)}{1296}. \label{si} \end{align} Notice that $\mathcal{D}= \mathbb{P}(\textnormal{Sym}^d \gamma_{3,4}^*)$, is a $\mathbb{P}^{n-1}$ bundle, where \begin{align} n:= 1 + \frac{d(d+3)}{2}. \label{n_dim_vec_sp} \end{align} Hence, we conclude (by the Leray Hirsch Theorem) that the cohomology ring structure of $\mathcal{D}$ is given by \begin{align} H^*(\mathcal{D}) & \approx \frac{\mathbb{Z}[a, \lambda]}{\langle a^4, ~~\lambda^n + s_1 a\lambda^{n-1} + s_2 a^2\lambda^{n-2} + s_3 a^3\lambda^{n-3}\rangle}, \label{ring_str} \end{align} where $\gamma_{\DD} \longrightarrow \mathbb{P}(\textnormal{Sym}^d \gamma_{3,4}^*)$ denotes the tautological line bundle and $\lambda:= c_1(\gamma_{\DD}^*)$.
\section{Intersection Numbers} Let $\gamma_{W}\longrightarrow \mathbb{P} W$ denote the tautological line bundle over the projective bundle $\mathbb{P}W \longrightarrow \mathcal{S}$. We denote $\lambda_{W} := c_1(\gamma_{W}^*)$
and $H$ to be the standard generator of $H^*(\mathbb{P}^3)$ (i.e. the class of a hyperplane in $\mathbb{P}^3$). \\
\hf \hf We are now in a position to define a few numbers.
Since we will primarily be dealing with planar degree $d$-curves in $\mathbb{P}^3$, we will usually use the prefix $N$ as opposed to the more elaborate $N_d^{\textnormal{Planar},\mathbb{P}^3}$. If there is a chance for confusion, we will use the latter notation.\\ \hf \hf We will occasionally be dealing with curves in $\mathbb{P}^2$. In such a case we will use the notation $N_d^{\mathbb{P}^2}$; we will never use $N$ in such a case. Let us now define \begin{align} \Num(A_1^{\delta} \mathfrak{X}, r, s, n_1, n_2, n_3) &:= \langle a^{n_1} \lambda^{n_2} \pi_{\delta+1}^*H^{n_3}, ~~[\overline{A_1^{\delta} \circ \mathfrak{X}}]\cap \mathcal{H}_L^r \cap \mathcal{H}_p^s \rangle,\\ \Num(A_1^{\delta} \PP \mathfrak{X}, r, s, n_1, n_2, n_3, \theta) &:= \langle a^{n_1} \lambda^{n_2} \pi_{\delta+1}^*H^{n_3} \lambda_{W}^{\theta}, ~~[\overline{A_1^{\delta} \circ \PP \mathfrak{X}}]\cap \mathcal{H}_L^r \cap \mathcal{H}_p^s \rangle \qquad \textnormal{and} \\ \Num(A_1^{\delta} \widehat{\mathfrak{X}}, r, s, n_1, n_2, n_3, \theta) &:= \langle a^{n_1} \lambda^{n_2} \pi_{\delta+1}^*H^{n_3} \lambda_W^{\theta}, ~~[\overline{A_1^{\delta} \circ \widehat{\mathfrak{X}}}]\cap \mathcal{H}_L^r \cap \mathcal{H}_p^s \rangle. \end{align} Here $\pi_i$ denotes the projection onto the $i^{\textnormal{th}}$-point.\\ \hf \hf Next, we note that if $\theta \geq 2$, then \begin{align} \Num(A_1^{\delta} \PP \mathfrak{\X}, r, s, n_1, n_2, n_3, \theta) & = -3 \Num(A_1^{\delta} \PP \mathfrak{\X}, r, s, n_1, n_2, n_3+1, \theta-1) \nonumber \\
& + \Num(A_1^{\delta} \PP \mathfrak{\X}, r, s, n_1+1, n_2, n_3, \theta-1)\nonumber \\
& -\Num(A_1^{\delta} \PP \mathfrak{\X}, r, s, n_1+2, n_2, n_3, \theta-2) \nonumber \\
& + 2\Num(A_1^{\delta} \PP \mathfrak{\X}, r, s, n_1+1, n_2, n_3+1, \theta-2) \nonumber \\
& -3 \Num(A_1^{\delta} \PP \mathfrak{\X}, r, s, n_1, n_2, n_3+2, \theta-2). \label{lm_reduce} \end{align} This is because \begin{align*} \lambda_{W}^2 & = -c_1(W) \lambda_W - c_2(W) \implies \lambda_{W}^2 = -(3H-a) \lambda_{W} -(a^2-2aH+ 3 H^2). \end{align*} \noindent The Chern classes $c_1(W)$ and $c_2(W)$ are given by \cref{ciW}. Next, we note that \begin{align} \Num(\A_1^{\delta} \mathfrak{X}, r, s, n_1, n_2, n_3) & = \frac{1}{\textnormal{deg}(\pi)} \Num(\A_1^{\delta} \PP \mathfrak{X}, r, s, n_1, n_2, n_3, 0), \label{deg_to_one_up_to_down} \end{align} where $\textnormal{deg}(\pi)$ is the degree of the projection map $\pi:\PP \mathfrak{X}\longrightarrow\mathfrak{X}$. We remind the reader that the degree is one when $\mathfrak{X} = A_{k \geq 2}$, it is two when $\mathfrak{X} = A_1$ and it is three when $\mathfrak{X} = D_4$. \\ We also note that \begin{align} \Num(A_1^{\delta} \widehat{\mathfrak{X}}, n_1, n_2, n_3, \theta) & = 0 \qquad \textnormal{if} \qquad \theta =0, \nonumber \\ \Num(A_1^{\delta} \widehat{\mathfrak{X}}, n_1, n_2, n_3, \theta) & = \Num(A_1^{\delta} \mathfrak{X}, n_1, n_2, n_3) \qquad \textnormal{if} \qquad \theta =1 \qquad \textnormal{and} \nonumber \\ \Num(A_1^{\delta} \widehat{\mathfrak{X}}, n_1, n_2, n_3, \theta) & = \Num(A_1^{\delta} \widehat{\mathfrak{X}}, n_1, n_2+1, n_3, \theta-1) -\Num(A_1^{\delta} \widehat{\mathfrak{X}}, n_1, n_2, n_3+1, \theta-2) \nonumber \\ & \qquad \qquad \textnormal{if} \qquad \theta>1. \label{deg_to_one_up_to_down_hat} \end{align} Finally, let us define
\begin{align} N(r,s,n_1, n_2) &:= \langle a^{n_1} \lambda^{n_2}, ~[\mathcal{D}]\cap \mathcal{H}_L^{r} \cap \mathcal{H}_p^s \rangle. \end{align} We now note that
\begin{align} \mathcal{H}_L & = \lambda + d a \qquad \textnormal{and} \qquad \mathcal{H}_p = \lambda a. \label{HL_Hp_class} \end{align} The reason why this is true is explained in \cite[Pages 18 and 19]{Zing_Notes}.
Now, using the ring structure of $\mathcal{D}$ (as given by \cref{ring_str}), we can compute $N(r,s,n_1, n_2)$ by extracting the coefficient of $a^3 \lambda^{n-1}$ from \begin{align*} (\lambda + d a)^r (\lambda a)^{s} a^{n_1} \lambda^{n_2}. \end{align*} Hence, $N(r,s,n_1, n_2)$ can be computed for any $r, s, n_1$ and $n_2$.
\section{Recursive Formulas} \label{recursive_formulas} We are now ready to state the recursive formulas. We have written a mathematica program to implement these recursive formulas and obtain the final formulas. The program is available on the second author's homepage \[ \textnormal{\url{https://www.sites.google.com/site/ritwik371/home}} \] For the convenience of the reader, we have explicitly written down the formulas
for $N(r,s,0,0)$ and $N(A_1^{\delta} \mathfrak{X}, r, s, 0,0)$ in \Cref{expfor}. Note that $N(r,s,0,0)$ is the number of planar degree-$d$ curves intersecting $r$ lines and passing through $s$ points. Our formulas for $N(A_1^{\delta},r, s, 0, 0, 0)$ agree with those obtained by Kleiman and Piene in \cite{KP2} and by Ties Laarakker in \cite{TL}. \\
\begin{thm} \label{na1_again} Consider the ring \begin{align*} \mathcal{R}&= \frac{\mathbb{Z}[a, H, \lambda]}{\langle a^4, ~~H^4, ~~\lambda^n + s_1 a\lambda^{n-1} + s_2 a^2\lambda^{n-2} + s_3 a^3\lambda^{n-3}\rangle}, \end{align*} where $s_1$, $s_2$, $s_3$ and $n$ are as defined in \cref{si} and \cref{n_dim_vec_sp}. Let \begin{align*} e & := (\lambda + H) (\lambda + d a)^r (\lambda a)^s a^{n_1} \lambda^{n_2} H^{n_3}(\lambda + dH)\Big((\lambda + dH)^2 -(3H-a)(\lambda + dH) + a^2 -2aH + 3H^2\Big). \end{align*} Then $N(A_1, r, s, n_1, n_2, n_3)$ is the coefficient of $\lambda^{n-1} a^3 H^3$ in the polynomial $e$, seen as an element of the ring $\mathcal{R}$. \end{thm}
\begin{rem} \Cref{na1_again} is true for all $d \geq 1$. \end{rem} Next, we will give a formula for $N(A_1^{\delta}A_1,r, s, n_1, n_2, n_3)$, when $1 \leq \delta \leq 3$. First let us make a couple of definitions.
\begin{align} \textnormal{Eul}(\delta, r, s, n_1, n_2, 0) & := (d-2d^2 + d^3) N(A_1^{\delta-1} A_1, r,s,n_1+1, n_2, 0) \nonumber \\
& + (3-6d + 3d^2) N(A_1^{\delta-1} A_1, r,s,n_1, n_2+1, 0) \nonumber \\ \textnormal{Eul}(\delta, r, s, n_1, n_2, 1) & := (d^2-d)N(A_1^{\delta-1} A_1, r,s,n_1+2, n_2, 0) \nonumber \\
& + (3d^2-4d+1)N(A_1^{\delta-1} A_1, r,s,n_1+1, n_2+1, 0) \nonumber \\
& + (3d-3) N(A_1^{\delta-1} A_1, r,s,n_1, n_2+2, 0), \nonumber \\ \textnormal{Eul}(\delta, r, s, n_1, n_2, 2) & := d N(A_1^{\delta-1} A_1, r,s,n_1+3, n_2, 0) \nonumber \\
& + (2d-1)N(A_1^{\delta-1} A_1, r,s,n_1+2, n_2+1, 0) \nonumber \\
& + (3d-2)N(A_1^{\delta-1} A_1, r,s,n_1+1, n_2+2, 0) \nonumber \\
& + N(A_1^{\delta-1} A_1, r,s,n_1, n_2+3, 0) \nonumber \\ \textnormal{Eul}(\delta, r, s, n_1, n_2, 3) & := N(A_1^{\delta-1} A_1, r,s,n_1+3, n_2+1, 0)\nonumber \\
& + N(A_1^{\delta-1} A_1, r,s,n_1+2, n_2+2, 0) \nonumber \\
& + N(A_1^{\delta-1} A_1, r,s,n_1+1, n_2+3, 0) \nonumber \\ \textnormal{Eul}(\delta, r, s, n_1, n_2, n_3) &= 0 \qquad \textnormal{if} ~~n_3 >3. \label{eul_num_na1} \end{align} \noindent We also define
\begin{align} \textnormal{B}(\delta, r, s, n_1, n_2, n_3) &:= \binom{\delta}{1}B_1 + \binom{\delta}{2}B_2 + \binom{\delta}{3}B_3, \qquad \textnormal{where} \nonumber \\ B_1 &:= \Big(N(A_1^{\delta-1}A_1, r, s, n_1, n_2+1, n_3) + dN(A_1^{\delta-1}A_1, r, s, n_1, n_2, n_3+1) \nonumber \\ & \qquad \qquad + 3 N(A_1^{\delta-1}\PP A_2, r, s, n_1, n_2, n_3, 0) \Big) \nonumber \\ B_2 &:= 4 \Big( N(A_1^{\delta-2}\PP A_3, r, s, n_1, n_2, n_3, 0) \Big) \nonumber \\ B_3 &:= \frac{18}{3}\Big( N(A_1^{\delta-3}\PP D_4, r, s, n_1, n_2, n_3, 0) \Big). \label{bd_na1}
\end{align}
We are now ready to state the formula for $N(A_1^{\delta}A_1,r, s, n_1, n_2, n_3)$.
\begin{thm} \label{na1_delta} Let $\textnormal{Eul}(\delta, r, s, n_1, n_2, n_3)$ and $\textnormal{B}(\delta, r, s, n_1, n_2, n_3)$ be defined as in \cref{eul_num_na1} and \cref{bd_na1} respectively. If $1 \leq \delta \leq 3$, then \begin{align*} N(A_1^{\delta}A_1,r, s, n_1, n_2, n_3) & = \textnormal{Eul}(\delta, r, s, n_1, n_2, n_3)-\textnormal{B}(\delta, r, s, n_1, n_2, n_3), \end{align*} provided $d\geq 2 \delta +1$. \end{thm}
We now state the remaining formulas.
\begin{thm} \label{npa1} If $0 \leq \delta \leq 2$, then \begin{align*} N(A_1^{\delta} \mathcal{P} A_1, r, s, n_1, n_2, n_3, 0) & = 2 N(A_1^{\delta} A_1, r, s, n_1, n_2, n_3), \\ N(A_1^{\delta} \mathcal{P} A_1, r, s, n_1, n_2, n_3, 1) & = N(A_1^{\delta} A_1, r, s, n_1, n_2+1, n_3)\\
& +(d-6)N(A_1^{\delta} A_1, r, s, n_1, n_2, n_3+1) \\
& + 2 N(A_1^{\delta} A_1, r, s, n_1+1, n_2, n_3) \\
&-2\binom{\delta}{2} N(A_1^{\delta-2} \PP D_4, r, s, n_1, n_2, n_3, 0),
\end{align*} provided $d \geq 2 \delta +2$.
\end{thm}
\begin{rem} To compute $N(A_1^{\delta} \mathcal{P} A_1, r, s, n_1, n_2, n_3, \theta)$ when $\theta \geq 2$, we use \cref{lm_reduce}. \end{rem}
\begin{thm} \label{npa2} Let $0 \leq \delta \leq 2$ and $\theta$ a non negative integer with the following property: if $\delta$ is either $0$ or $1$, then $\theta$ can be anything, but if $\delta=2$, then $\theta=0$. Then, \begin{align*} N(A_1^{\delta} \mathcal{P} A_2, r, s, n_1, n_2, n_3, \theta) & = N(A_1^{\delta} \mathcal{P} A_1, r, s, n_1+1, n_2, n_3, \theta) \\
&+ N(A_1^{\delta} \mathcal{P} A_1, r, s, n_1, n_2+1, n_3, \theta) \\
&+ (d-3)N(A_1^{\delta} \mathcal{P} A_1, r, s, n_1, n_2, n_3+1, \theta) \\
&-2\binom{\delta}{1} N(A_1^{\delta-1} \mathcal{P} A_3, r, s, n_1, n_2, n_3, \theta) \\
&-3 \binom{\delta}{1} N(A_1^{\delta-1} \widehat{D}_4, r, s, n_1, n_2, n_3, \theta) \\
&-4\binom{\delta}{2} N(A_1^{\delta-2} \mathcal{P} D_4, r, s, n_1, n_2, n_3, \theta),
\end{align*} provided $d \geq 2 \delta +2$. \end{thm}
\begin{rem} If $\delta =2$ and $\theta>0$, then the formula given by Theorem \ref{npa2} is not valid; there is a further correction term (the interested reader can refer to \cite{BM8} to see what the extra correction term is). However, to compute $N(A_1^{2}A_2, r, s, n_1, n_2, n_3)$ we only need to know what is $N(A_1^{2} \mathcal{P} A_2, r, s, n_1, n_2, n_3, 0)$ and hence for the purposes of this paper, this Theorem is sufficient. We would require $N(A_1^{2} \mathcal{P} A_2, r, s, n_1, n_2, n_3, \theta)$ for $\theta>0$ if we were computing any of the codimension five (or higher) numbers; in this paper we are computing numbers till codimension four. \end{rem}
\begin{thm} \label{npa3} If $0 \leq \delta \leq 1$, then \begin{align*} N(A_1^{\delta} \mathcal{P} A_3, r, s, n_1, n_2, n_3, \theta) &= N(A_1^{\delta} \mathcal{P} A_2, r, s, n_1, n_2+1, n_3, \theta) \\
&+ 3 N(A_1^{\delta} \mathcal{P} A_2, r, s, n_1, n_2, n_3, \theta+1) \\
&+ d N(A_1^{\delta} \mathcal{P} A_2, r, s, n_1, n_2, n_3+1, \theta) \\
&-2\binom{\delta}{1}N(A_1^{\delta-1} \mathcal{P} A_4, r, s, n_1, n_2, n_3, \theta), \end{align*} provided $d \geq 2 \delta +3$. \end{thm}
\begin{thm} \label{npa4} If $d \geq 4$, then \begin{align*} N(\mathcal{P} A_4, r, s, n_1, n_2, n_3, \theta) &= 2N(\mathcal{P} A_3, r, s, n_1, n_2+1, n_3, \theta) \\
&+2N(\mathcal{P} A_3, r, s, n_1, n_2, n_3, \theta+1) \\
&+2N(\mathcal{P} A_3, r, s, n_1+1, n_2, n_3, \theta)
&+(2d-6)N(\mathcal{P} A_3, r, s, n_1, n_2, n_3+1, \theta) \end{align*} \end{thm}
\begin{thm} \label{npd4} If $d \geq 3$, then \begin{align*} N(\mathcal{P} D_4, r, s, n_1, n_2, n_3, \theta) &= N(\mathcal{P} A_3, r, s, n_1, n_2+1, n_3, \theta) \\
&-2N(\mathcal{P} A_3, r, s, n_1, n_2, n_3, \theta+1) \\
&+2N(\mathcal{P} A_3, r, s, n_1+1, n_2, n_3, \theta)
&+(d-6)N(\mathcal{P} A_3, r, s, n_1, n_2, n_3+1, \theta) \end{align*} \end{thm}
We will now prove these recursive formulas.
\section{Proof of the recursive formulas} We are now ready to prove the formulas stated in section \ref{recursive_formulas}. We will use a topological method to compute the degenerate contribution to the Euler class. Our method is an extension of the method that originates in the paper by A.~Zinger (\cite{Zin}) and which is further pursued by S.~Basu and the second author in \cite{R.M}, \cite{BM13_2pt_published} and \cite{BM8}. \\ \hf \hf When there is no cause for confusion, we will sometimes abbreviate $N(A_1^{\delta}A_1,r, s, n_1, n_2, n_3)$ and \newline $N(A_1^{\delta}\mathcal{P}\mathfrak{X},r, s, n_1, n_2, n_3, \theta)$ as $N(A_1^{\delta}A_1)$ and $N(A_1^{\delta} \mathcal{P}\mathfrak{X})$ (for the sake of notational simplicity).
\subsection{Proof of \Cref{na1_again} and \ref{na1_delta}: computation of $N(A_1^{\delta}A_1)$ when $0\leq \delta \leq 3$} \label{na1_delta_proof} \verb+ +\\
\hf \hf We will justify our formula for $N(A_1^{\delta} A_1,r, s, n_1, n_2, n_3)$, when $0 \leq \delta \leq 3$.
Recall that in \Cref{notation}, we have defined
\begin{align*} \A^{\delta}_1 \circ \mathcal{S}_{\DD} := \{ ([f], [\eta]; q_1, \ldots, q_{\delta}, q_{\delta+1}) & \in \DD \times (\mathbb{P}^3)^{\delta +1}: \eta (q_i) = 0, \qquad \forall i = 1 ~~\textnormal{to} ~~ \delta+1, \\ &\textnormal{$f$ has a singularity of type $\A_1$ at $q_1, \ldots, q_{\delta}$}, \\
& \textnormal{$q_1, \ldots, q_{\delta+1}$ all distinct}\}. \end{align*} Let $\mu$ be a generic cycle, representing the class \begin{align*} [\mu] = \mathcal{H}_L^r \cdot \mathcal{H}_p^s \cdot a^{n_1} \lambda^{n_2} (\pi_{\delta+1}^*H)^{n_3}. \end{align*} Here $\pi_i$ denotes the projection onto the $i^{\textnormal{th}}$ point.
We will often omit writing down $\pi_{\delta+1}^*$, if there is no cause for confusion.
We now consider sections of the following two bundles that are induced by the evaluation map and the vertical derivative at the last point, namely: \begin{align*} \Psi_{\A_0}: A_1^{\delta} \circ \mathcal{S}_{\DD} \longrightarrow \mathcal{L}_{\A_0} & := \gamma_{\DD}^* \otimes \pi_{\delta+1}^*\gamma_{\mathbb{P}^3}^{* d}, \qquad \qquad \{\Psi_{\A_0}([f], [\eta], q_1, \ldots, q_{\delta+1})\}(f) := f(q_{\delta+1}) \qquad \textnormal{and} \\ \Psi_{\A_1} : \psi_{\A_0}^{-1}(0) \longrightarrow \mathcal{V}_{\A_1} &:= \gamma_{\D}^*\otimes \pi_{\delta+1}^*W^* \otimes \pi_{\delta+1}^*\gamma_{\mathbb{P}^3}^{* d}, \qquad
\{\Psi_{\A_1}([f], [\eta], q_1, \ldots, q_{\delta+1})\}(f) := \nabla f|_{q_{\delta+1}}.
\end{align*} We will show shortly that $\psi_{\A_0}$ and $\psi_{A_1}$ are transverse to zero, provided $d \geq 2 \delta +1$. \\ \hf \hf Next, let us define \begin{align*} \mathcal{B} &:= \overline{\A^{\delta}_1 \circ \mathcal{S}}_{\DD}- \A^{\delta}_1 \circ \mathcal{S}_{\DD}. \end{align*}
Hence \begin{align} \langle e(\mathcal{L}_{\A_0}) e(\mathcal{V}_{\A_1}), ~~[\overline{\A^{\delta}_1 \circ \mathcal{S}_{\DD}}] \cap [\mu] \rangle & = \N(\A_1^{\delta}\A_1, r, s, n_1, n_2, n_3) + \mathcal{C}_{\mathcal{B} \cap \mu},\label{euler_na1_delta}
\end{align} where $e$ denote the Euler class and $\mathcal{C}_{\mathcal{B} \cap \mu}$ denotes the contribution of the section to the Euler class from the points of $\mathcal{B} \cap \mu$.\\ \hf \hf Let us first explain how to compute the left hand side of \cref{euler_na1_delta} (i.e. the Euler class).
From equations \eqref{HL_Hp_class} and \eqref{s_ev}, we note that \begin{align*} \mathcal{H}_L & = \lambda + d a, \qquad \mathcal{H}_p = \lambda a \qquad \textnormal{and} \qquad [\pi_{\delta+1}^*\mathcal{S}_{\DD}] = \lambda + \pi_{\delta+1}^*H. \end{align*}
Next, we need to compute the Chern classes of $W$. We note that over $\mathcal{S}$, we have the following short exact sequence of vector bundles: \begin{align*} 0&\longrightarrow W \longrightarrow T \mathbb{P}^3 \longrightarrow \gamma^{*}_{\widehat{\mathbb{P}}^3} \otimes \gamma^{*}_{\mathbb{P}^3} \longrightarrow 0. \end{align*}
Here the first map is the inclusion map and the second map is $\nabla \eta|_q$. Hence, \begin{align} c(W) c(\gamma^{*}_{\widehat{\mathbb{P}}^3} \otimes \gamma^{*}_{\mathbb{P}^3}) & = c(T\mathbb{P}^3) ~~ \implies c_1(W) = 3H-a \qquad \textnormal{and} \qquad c_2(W) = a^2-2aH + 3H^2. \label{ciW} \end{align}
Next, using the splitting principle, we conclude that
\begin{align} e(\gamma_{\mathcal{D}}^*\otimes \gamma_{\mathbb{P}^3}^{* d})e(\gamma_{\mathcal{D}}^*\otimes W^*\otimes \gamma_{\mathbb{P}^3}^{* d}) &= (\lambda + dH)((\lambda + dH)^2 -c_1(W)(\lambda + dH) + c_2(W)). \label{Eul_ring} \end{align} Note that we have made an abuse of notation by omitting to write down $\pi_{\delta+1}^*$; henceforth we will make this abuse of notation. Now, suppose $\delta =0$. Then, using the ring structure of $\mathcal{D}$ (as given by \cref{ring_str}) and by extracting the coefficient of $\lambda^{n-1} a^3 H^3$ from \begin{align*} (\lambda + H)(\lambda + dH)((\lambda + dH)^2 -c_1(W)(\lambda + dH) + c_2(W))(\lambda+d a)^r (\lambda a)^s a^{n_1} \lambda^{n_2} H^{n_3}, \end{align*} we obtain the Euler class. When $\delta=0$, using \cref{ciW}, we get
the formula of Theorem \ref{na1_again}. When $\delta>0$, we get $\textnormal{Eul}(\delta, r, s, n_1, n_2, n_3)$ as defined in \cref{eul_num_na1}.\\ \hf \hf Let us now explain how to compute $\mathcal{C}_{\mathcal{B} \cap \mu}$, the degenerate contribution to the Euler class. When $\delta=0$, the boundary $\mathcal{B}$ is empty and hence we get the result of Theorem \ref{na1_again}. Let us now consider the case when $\delta \geq 1$. Given $k$ distinct integers $i_1, i_2, \ldots, i_k \in [1, \delta+1]$, let us define \begin{align*} \Delta_{i_1, \ldots, i_{k}} & := \{ ([f], [\eta]; q_1, \ldots, q_{\delta}, q_{\delta+1}) \in \mathcal{S}_{\mathcal{D}_{\delta+1}}: ~~q_{i_1}= q_{i_2}=\ldots = q_{i_k}\} \qquad \textnormal{and} \\ \mathcal{B}(q_{i_1}, \ldots, q_{i_k}) & := \mathcal{B}\cap \Delta_{i_1, \ldots, i_k}. \end{align*} Let us now consider $\mathcal{B}(q_i, q_{\delta+1})$. We claim that \begin{align} \mathcal{B}(q_i, q_{\delta+1}) &\approx \overline{A_1^{\delta-1}\circ A}_1 \qquad \forall ~~i=1 ~\textnormal{to} ~\delta, \label{a1_a0_bdry_cmp} \end{align} where $\mathcal{B}(q_i, q_{\delta+1})$ is identified as a subset of $\mathcal{S}_{\mathcal{D}_{\delta}}$ in the obvious way (namely via the inclusion of $\mathcal{S}_{\mathcal{D}_{\delta}}$ inside $\mathcal{S}_{\mathcal{D}_{\delta+1}}$ where the $(\delta+1)^{\textnormal{th}}$ point is equal to the $i^{\textnormal{th}}$ point). Next, we claim that the contribution from $\mathcal{B}(q_i, q_{\delta+1})\cap \mu$ is given by \begin{align} \langle e(\mathcal{L}_{A_0}), ~[\overline{A_1^{\delta-1}\circ A}_1]\cap [\mu] \rangle + 3N(A_1^{\delta-1}\mathcal{P}A_2, r,s,n_1, n_2, n_3, 0). \label{Eul_deg_node_bdry} \end{align} We will explain the reason for both the claims shortly. The expression given by \cref{Eul_deg_node_bdry} is precisely equal to $\mathrm{B}_1$ as defined in \cref{bd_na1}. Hence, the sum total of the contribution from $\mathcal{B}(q_i, q_{\delta+1})$ for $i=1$ to $\delta$ is $\binom{\delta}{1} \mathrm{B}_1$.\\ \hf \hf Next, let us assume $\delta \geq 2$ and consider $\mathcal{B}(q_{i_1}, q_{i_2}, q_{\delta+1})$. We claim that \begin{align} \mathcal{B}(q_{i_1}, q_{i_2}, q_{\delta+1}) &\approx \overline{A_1^{\delta-2}\circ A}_3 \label{a1_a1_is_a3} \end{align} for all distinct pairs $(i_1, i_2)$. We also claim that the contribution from each of the points of \[\mathcal{B}(q_{i_1}, q_{i_2}, q_{\delta+1})\cap \mu\] is $4$. We will justify both these claims shortly. Hence the sum total of the contribution as we vary over all $(i_1, i_2)$ is precisely $\binom{\delta}{2}\mathrm{B_2}$, where $\mathrm{B}_2$ is as defined in \cref{bd_na1}.\\ \hf \hf Finally, let us assume $\delta \geq 3$ and consider $\mathcal{B}(q_{i_1}, q_{i_2}, q_{i_3}, q_{\delta+1})$. We claim that \begin{align} \mathcal{B}(q_{i_1}, q_{i_2}, q_{i_3}, q_{\delta+1}) &\approx \overline{A_1^{\delta-3}\circ A}_5 \cup \overline{A_1^{\delta-3}\circ D}_4\label{three_node_triple_point} \end{align} for all distinct triples $(i_1, i_2, i_3)$. Note that $\overline{A_1^{\delta-3}\circ A}_5 \cap \mu$ is empty, since the sum of their dimensions is one less than the dimension of the ambient space where we are intersecting them. Hence, we get no contribution from $\overline{A_1^{\delta-3}\circ A}_5 \cap \mu$. Finally, we claim that the contribution from each of the points of $\overline{A_1^{\delta-3}\circ D}_4 \cap \mu$ is $18$. Hence the sum total of the contribution as we vary over all $(i_1, i_2, i_3)$ is precisely $\binom{\delta}{3}\mathrm{B_3}$, where $\mathrm{B}_3$ is as defined in \cref{bd_na1}. \\ \hf \hf Let us now prove the claims regarding transversality and degenerate contributions to the Euler class. We will start by proving transversality. Note that we need to prove $A_1^{\delta+1}$ is a smooth complex submanifold of $\mathcal{S}_{\mathcal{D}_{\delta+1}}$ (provided $d\geq 2\delta+1$). We will prove a stronger statement: we will show that $\overline{A}_1^{\delta+1}$ is a smooth complex submanifold of $\mathcal{S}_{\mathcal{D}_{\delta+1}}$ and the sections $\Psi_{A_0}$ and $\Psi_{A_1}$, defined on $\overline{A}_1^{\delta}\circ \mathcal{S}_{\delta}$ and $\Psi_{A_0}^{-1}(0)$ respectively, are transverse to zero. Our desired claim follows immediately from this statement since $A_1^{\delta+1}$ is an open subset of $\overline{A}_1^{\delta+1}$.\\ \hf \hf Let us begin by showing that $\Psi_{A_0}$ is transverse to zero if $d \geq 2\delta +1$. Suppose \begin{align*} \Psi_{A_0}([f], [\eta], q_1, \ldots, q_{\delta+1})&= 0. \end{align*} Without loss of generality, let us assume that $[\eta]$ determines the plane where the last coordinate is zero, and $q_{\delta+1}$ is the point where only the third coordinate is nonzero and the rest are zero, i.e.
\begin{align*} \mathbb{P}^2_{\eta} & \approx \{[X,Y,Z, W] \in \mathbb{P}^3: W=0\} \qquad \textnormal{and} \qquad q_{\delta+1}:= [0,0,1,0]. \end{align*} Assume that the remaining points
are given by \begin{align*} q_i&:= [X_i, Y_i, Z_i, 0] \qquad \textnormal{for}~~i=1~~\textnormal{to}~~\delta. \end{align*} For simplicity, we can assume that all $Z_i$ are nonzero. Furthermore, since all the $q_i$ are distinct, we conclude that $X_i$ and $Y_i$ can not both be zero; for simplicity let us assume $X_i$ is nonzero for each $i$ (from $1$ to $\delta$). Consider the homogeneous degree $d$ polynomial, given by
\begin{align*} \rho_{00}&:= (X-X_1)^2(X-X_2)^2\ldots \cdot (X-X_{\delta})^2 Z^{d-2\delta}.
\end{align*} We note the following facts about $\rho_{00}$: \begin{align} \rho_{00}(q_i) &= 0 \qquad \forall ~~i=1~~\textnormal{to}~~\delta, \label{rho00}\\
\nabla \rho_{00}|_{q_i} &= 0 \qquad \forall ~~i=1~~\textnormal{to}~~\delta \qquad \textnormal{and} \label{nabla_rho_00}\\ \rho_{00}(q_{\delta+1})& \neq 0. \label{rho00_last_pt} \end{align} Now consider the curve $\gamma:(-\varepsilon, \varepsilon) \longrightarrow \mathcal{S}_{\mathcal{D}_{\delta+1}}$, given by \begin{align*} \gamma(t)&:= ([f+t \rho_{00}], [\eta], q_1, \ldots, q_{\delta+1}). \end{align*} Because of \cref{rho00} and \cref{nabla_rho_00}, we conclude that this curve lies in $\overline{A}_1^{\delta}\circ \mathcal{S}_{\mathcal{D}}$. We now note that \begin{align}
\{\{\nabla \Psi_{A_0}|_{([f], [\eta], q_1, \ldots, q_{\delta+1})}\}(\gamma^{\prime}(0))\}(f)& = \rho_{00}(q_{\delta+1}). \label{nabla_Psi_A0} \end{align} Using \cref{rho00_last_pt}, we conclude that the right hand side of \cref{nabla_Psi_A0} is nonzero, whence $\Psi_{A_0}$ is transverse to zero. Next, let us prove transversality for the section $\Psi_{A_1}$. Consider the polynomials, \begin{align*} \rho_{10}&:= (X-X_1)^2(X-X_2)^2\ldots \cdot (X-X_{\delta})^2X Z^{d-2\delta-1} \qquad \textnormal{and} \\ \rho_{01}&:= (X-X_1)^2(X-X_2)^2\ldots \cdot (X-X_{\delta})^2Y Z^{d-2\delta-1}. \end{align*} We note that $\rho_{10}$ and $\rho_{01}$ satisfy \cref{rho00} and \cref{nabla_rho_00} (with $\rho_{00}$ replaced with $\rho_{10}$ and $\rho_{01}$ respectively). Furthermore, \begin{align} \rho_{10}(q_{\delta+1})&=0 \qquad \textnormal{and} \qquad \rho_{01}(q_{\delta+1})= 0. \label{rho_10_and_01} \end{align} Construct the curves \begin{align*} \gamma_{10}(t)&:= ([f+t \rho_{10}], [\eta], q_1, \ldots, q_{\delta+1}) \qquad \textnormal{and} \qquad \gamma_{01}(t):= ([f+t \rho_{01}], [\eta], q_1, \ldots, q_{\delta+1}). \end{align*} Because of \cref{rho00} and \cref{nabla_rho_00} (with $\rho_{00}$ replaced with $\rho_{10}$ and $\rho_{01}$ respectively) and \cref{rho_10_and_01}, these curves lie inside $\Psi_{A_0}^{-1}(0)$. We now note that \begin{align*}
\{\{\nabla \Psi_{A_1}|_{([f], [\eta], q_1, \ldots, q_{\delta+1})}\}(\gamma_{10}^{\prime}(0))\}(f)& = \lambda Z^{d-2\delta-1} \nabla X|_{[0,0,1,0]} \qquad \textnormal{and} \\
\{\{\nabla \Psi_{A_1}|_{([f], [\eta], q_1, \ldots, q_{\delta+1})}\}(\gamma_{01}^{\prime}(0))\}(f)& = \lambda Z^{d-2\delta-1} \nabla Y|_{[0,0,1,0]}, \qquad \textnormal{where} \qquad \lambda := (-X_1)^2 \ldots (-X_{\delta})^2. \end{align*}
Since $\nabla X|_{[0,0,1,0]}$ and $\nabla Y|_{[0,0,1,0]}$ are two linearly independent vectors of $T\mathbb{P}^2_{\eta}|_{[0,0,1,0]}$, we conclude that $\Psi_{A_1}$ is transverse to zero. \\ \hf \hf Let us now justify the closure and multiplicity claims. We will start by giving the reason for \cref{a1_a0_bdry_cmp} and \cref{Eul_deg_node_bdry}. This follows from the argument given in the proof \cite[Lemma 6.3 (1), Page 685]{BM13_2pt_published} and \cite[Corollary 6.6, Page 689]{BM13_2pt_published}. The proof is the same. \\ \hf \hf Next, let us justify \cref{a1_a1_is_a3}. Without loss of generality, it suffices to justify it when $i_1:= \delta-1$ and $i_2:= \delta$. Hence, we need to show that \begin{align} \{([f], [\eta], q_1, \ldots,q_{\delta+1}) \in \overline{A_1^{\delta}\circ \mathcal{S}}_{\mathcal{D}}: q_{\delta-1} = q_{\delta} = q_{\delta+1}\}&= \overline{A_1^{\delta-2}\circ A}_3. \label{a1_a1_a3_exp} \end{align} Before proceeding further, let us make a simple observation. Notice that the left hand side of \cref{a1_a1_a3_exp} is the same as \begin{align} \{([f], [\eta], q_1, \ldots,q_{\delta+1}) \in \overline{A_1^{\delta}}: q_{\delta-1} = q_{\delta}\}. \label{temp_nat} \end{align}
Hence, an equivalent way of stating \cref{a1_a1_a3_exp} is \begin{align} \{([f], [\eta], q_1, \ldots,q_{\delta}) \in \overline{A_1^{\delta}}: q_{\delta-1} = q_{\delta}\} & = \overline{A_1^{\delta-2}\circ A}_3. \label{a1_a1_a3_exp_ag2} \end{align} Following \cite[Equation 6.4, Page 685]{BM13_2pt_published}, we conclude that \begin{align} \Big(\{([f], [\eta], q_1, \ldots, q_{\delta-1}, q_{\delta}, q_{\delta+1}) \in \overline{A_1^{\delta}\circ \mathcal{S}}_{\mathcal{D}}: q_{\delta-1} = q_{\delta} = q_{\delta+1}\}\Big)\cap \Big(A_1^{\delta-2}\circ A_2\Big) &= \emptyset. \label{two_nodes_No_cusp} \end{align} Equation \cref{two_nodes_No_cusp} is saying that if two nodes come together, then the singularity has to be more degenerate than a cusp. Hence, the singularity has to be at least as degenerate as a tacnode (since $\overline{A}_2 = A_2 \cup \overline{A}_3$).
Hence, the left hand side of \cref{a1_a1_a3_exp} is a subset of its right hand side. We will now prove the converse.
We will simultaneously prove the following four statements: \begin{align} \{([f], [\eta], q_1, \ldots, q_{\delta+1}) &\in \overline{A_1^{\delta}\circ \mathcal{S}}_{\mathcal{D}}: q_{\delta-1} = q_{\delta} = q_{\delta+1}\} \supset A_1^{\delta-2}\circ A_3,\label{a1_a1_a3_subset}\\ \Big(\{([f], [\eta], q_1, \ldots, q_{\delta+1}) & \in \overline{A_1^{\delta}\circ A}_1: q_{\delta-1} = q_{\delta}= q_{\delta+1}\}\Big)\cap \Big(A_1^{\delta-2}\circ A_3\Big) = \emptyset, \label{a1_a1_a1_cap_a3_is_empty} \\ \Big(\{([f], [\eta], q_1, \ldots, q_{\delta+1}) & \in \overline{A_1^{\delta}\circ A}_1: q_{\delta-1} = q_{\delta}= q_{\delta+1}\}\Big)\cap \Big(A_1^{\delta-2}\circ A_4\Big) = \emptyset \qquad \textnormal{and}\label{three_nodes_canot_be_a4} \\ \{([f], [\eta], q_1, \ldots, q_{\delta+1}) &\in \overline{A_1^{\delta}\circ A}_1: q_{\delta-1} = q_{\delta} = q_{\delta+1}\} \supset A_1^{\delta-2}\circ A_5. \label{three_nodes_A5} \end{align}
Since $\overline{A_1^{\delta}\circ \mathcal{S}}_{\mathcal{D}}$ is a closed set, \cref{a1_a1_a3_subset} implies that the right hand side of \cref{a1_a1_a3_exp} is a subset of its left hand side. Before we prove the above four statements, let us explain intuitively the significance of each of the statements.\\ \hf \hf The first statement, \cref{a1_a1_a3_subset} is saying that every tacnode is in the closure of two nodes (we remind the reader that the left hand side of \cref{a1_a1_a3_subset} is same as the expression given by \cref{temp_nat}). Geometrically, figure \ref{two_nodes_tacnode_pic} explains the meaning of \cref{a1_a1_a3_subset}.
\begin{figure}
\caption{Two nodes colliding into a tacnode}
\label{two_nodes_tacnode_pic}
\end{figure} \newline \hf\hf The second statement, \cref{a1_a1_a1_cap_a3_is_empty} is saying that in the closure of three nodes, we get a singularity more degenerate than a tacnode. The third statement, \cref{three_nodes_canot_be_a4} is saying that in the closure of three nodes, we get a singularity more degenerate than an $A_4$ singularity. Finally, \cref{three_nodes_A5} is saying that every $A_5$ singularity is in the closure of three nodes. Geometrically, figure \ref{fig_3_nodes_a5_ag} explains the meaning of \cref{three_nodes_A5} \begin{figure}
\caption{Three nodes colliding into an $\A_5$-singularity}
\label{fig_3_nodes_a5_ag}
\end{figure} \newline \hf \hf We are now ready to prove the above statements. Let us prove the following two claims: \begin{claim} \label{two_and_three_node_claim} Let $([f], [\eta], q_1, \ldots, \ldots, q_{\delta}) \in A_1^{\delta-2}\circ A_3$. Then there exists points \begin{align*} \big([f_t], [\eta_t], q_1(t), \ldots, q_{\delta-2}(t); q_{\delta-1}(t), q_{\delta}(t), q_{\delta+1}(t)\big) \in A_1^{\delta-2}\circ \mathcal{S}_{\mathcal{D}}^{3} \end{align*} sufficiently close to $([f], [\eta], q_1, \ldots, \ldots, q_{\delta-2}; q_{\delta}, q_{\delta}, q_{\delta})$, such that \begin{align}
f_t(q_{i}(t)) &=0, ~~~\nabla f_t|_{q_{i}(t)} = 0 \qquad \textnormal{for} ~~~i=\delta-1 ~~~\textnormal{and} ~~~\delta. \qquad \label{two_nodes_eqn} \end{align} Furthermore, \textit{every} such solution satisfies the condition \begin{align}
\Big(f_t(q_{\delta+1}(t)) , ~\nabla f_t|_{q_{\delta+1}(t)}\Big) &\neq (0, 0), \label{three_node_int_tacnode_empty} \end{align} i.e. $\big([f_t], [\eta_t], q_1(t), \ldots, q_{\delta-2}(t); q_{\delta-1}(t), q_{\delta}(t), q_{\delta+1}(t)\big) \not\in A_1^{\delta} \circ A_1$. In fact, if \[([f], [\eta], q_1, \ldots, \ldots, q_{\delta}) \in A_1^{\delta-2}\circ A_4,\] then there does not exist any point \begin{align*} \big([f_t], [\eta_t], q_1(t), \ldots, q_{\delta-2}(t); q_{\delta-1}(t), q_{\delta}(t), q_{\delta+1}(t)\big) \in A_1^{\delta-2}\circ \mathcal{S}_{\mathcal{D}}^{3} \end{align*} sufficiently close to $([f], [\eta], q_1, \ldots, \ldots, q_{\delta-2}; q_{\delta}, q_{\delta}, q_{\delta})$, such that \begin{align}
f_t(q_{i}(t)) &=0, ~~~\nabla f_t|_{q_{i}(t)} = 0 \qquad \textnormal{for} ~~~i=\delta-1, ~\delta ~~\textnormal{and} ~~\delta+1. \qquad \end{align} \end{claim}
\begin{claim} \label{three_nodes_A5_claim} Let $([f], [\eta], q_1, \ldots, \ldots, q_{\delta}) \in A_1^{\delta-2}\circ A_5$. Then there exists points \begin{align*} \big([f_t], [\eta_t], q_1(t), \ldots, q_{\delta-2}(t); q_{\delta-1}(t), q_{\delta}(t), q_{\delta+1}(t)\big) \in A_1^{\delta-2}\circ A_1^{3} \end{align*} sufficiently close to $([f], [\eta], q_1, \ldots, \ldots, q_{\delta-2}; q_{\delta}, q_{\delta}, q_{\delta})$. \end{claim}
\begin{rem} We note claim \ref{two_and_three_node_claim} proves \cref{a1_a1_a3_subset}, \cref{a1_a1_a1_cap_a3_is_empty} and \cref{three_nodes_canot_be_a4} simultaneously. We also note that claim \ref{three_nodes_A5_claim} proves \cref{three_nodes_A5}. \end{rem}
\begin{rem} \label{not_big_O} Before proceeding with the proof, let us make a shorthand notation. We denote \begin{align*}
O(|(x_1, x_2, \ldots, x_n)|^k) \end{align*} to be a \textbf{holomorphic} function (in the variables $x_1, \ldots, x_n$), defined in a neighborhood of the origin in $\mathbb{C}^n$,
whose order of vanishing is at least $k$ (i.e. all the terms of degree lower than $k$ are absent in the Taylor expansion of the function around the origin). We say that such an expressions is of order $k$.
For example, $x_1^4 + x_1 x_2 x_3^2 + x_2^2 x_3^3$ is a term of order $4$ and we will denote it by $O(|(x_1, x_2, x_3)|^4)$. Note that we are always dealing with holomorphic functions. Hence, suppose a function (in say one variable) is of type
$O(|x|^2)$, it means, its Taylor expansion is of the type \[ f(x) = a_2 x^2 + a_3 x^3 + \ldots. \]
It does not mean that there are terms of type $x \overline{x}$ (although the $|x|^2$ in the $O(|x|^2)$ might suggest that).
Henceforth, it will be understood that $O(|x|^n)$ and $O(x^n)$ mean the same thing in our paper (the latter is the standard notation in one variable). \end{rem}
\noindent \textbf{Proof of claims \ref{two_and_three_node_claim} and \ref{three_nodes_A5_claim}:}
Let us define \begin{align*} \mathbb{C}^2_{z}&:= \{(x,y,z) \in \mathbb{C}^3: z=0\}. \end{align*} We will now work in an affine chart where we send the plane $\mathbb{P}^2_{\eta_t}$ to $\mathbb{C}^2_{z}$ and the point $q_{\delta}(t) \in \mathbb{P}^2_{\eta_t}$ to
$(0,0,0) \in \mathbb{C}^2_{z}$.
Using this chart, let us write down the Taylor expansion of $f_t$ around the point $(0,0)$, namely \begin{align*} f_t(x,y)&= \frac{f_{t_{20}}}{2} x^2 + f_{t_{11}}xy + \frac{f_{t_{02}}}{2} y^2 + \ldots \end{align*} Note that since \cref{two_nodes_eqn} holds (for $i=\delta$), we conclude that $f_{t_{00}}, f_{t_{10}}$ and $f_{t_{01}}$ are zero.\\ \hf \hf Next, since $([f], [\eta], q_{\delta}) \in A_3$, we conclude that
$f_{t_{20}}$ and $f_{t_{02}}$ can not both be zero; let us assume $f_{t_{02}} \neq 0$. Hence, $f_t(x,y)$ can be re-written as \begin{align*} f_t(x,y)&= A_0(x) + A_1(x)y + A_2(x) y^2 + \ldots \qquad \textnormal{where} \qquad A_2(0) \neq 0. \end{align*} We will now make a change of coordinates; let us define \begin{align*} \hat{y}&:= y- B(x) \end{align*} where $B(x)$ is a function that is to be determined. We claim that there exists a unique holomorphic $B(x)$ (vanishing at the origin) such that after plugging it in $f_t(x, y)$ we get \begin{align*} f_t (x, y(x, \hat{y})) &= \hat{A}_0(x) + \hat{A}_2 (x) \hat{y}^2 + \hat{A}_3 (x) \hat{y}^3 + \ldots \end{align*} In other words, we want $\hat{A}_1(x) \equiv 0$. This is possible if $B(x)$ satisfies the equation \begin{align} A_1(x) + 2A_2(x) B(x) + 3A_3(x) B(x)^2 + \ldots & = 0. \label{psconvgg2} \end{align} Since $A_2(0) = \frac{f_{t_{02}}}{2} \neq 0$, $B(x)$ exists by the Implicit Function Theorem and we can compute $B(x)$ explicitly as a power series using \eqref{psconvgg2} and then compute $\hat{A}_0(x)$. Hence, \begin{align*} f_t (x, y(x, \hat{y})) &= \varphi(x,\hat{y})\hat{y}^2 + \frac{\mathcal{B}_2^{f_t}}{2!} x^2 + \frac{\mathcal{B}_3^{f_t}}{3!} x^3 + \frac{\mathcal{B}_4^{f_t}}{4!} x^4 + \mathcal{R}(x)x^5, \end{align*} where \begin{align*} \B^{f_t}_2 & := f_{t_{20}} - \frac{f_{t_{11}}^2}{f_{t_{02}}}, \qquad \B^{f_t}_3 := \frac{f_{t_{30}}}{6} +\frac{f_{t_{11}}^2 f_{t_{12}}}{f_{t_{02}}^2}, \ldots \ldots, \qquad \varphi(0,0) \neq 0 \end{align*} and $\mathcal{R}(x)$ is a holomorphic function defined in a neighborhood of the origin. Since $([f], [\eta], q_{\delta}) \in A_3$, we conclude that $\B^{f_t}_2$ and $\B^{f_t}_3$ are small (close to zero) and $\B^{f_t}_4$ is nonzero. Let us make a further change of coordinates and denote \begin{align*} \hat{\hat{y}}&:= \sqrt{\varphi(x, \hat{y})} \hat{y}. \end{align*} Note that we can choose a branch of the square root since $\varphi(0,0) \neq 0$. Next, for notational convenience, let us now define \begin{align} \hat{f}_t(x, \hat{\hat{y}}) &:= f_t (x, y(x, \hat{y}(\hat{\hat{y}})))), \label{hat_hat_f_defn} \end{align} i.e. $\hat{f}_t$ is basically $f_t$ written in the new coordinates (namely $x$ and $\hat{\hat{y}}$). Hence, \begin{align*} \hat{f}_t(x, \hat{\hat{y}}) & = \hat{\hat{y}}^2 + \frac{\mathcal{B}_2^{f_t}}{2!} x^2 + \frac{\mathcal{B}_3^{f_t}}{3!} x^3 + \frac{\mathcal{B}_4^{f_t}}{4!} x^4 + \mathcal{R}(x)x^5. \end{align*} We will now solve \cref{two_nodes_eqn} for $i=\delta-1$. We note that this is amounts to solving for the set of equations
\begin{align} \hat{f}_t(u, v) = 0, \qquad \hat{f}_{t_x}(u, v) = 0 \qquad \textnormal{and} \qquad \hat{f}_{t_{\hat{\hat{y}}}}(u, v) = 0 \quad (u, v)\neq (0,0) ~~~\textnormal{but small}, \label{hathatf_a2} \end{align} \textbf{and} requiring $\hat{f}_t$ to have $\delta-2$ more nodes (all distinct from each other and distinct from $(0,0)$ and $(u, v)$). The solutions to \cref{hathatf_a2} are given by \begin{align} v =0, \qquad \mathcal{B}^{f_t}_2 & = \frac{\mathcal{B}_4^{f_t}}{12} u^2 + 4 u^3 \mathcal{R}(u) + 2 u^4 \mathcal{R}^{\prime}(u) \qquad \textnormal{and} \nonumber \\ \mathcal{B}^{f_t}_3 & = -\frac{\mathcal{B}_4^{f_t}}{2} u -18 u^2 \mathcal{R}(u) -6u^3 \mathcal{R}^{\prime}(u). \label{soln_two_node_tacnode} \end{align} To see how, we first use the third equation of \cref{hathatf_a2} to get $v=0$. Then we use the second and first equations of \cref{hathatf_a2} to get the value of $\mathcal{B}^{f_t}_2$ and $\mathcal{B}^{f_t}_3$.\\ \hf \hf We now require the curve to have $\delta-2$ more nodes. To do that, first construct a degree $4$ curve that satisfies \cref{soln_two_node_tacnode}; we can do that since $\B^{f_t}_4$ only depends on the fourth order derivatives of $f_t$. Call this degree $4$ curve $g$. Let us now assume that the points $q_1, q_2, \ldots, q_{\delta-2}$ correspond to $(x_1, y_1), \ldots, (x_{\delta-2}, y_{\delta-2})$ under the affine chart we are considering. Define \begin{align*} f_t &:= g(x,y)\cdot ((x-x_1)^2+ (y-y_1)^2)\ldots ((x-x_{\delta-2})^2+ (y-y_{\delta-2})^2). \end{align*} This curve $f_t$ satisfies \cref{hathatf_a2} and has $\delta-2$ nodes. This argument works provided the degree of the curve is at least $4+2(\delta-2)$. Hence, solutions to \cref{two_nodes_eqn} exist, if $d \geq 4 + 2(\delta-2)$.\\ \hf \hf Next, let us prove \cref{three_node_int_tacnode_empty}, i.e. we have to show that in a neighborhood of a tacnode, we can not have a curve with three distinct nodes. More precisely, we need to show that there can not be any solutions to the set of equations \begin{align} \hat{f}_t(u_1, v_1) =0, \quad &\hat{f}_{t_x}(u_1, v_1)=0, \quad \hat{f}_{t_{\hat{\hat{y}}}}(u_1, v_1) =0, \label{three_node_uv1} \\ \hat{f}_t(u_2, v_2) =0, \quad &\hat{f}_{t_x}(u_2, v_2)=0, \quad \hat{f}_{t_{\hat{\hat{y}}}}(u_2, v_2) =0, \label{three_node_uv2}\\ & (0,0), ~~(u_1, v_1) ~~\textnormal{and} ~~(u_2, v_2) ~~\textnormal{all distinct (but small)}. \nonumber
\end{align} Let us try to solve for the above set of equations. Let us first explicitly write down $\hat{f}_t(x, \hat{\hat{y}})$ as \begin{align} \hat{f}_t(x, \hat{\hat{y}})&= \hat{\hat{y}}^2 + \frac{\mathcal{B}_2^{f_t}}{2!} x^2 + \frac{\mathcal{B}_3^{f_t}}{3!} x^3 + \frac{\mathcal{B}_4^{f_t}}{4!} x^4 + \frac{\mathcal{B}_5^{f_t}}{5!} x^5 +\frac{\mathcal{B}_6^{f_t}}{6!} x^6 + \ldots \label{hat_f_defn} \end{align} To begin with, we unwind \cref{three_node_uv1} using the expression for $\hat{f}_t$ as given by \cref{hat_f_defn} and solve for $\B^{f_t}_2$ and $\B^{f_t}_3$ in terms of $u_1, v_1$, $\B^{f_t}_4, \B^{f_t}_5$ and $\B^{f_t}_6$. We then plug in these values for $\B^{f_t}_2$ and $\B^{f_t}_3$ in \cref{hat_f_defn} and plug it in \cref{three_node_uv2}. Now we can solve for $\B^{f_t}_4$ and $\B^{f_t}_5$ in terms of $\B^{f_t}_6$ and then plugging back those values in the previous expressions for $\B^{f_t}_2$ and $\B^{f_t}_3$, gives us their values in terms of $\B^{f_t}_6$. Doing that, we get
\begin{align} v_1, v_2 & =0, \nonumber \\
\B^{f_t}_2 & = \frac{1}{360} \B^{f_t}_6 u_1^2 u_2^2 + O(|(u_1, u_2)|^5), \qquad
\B^{f_t}_3 = -\frac{1}{60} \B^{f_t}_6(u_1^2 u_2 + u_1 u_2^2)+ O(|(u_1, u_2)|^4), \nonumber \\
\B^{f_t}_4 &= \frac{1}{30}\B^{f_t}_6(u_1^2+ 4 u_1 u_2 + u_2^2)+ O(|(u_1, u_2)|^3) \qquad \textnormal{and} \qquad
\B^{f_t}_5 = -\frac{1}{3}\B^{f_t}_6 (u_1+u_2)+ O(|(u_1, u_2)|^2), \label{A6_nhbd} \end{align}
where $O(|(u_1, u_2)|^n)$ is as defined in \cref{not_big_O}. Hence, $\B^{f_t}_4$ is close to zero, which is a contradiction, since $([f], [\eta], q_{\delta}) \in A_3$. Since $\B^{f_t}_5$ is also close to zero, we get the last part of the claim \ref{two_and_three_node_claim} (i.e. \cref{three_nodes_canot_be_a4}). Finally, we note that the solutions constructed in \cref{A6_nhbd} immediately prove claim \ref{three_nodes_A5_claim} (in fact these are the \textit{only} possible solutions). This finishes the proof of claims \ref{two_and_three_node_claim} and \ref{three_nodes_A5_claim}. \\ \hf \hf Next,
we claim that each point of $(A_1^{\delta-2}\circ A_3)\cap \mu$ contributes $4$ to the Euler class in \cref{euler_na1_delta}. Using \cref{soln_two_node_tacnode}, we conclude that the multiplicity is the number of small solutions $(x, \hat{\hat{y}}, u)$ to the following set of equations \begin{align*} \hat{f}_t(x, \hat{\hat{y}}) & := \hat{\hat{y}}^2 + \frac{\mathcal{B}_2^{f_t}}{2!} x^2 + \frac{\mathcal{B}_3^{f_t}}{3!} x^3 + \frac{\mathcal{B}_4^{f_t}}{4!} x^4 + \mathcal{R}(x) x^5 =\varepsilon_0, \\ \hat{f_t}_x(x, \hat{\hat{y}}) & := \B^{f_t}_2 x + \frac{\B^{f_t}_3}{2} x^2 + \frac{\B^{f_t}_4}{12} x^3 + 5 x^4 \mathcal{R}(x) + \mathcal{R}^{\prime}(x)x^5 =\varepsilon_1, \qquad \hat{f_t}_{\hat{\hat{y}}}(x, \hat{\hat{y}}):= 2 \hat{\hat{y}} = \varepsilon_2, \\ \B^{f_t}_2& = \frac{\mathcal{B}_4^{f_t}}{12} u^2 + 4 u^3 \mathcal{R}(u) + 2 u^4 \mathcal{R}^{\prime}(u) \qquad \textnormal{and} \qquad \mathcal{B}^{f_t}_3 = -\frac{\mathcal{B}_4^{f_t}}{2} u -18 u^2 \mathcal{R}(u) -6u^3 \mathcal{R}^{\prime}(u), \end{align*} where $(\varepsilon_0, \varepsilon_1, \varepsilon_2) \in \mathbb{C}^3$ is small and generic. Let us write $u:= h+x$ and Taylor expand $\mathcal{R}(x+h)$ and $\mathcal{R}^{\prime}(x+h)$ around $h=0$, i.e. \begin{align} \mathcal{R}(x+h)& = \mathcal{R}(x) + h \mathcal{R}^{\prime}(x) + \frac{h^2}{2} \mathcal{R}^{\prime \prime}(x) + \ldots \qquad \textnormal{and} \qquad \mathcal{R}^{\prime}(x+h) = \mathcal{R}^{\prime}(x) + h \mathcal{R}^{\prime \prime}(x) + \ldots \end{align} Hence, substituting the values of $\B^{f_t}_2$, $\B^{f_t}_3$, $\mathcal{R}(x+h)$ and $\mathcal{R}^{\prime}(x+h)$ we conclude that we need to find the number of small solutions $(x,h)$ to the following set of equations
\begin{align}
\frac{(x^2 h^2)\Big( \mathcal{B}_4^{f_t} + O(|(x,h)|) \Big)}{24} & = \varepsilon_3 \qquad \textnormal{and} \label{ep3} \\
\frac{(xh)\Big(\mathcal{B}_4^{f_t}h-\mathcal{B}_4^{f_t}x + O(|(x,h)|^2) \Big)}{12} & = \varepsilon_1, \label{ep2} \end{align} where $\varepsilon_3:= \varepsilon_0 - \frac{\varepsilon_2^2}{4}$.
We claim that we can set $\varepsilon_1$ to be $0$; that is justified in \cref{local_degree}. Assuming that claim, we use \cref{ep2} to solve for $x$ in terms of $h$ and conclude that \begin{align} x&= h + O(h^2). \label{x_h} \end{align}
This is because $x=0$ and $h=0$ can not be solutions to \cref{ep2} (since if we plug it back in \cref{ep3}, we will get $0$ and not $\varepsilon_3$). Plugging in the value of $x$ from \cref{x_h} into \cref{ep3}, we get \begin{align} \frac{\mathcal{B}_4^{f_t}}{24} h^4 + O(h^5) &= \varepsilon_3. \label{h4} \end{align} Equation \eqref{h4} clearly has $4$ solutions.\\ \hf \hf Finally, we need to
justify \cref{three_node_triple_point} and the corresponding contribution to the Euler class.
More precisely, we are going to show that \begin{align} \{([f], [\eta], q_1, \ldots,q_{\delta+1}) \in \overline{A_1^{\delta}\circ \mathcal{S}}_{\mathcal{D}}: q_{\delta-2}= q_{\delta-1} = q_{\delta} = q_{\delta+1}\}&= \overline{A_1^{\delta-3}\circ A}_5 \cup \overline{A_1^{\delta-3}\circ D}_4. \label{three_nodes_collide_D4_yy} \end{align} Just like \cref{a1_a1_a3_exp} is equivalent to \cref{a1_a1_a3_exp_ag2}, we similarly conclude
that \cref{three_nodes_collide_D4_yy} can be equivalently stated as \begin{align} \{([f], [\eta], q_1, \ldots,q_{\delta}) \in \overline{A_1^{\delta}}: q_{\delta-2}= q_{\delta-1} = q_{\delta}\}&= \overline{A_1^{\delta-3}\circ A}_5 \cup \overline{A_1^{\delta-3}\circ D}_4. \label{three_nodes_collide_D4_ag2} \end{align} Let us define \begin{align}
W_1&:= \{ ([f], [\eta], q_1, \ldots,q_{\delta+1}) \in \mathcal{S}_{\mathcal{D}_{\delta+1}}: f(q_{\delta+1})=0, ~\nabla f|_{q_{\delta+1}} =0, ~~
\nabla^2f|_{q_{\delta+1}} \neq 0\} \qquad \textnormal{and} \nonumber \\
W_2&:= \{ ([f], [\eta], q_1, \ldots,q_{\delta+1}) \in \mathcal{S}_{\mathcal{D}_{\delta+1}}: f(q_{\delta+1})=0, ~\nabla f|_{q_{\delta+1}} =0, ~~
\nabla^2f|_{q_{\delta+1}} = 0\}. \label{W12_defn} \end{align} In order to prove \cref{three_nodes_collide_D4_yy}, it suffices to show that \begin{align} \Big(\{([f], [\eta], q_1, \ldots,q_{\delta+1}) \in \overline{A_1^{\delta}\circ \mathcal{S}}_{\mathcal{D}}: q_{\delta-2}= q_{\delta-1} = q_{\delta} = q_{\delta+1}\}\Big)\cap W_1 & = \Big(\overline{A_1^{\delta-3}\circ A}_5\Big)\cap W_1 ~~~ \textnormal{and} \label{three_node_cap_w1} \\ \Big(\{([f], [\eta], q_1, \ldots,q_{\delta+1}) \in \overline{A_1^{\delta}\circ \mathcal{S}}_{\mathcal{D}}: q_{\delta-2}= q_{\delta-1} = q_{\delta} = q_{\delta+1}\}\Big)\cap W_2 & = \overline{A_1^{\delta-3}\circ D}_4. \label{three_node_cap_w2} \end{align}
\hf \hf Note that the right hand side of \cref{three_node_cap_w2} is a subset of $W_2$; hence we didn't write down $\cap W_2$ on the right hand side of \cref{three_node_cap_w2}.
Let us first justify \cref{three_node_cap_w1}. Equations
\eqref{a1_a1_a1_cap_a3_is_empty} and \eqref{three_nodes_canot_be_a4}, show that the the left hand side of \eqref{three_node_cap_w1} is a subset of its right hand side.
Furthermore, \cref{three_nodes_A5} shows that
the right hand side of \eqref{three_node_cap_w1} is a subset of its left hand side; hence \cref{three_node_cap_w1} is true. \\
\hf \hf We will now prove \cref{three_node_cap_w2}. Equation \eqref{a1_a1_a1_cap_a3_is_empty} shows that the left hand side of \cref{three_node_cap_w2} is a subset of its right hand side. Hence, what remains is to show that the right hand side of \cref{three_node_cap_w2} is a subset of its left hand side. Before we start the proof of that assertion, let us give an intuitive idea about the significance of that statement.
The statement is saying that every triple point is in the closure of three nodes.
\begin{figure}
\caption{Three nodes colliding into a triple point}
\label{fig_3_nodes_triple_pt_ag}
\end{figure}
To summarize, the geometric significance of \cref{three_node_cap_w1} is given by figure \ref{fig_3_nodes_a5_ag} while the geometric significance of \cref{three_node_cap_w2} is given by figure \ref{fig_3_nodes_triple_pt_ag}. Equation \eqref{three_nodes_collide_D4_yy} says that these are the \textbf{only} two pictures that can occur.\\ \hf \hf Let us now prove \cref{three_node_cap_w2}. We will prove the following claim:
\begin{claim} \label{three_node_claim_d4} Let $([f], [\eta], q_1, \ldots, \ldots, q_{\delta}) \in A_1^{\delta-3}\circ D_4$. Then, there exists points \begin{align*} \big([f_t], [\eta_t], q_1(t), \ldots, q_{\delta-3}(t); q_{\delta-2}(t), q_{\delta-1}(t), q_{\delta}(t), q_{\delta+1}(t)\big) \in A_1^{\delta-3}\circ \mathcal{S}_{\mathcal{D}}^{4} \end{align*} sufficiently close to $([f], [\eta], q_1, \ldots, \ldots, q_{\delta-3}; q_{\delta}, q_{\delta}, q_{\delta}, q_{\delta})$, such that \begin{align}
f_t(q_{i}(t)) &=0, ~~~\nabla f_t|_{q_{i}(t)} = 0 \qquad \textnormal{for} ~~~i=\delta-2, ~~\delta-1 ~~~\textnormal{and} ~~~\delta. \label{D4_nhbd_eqn} \end{align} \end{claim}
\begin{rem} We note that claim \ref{three_node_claim_d4} implies that the right hand side of \cref{three_node_cap_w2} is a subset of the left hand side.
\end{rem}
\noindent \textbf{Proof:} Following the setup of the proof of claim \ref{two_and_three_node_claim}, we will now work in an affine chart, where we send the plane $\mathbb{P}^2_{\eta_t}$ to $\mathbb{C}^2_{z}$ and the point $q_{\delta}(t) \in \mathbb{P}^2_{\eta_t}$ to $(0,0,0) \in \mathbb{C}^2_{z}$. Using this chart, let us write down the Taylor expansion of $f_t$ around the point $(0,0)$, namely \begin{align} f_t(x,y)&= \frac{f_{t_{20}}}{2} x^2 + f_{t_{11}}xy + \frac{f_{t_{02}}}{2} y^2 + \frac{f_{t_{30}}}{6} x^3 + \frac{f_{t_{21}}}{2} x^2y + \frac{f_{t_{12}}}{2} x y^2 + \frac{f_{t_{03}}}{6} y^3+\ldots \label{eqf_t} \end{align} Since $([f], [\eta], q_{\delta}) \in D_4$, we conclude that $f_{t_{20}}, f_{t_{11}}$ and $f_{t_{02}}$ are all small (close to zero). Let us now construct solutions to \cref{D4_nhbd_eqn}. Let us assume that the points $q_{\delta-1}(t)$ and $q_{\delta-2}(t)$ are sent to $(x_1, y_1, 0)$ and $(x_2, y_2, 0)$ under the affine chart we are considering. Hence, constructing solutions to \cref{D4_nhbd_eqn} is same as constructing solutions to the set of equations \begin{align} f_t(x_1, y_1)&=0, \qquad f_{t_x}(x_1, y_1)=0, \qquad f_{t_{y}}(x_1, y_1) =0 \qquad \textnormal{and} \label{eq1} \\ f_t(x_2, y_2)&=0, \qquad f_{t_x}(x_2, y_2)=0, \qquad f_{t_{y}}(x_2, y_2) =0, \label{eq2} \end{align} where $(0,0), (x_1, y_1)$ and $(x_2, y_2)$ are all distinct (but close to each other).\\ \hf \hf Next, let us define \begin{align} g_t(x,y) & := x f_{t_{x}}(x,y) + y f_{t_{y}}(x,y)- 2 f_t(x,y). \label{g_defn} \end{align} The quantity $g(x,y)$ is similarly defined with $f_t$ replaced by $f$. We note that solving \cref{eq1} and \cref{eq2} is equivalent to solving \begin{align} g_t(x_1, y_1)&=0, \qquad f_{t_x}(x_1, y_1)=0, \qquad f_{t_{y}}(x_1, y_1) =0 \qquad \textnormal{and} \label{eq3} \\ g_t(x_2, y_2)&=0, \qquad f_{t_x}(x_2, y_2)=0, \qquad f_{t_{y}}(x_2, y_2) =0, \label{eq4} \end{align} where $(0,0), (x_1, y_1)$ and $(x_2, y_2)$ are all distinct (but close to each other). We now note that $g_t(x,y)$ and $f_t(x,y)$ have exactly the same cubic term in the Taylor expansion. Furthermore, $g_t(x,y)$ has no quadratic term. \\
\hf \hf Let us now study the cubic term of the Taylor expansion of $f$ carefully.
Let us assume first $f_{{30}} \neq 0$.
Since $([f], [\eta], q) \in D_4$, we conclude that the cubic term factors into three \textbf{distinct} linear factors. Hence, the cubic term can be written as
\begin{align} \frac{f_{30}}{6} (x-A_1(0) y) (x-A_2(0)y) (x-A_3(0) y), \label{cubic_defn_0} \end{align} where $A_1(0), A_2(0)$ and $A_3(0)$ are all distinct.
Note that $A_1(0), A_2(0)$ and $A_3(0)$ are explicit expressions involving the coefficients $f_{ij}$. If $f_{{30}}=0$, then the cubic term will be of the type \begin{align*} \frac{f_{{21}}}{2}y(x-A_1(0) y)(x-A_2(0) y), \end{align*} where $A_1(0)$ and $A_2(0)$ are distinct and $f_{{21}}$ is nonzero. We will assume that $f_{{30}} \neq 0$; the case $f_{{30}}=0$ can be dealt with similarly. Hence, $g_t$ (or equivalently $f_t$) can be written as \begin{align}
g_t(x,y)& = \frac{f_{t_{30}}}{6} (x-A_1 y) (x-A_2 y) (x-A_3 y) + O(|(x,y)|^4), \label{cubic_defn_0} \end{align} where $A_i$ are the same as $A_i(0)$, but with the $ f_{ij}$ replaced by $ f_{t_{ij}}$. For notational simplicity, we denoted these quantities by the letter $A_i$ and not $A_i(t)$. \\
\hf \hf Let us now make a change of coordinates \begin{align}
x & := \hat{x} + O(|(\hat{x}, \hat{y})|^2) \qquad \textnormal{and} \qquad y:= \hat{y} + O(|(\hat{x}, \hat{y})|^2), \label{x_y_hat} \end{align}
such that \begin{align} g_t& = \frac{f_{t_{30}}}{6} (\hat{x}-A_1 \hat{y}) (\hat{x}-A_2\hat{y}) (\hat{x}-A_3 \hat{y}). \label{g_hat} \end{align} Hence, $g_t=0$ has three distinct solutions, given by $\hat{x} = A_i \hat{y}$
for $i=1,2$ and $3$. Converting back in terms of $x$, we conclude that the solutions to $g_t(x,y)=0$ (where $(x,y)$ is small but nonzero) are given by
\begin{align} y&= u \qquad \textnormal{and} \qquad x = A_i u + E_i(u), \label{g_soln_2} \end{align}
where $E_i(u)$ is a second order term in $u$ (and $u$ is small but nonzero).\\
\hf \hf Next, for notational simplicity we will denote $f_{t_{02}}$ by the letter $w$.
Let us consider the solution $y:= u$ and $x = A_1 u + E_1(u)$ of the equation $g_t(x,y)=0$. Plugging this in $f_{t_{x}}(x, y) =0$ and $f_{t_{y}}(x, y) =0$ and solving for $A_1 f_{t_{11}}$ and $A_1^2 f_{t_{20}}$, we conclude that \begin{align}
A_1 f_{t_{11}} &= \frac{A_1f_{t_{30}}}{6} (A_1-A_2)(A_1-A_3)u - w + O(|(u,w)|^2) \qquad \textnormal{and} \nonumber \\
A_1^2 f_{t_{20}} & = -\frac{A_1 f_{t_{30}}}{3} (A_1-A_2)(A_1-A_3)u + w + O(|(u,w)|^2). \label{3_node_sol_1} \end{align}
Let us now consider a second solution to $g_t(x,y)=0$ (where $(x,y)$ is small but nonzero). This will be given by
$y:= v$ and $x:= A_2 v+ E_2(v)$, where $v$ is small but nonzero (or the analogous thing with $A_2$ replaced by $A_3$). Using \cref{3_node_sol_1} to express the values of $f_{t_{11}}$ and $f_{t_{20}}$ in terms of $u$ and $w$ and then using $f_{t_{x}}(x, y) =0$, we conclude that \begin{align} w & = \frac{f_{t_{30}}}{6} \Big(A_1^3 -2 A_1^2 A_2-A_1^2 A_3 +2 A_1
A_2 A_3\Big)u + \frac{f_{t_{30}}}{6}\Big(A_1^2 A_3- A_1^2 A_2\Big)v + O(|(u,w)|^2). \label{w1} \end{align} Similarly, using \cref{3_node_sol_1} to express the values of $f_{t_{11}}$ and $f_{t_{20}}$ in terms of $u$ and $w$ and then using $f_{t_{y}}(x, y) =0$, we conclude that \begin{align} w & = \frac{f_{t_{30}}}{6}\Big(-A_1^2 A_2 + A_1 A_2 A_3 \Big)u +
\frac{f_{t_{30}}}{6}\Big(-A_1 A_2^2 + A_1 A_2 A_3\Big)v + O(|(u,w)|^2). \label{w2} \end{align} Equating the right hand sides of \cref{w1} and \cref{w2}, we conclude that
\begin{align}
\frac{f_{t_{30}}}{6} A_1 (A_1-A_2)(A_1-A_3)u -\frac{f_{t_{30}}}{6} A_1(A_1-A_2)(A_2-A_3)v + O(|(u,v,w)|^2) & = 0. \label{Q} \end{align}
From \cref{Q}, we can further conclude that \begin{align}
A_1 v & = \Big(\frac{A_1-A_3}{A_2-A_3}\Big)(A_1 u) + O(|(u,w)|^2). \label{v_value} \end{align}
Finally, substituting the value for $v$ from \cref{v_value} into $w$ in \cref{w1}, we get that \begin{align}
w &= -\frac{f_{t_{30}}}{3} A_1 A_2 (A_1 - A_3) u + O(|(u,w)|^2)
~~ \implies ~~w = -\frac{f_{t_{30}}}{3} A_1 A_2 (A_1 - A_3) u + O(|u|^2). \label{w_value_final} \end{align}
Plugging the value of $w$ from \cref{w_value_final} in \cref{3_node_sol_1}, we conclude that \begin{align*}
f_{t_{11}} &= \frac{f_{t_{30}}}{6}(A_1 + A_2) (A_1 - A_3) u + O(|u|^2) \qquad \textnormal{and} \qquad
f_{t_{20}} = -\frac{f_{t_{30}}}{3} (A_1 - A_3) u + O(|u|^2). \end{align*} Hence, solutions to \cref{eq1} and \cref{eq2} exist, given by \begin{align} (x_1, y_1) & = (A_1 u + E_1(u), u), \qquad (x_2, y_2) = \Big(A_2\frac{(A_1-A_3)}{(A_2-A_3)}u + E_2(u), ~\frac{(A_1-A_3)}{(A_2-A_3)}u + E_4(u)\Big), \nonumber \\ f_{t_{11}} &= \frac{f_{t_{30}}}{6}(A_1 + A_2) (A_1 - A_3) u + E_5(u), \qquad f_{t_{20}} = -\frac{f_{t_{30}}}{3} (A_1 - A_3) u + E_6(u) \qquad \textnormal{and} \nonumber \\ f_{t_{02}} & = -\frac{f_{t_{30}}}{3} A_1 A_2 (A_1 - A_3) u + E_7(u), \label{d4_three_nodes_soln} \end{align} where $u$ is small and nonzero and the $E_i$ are all second order terms. Furthermore, there are \textbf{exactly} $6$ distinct solutions, that corresponds to $(A_1, A_2)$ being replaced with $(A_i, A_j)$, where the $(A_i, A_j)$ are ordered (or alternatively, we can think of this this way; the $(A_i, A_j)$ is unordered as far as the construction of $f_t$ is concerned, but we can permute the values of $(x_1, y_1)$ and $(x_2, y_2)$).
This proves \cref{three_node_claim_d4} and hence proves \cref{three_node_triple_point}. \qed \\
\hf \hf Let us now justify the multiplicity. We claim that each point of $(A_1^{\delta-3}\circ D_4)\cap \mu$ contributes $18$ to the Euler class in \cref{euler_na1_delta}. As we just explained, there are exactly $6$ distinct solutions to \cref{eq1} and \cref{eq2}; we will call each distinct solution of \cref{eq1} and \cref{eq2} a \textbf{branch} of a neighborhood of $A_1^{\delta-3}\circ D_4$ inside $\overline{A_1^{\delta}}$.
Since there are $6$ branches, it suffices to show that the multiplicity from each branch is $3$ (in which case the total contribution to the Euler class will be $18$). Let us now compute the multiplicity from each branch.\\
\hf \hf Let us consider the branch given by \cref{d4_three_nodes_soln}. The multiplicity from this branch
is the number of small solutions $(x, y, u)$ to the following set of equations \begin{align} f_t(x,y)& = \varepsilon_0, \qquad f_{t_x}(x,y) = \varepsilon_1 \qquad \textnormal{and} \qquad f_{t_y}(x,y) = \varepsilon_2 \label{epp12} \end{align} where $(\varepsilon_0, \varepsilon_1, \varepsilon_2) \in \mathbb{C}^3$ is small and generic and $f_{t_{20}}, f_{t_{11}}$ and $f_{t_{02}}$ are as given in \cref{d4_three_nodes_soln}. We claim that we can set $\varepsilon_1$ and $\varepsilon_2$ to be zero; this is justified in section \ref{local_degree}. Hence, we need to find the number of small solutions $(x,y,u)$ to the set of equations \begin{align*} f_t(x,y)& = \varepsilon_0, \qquad f_{t_x}(x,y) = 0 \qquad \textnormal{and} \qquad f_{t_y}(x,y) = 0. \end{align*} This is same as the number of small solutions $(x,y,u)$ to the set of equations \begin{align} g_t(x,y)& = -2\varepsilon_0, \label{g_only}\\ f_{t_x}(x,y) & = 0 \qquad \textnormal{and} \qquad f_{t_y}(x,y) = 0, \label{g_eqn_solve} \end{align} where $g_t(x,y)$ is as defined in \cref{g_defn}. Let us start by solving only the two equations in \cref{g_eqn_solve}.
Plugging in the values for $f_{t_{20}}, f_{t_{11}}$ and $f_{t_{02}}$ as given in \cref{d4_three_nodes_soln} and solving the equation $f_{t_x}(x,y)=0$, we conclude that \begin{align}
\Big(-2 x + (A_1 + A_2) y\Big) \Big(u+ O(|u|^2)\Big) & = \frac{(3 x^2 - 2(A_1+A_2+A_3)x y + (A_1 A_2 + A_1 A_3 + A_2 A_3) y^2)}{(A_3-A_1)} \nonumber \\
& ~~ + O(|(x,y)|^3). \label{ou} \end{align}
Similarly, plugging in the values for $f_{t_{20}}, f_{t_{11}}$ and $f_{t_{02}}$ as given in \cref{d4_three_nodes_soln} and solving the equation $f_{t_y}(x,y)=0$, we conclude that \begin{align}
\Big( (A_1+A_2)x -2A_1 A_2 y \Big) \Big(u+ O(|u|^2)\Big) & = -\frac{((A_1+A_2+A_3)x^2 -2(A_1 A_2 + A_1 A_3 + A_2 A_3)xy +3A_1A_2 A_3 y^2)}{(A_3-A_1)} \nonumber \\
& ~~ + O(|(x,y)|^3). \label{ou3} \end{align} Multiplying \cref{ou} by $(A_1+A_2)x -2A_1 A_2 y$ and multiplying \cref{ou3} by $(-2 x + (A_1 + A_2) y)$, we conclude that
\begin{align} \Big(x -A_1 y\Big) \Big(x - A_2 y\Big)\Big( (A_1+A_2-2A_3)x -(2A_1 A_2-A_1A_3 - A_2 A_3) y \Big)+
O(|x,y|^4) & = u^2 O(|(x,y)|^2). \label{x_A_y_soln} \end{align}
Let us now solve \cref{x_A_y_soln}. Let us make a change of coordinates \begin{align*}
x & = \hat{x} + O(|(\hat{x}, \hat{y})|^2) \qquad \textnormal{and} \qquad y = \hat{y} + O(|(\hat{x}, \hat{y})|^2) \end{align*} such that \cref{x_A_y_soln} can be rewritten as \begin{align}
\Big(\hat{x} -A_1 \hat{y}\Big) \Big(\hat{x} - A_2 \hat{y}\Big)\Big( (A_1+A_2-2A_3)\hat{x} -(2A_1 A_2-A_1A_3 - A_2 A_3) \hat{y} \Big) & = u^2 O(|(\hat{x}, \hat{y})|^2) \label{x_A_y_soln_hat} \end{align} Using \cref{x_A_y_soln_hat}, we solve for $\hat{x}$ in terms for $\hat{y}$ and $u$ and convert back to $x$ and $y$ to conclude that the only possible solutions are given by
\begin{align} x & = A_1 y + E_8(y,u)\qquad \textnormal{or} \qquad x = A_2 y + E_9(y,u) \qquad \textnormal{or} \nonumber \\ (A_1+A_2-2A_3)x &= (2A_1 A_2-A_1A_3 - A_2 A_3) y + E_{10}(y,u), \label{xy_soln} \end{align}
such that $E_i(y,0) = O(|y|^2)$, for $i=8,9$ and $10$.
Plugging the three solutions obtained in \cref{xy_soln} into \cref{ou}, solving for $y$ in terms of $u$ and then plugging that back into \cref{xy_soln} to express $x$ in terms of $u$, we conclude that the only possible solutions to \cref{g_eqn_solve} are given by \begin{align} (x, y)& = \Big(A_1u + \widetilde{E}_1(u), u+ \widehat{E}_1(u) \Big) \qquad \textnormal{or} \label{sol1}\\ (x,y) & = \Big(A_2\Big(\frac{A_1-A_3}{A_2-A_3}\Big)u + \widetilde{E}_2(u), \Big(\frac{A_1-A_3}{A_2-A_3}\Big)u+\widehat{E}_2(u)\Big) \qquad \textnormal{or} \label{sol2} \\ (x,y) & = \Big( \frac{(2A_1 A_2-A_1 A_3-A_2 A_3)}{3(A_2-A_3)} u + \widetilde{E}_3(u) , \frac{(A_1+A_2-2A_3)}{3(A_2-A_3)} u + \widehat{E}_3(u)\Big), \label{sol3} \end{align} where $\widetilde{E}_i(u)$ and $\widehat{E}_i(u)$ are second order terms (for $i=1,2$ and $3$). From \cref{d4_three_nodes_soln}, we conclude that the solutions in \cref{sol1} and \cref{sol2} with $\widetilde{E}_i(u)$ replaced by $E_i(u)$ and $\widehat{E}_i(u)$ replaced by $0$ (for $i=1$ and $2$) is a solution to \cref{g_eqn_solve}. Since the solutions in \cref{sol1} and \cref{sol2} are the \textbf{only} solutions to \cref{g_eqn_solve}, we conclude that $\widetilde{E}_i(u) = E_i(u)$ and $\widehat{E}_i(u) =0$ (for $i=1$ and $2$). Hence, if we plug the solutions obtained from \cref{sol1} and \cref{sol2} into $f_t(x,y)$ (or equivalently $g_t(x,y)$), we will get $0$ and not $\varepsilon_0$. Hence, we reject the solutions given by \cref{sol1} and \cref{sol2}. \\ \hf \hf It remains to consider the solution given by \cref{sol3}.
Plugging in the expression for $x$ and $y$ from \cref{sol3} into $g_t(x,y)$ gives us
\begin{align} g_t(x,y) & = \Big(\frac{(A_2-A_1)^2 (A_3-A_1)^2}{162 (A_2 - A_3)}\Big) u^3 + O(u^4). \label{g_3_mult} \end{align} From \cref{g_3_mult}, we conclude that $g_t(x,y) = -2\varepsilon_0$ has $3$ solutions. This justifies the multiplicity and concludes the proof of \cref{na1_delta}. \qed
\subsubsection{Local degree of a smooth map} \label{local_degree} It remains to show why we could set $\varepsilon_2$ to be $0$ in \cref{ep2} and set $(\varepsilon_1, \varepsilon_2)$ to be $(0,0)$ in \cref{epp12}. Let us first recall the definition of the local degree of a smooth map around a given point. We will follow the discussion and theory developed in \cite{Kes}.\\ \hf \hf Let us begin with the proposition 2.1.2 of \cite{Kes}. The statement is as follows: \begin{prp}\label{local_degree_kes} Let $f \in C^2(\bar{\Omega}, \mathbb{R}^n)$ where $\Omega$ is an open subset of $\mathbb{R}^n$ and let $b \notin f(\partial \Omega)$. Let $\rho_0$ be the distance between $b$ and $f(\partial \Omega)$ with $\rho_0>0$. Let $b_1,b_2 \in B(b;\rho_0),$ the ball of radius $\rho_0$ with center $b$. If $b_1,b_2$ both are regular values of $f$, then $\textnormal{deg}(f,\Omega,b_1)=\textnormal{deg}(f,\Omega,b_2)$ where $\textnormal{deg}(f,\Omega,y)$ represent the degree of $f$ at $y$ (i.e. the number of solutions to the equation $f(x)=y$ in $\Omega$). \end{prp} \hf \hf Let us first justify the assertion for \cref{ep2}. Let $U$ be
an open ball in $\mathbb{C}^2$ with center $(0,0)$ and radius $r$, where $r$ is sufficiently small and positive real number. Consider the map $\varphi: U \longrightarrow \mathbb{C}^2$, given by \begin{align*} \varphi(x,h)& = (\varphi_{1}(x,h), \varphi_2(x,h))\\
&:= \Big(\frac{(x^2 h^2)\Big( \mathcal{B}_4^{f_t} + O(|(x,h)|) \Big)}{24},
~~\frac{(xh)\Big(\mathcal{B}_4^{f_t}h - \mathcal{B}_4^{f_t}x + O(|(x,h)|^2)\Big)}{12}\Big). \end{align*}\\ Before proceeding, let us first prove the following claim: \begin{claim} If $\varepsilon \neq 0,$ then the point $(\varepsilon,0)$ is a regular value of $\varphi$. \end{claim} \begin{proof} Let us assume $\varphi(x, h) = (\varepsilon, 0)$. Using the fact that $\varphi_2(x,h) =0$, we conclude that \[ x(h) = h + O(h^2).\] Plugging in this value of $x$ in $\varphi_1(x,h)$, we conclude that \begin{align}
h^4\Big(\frac{\mathcal{B}^{f_t}_4}{24} + O(h) \Big) & = \varepsilon. \label{ze1} \end{align} Note that if $h$ is sufficiently small, then $\frac{\mathcal{B}^{f_t}_4}{24} + O(h)$ is nonzero, since $\mathcal{B}^{f_t}_4$ is nonzero. We also note that since $\varepsilon$ is nonzero, \cref{ze1} implies that $h$ is nonzero. \\ \hf \hf Next, let us compute the determinant of the differential of $\varphi$ at $(x(h), h)$. It is given by
\begin{align} M &:= \textnormal{det}\begin{pmatrix} \varphi_{1_x}& \varphi_{1_h} \\ \varphi_{2_x}& \varphi_{2_h} \\
\end{pmatrix}\Big|_{(x(h), h)} = h^5\Big(\frac{(\mathcal{B}_4^{f_t})^2}{72} + O(h)\Big) \label{ze2} \end{align} Using \cref{ze1} and \cref{ze2}, we conclude that \begin{align} M & = h^4 \cdot h\Big(\frac{(\mathcal{B}_4^{f_t})^2}{72} + O(h) \Big) \nonumber \\
& = \varepsilon \frac{h\Big(\frac{(\mathcal{B}_4^{f_t})^2}{72} + O(h) \Big)}{\Big(\frac{\mathcal{B}^{f_t}_4}{24} + O(h) \Big)}. \label{ze3} \end{align} Since, $\mathcal{B}^{f_t}_4$ is nonzero, $h$ is small and nonzero and $\varepsilon$ is nonzero, we conclude from \cref{ze3} that $M$ is nonzero. Hence, $(\varepsilon,0)$ is a regular value of $\varphi$. \end{proof}
\hf \hf Next, we note that if $S$ is a non empty subset of $\mathbb{C}^2$, then
the distance function $d_S: \mathbb{C}^2 \longrightarrow \mathbb{R}$
is a continuous function. Hence, the set \begin{align*} V & :=(d_{\varphi(\partial U)}-d_{X})^{-1}(0,\infty) \\
& = \{(\varepsilon_1,\varepsilon_2)\in \mathbb{C}^2\mid d_{\varphi(\partial U)}(\varepsilon_1,\varepsilon_2) > d_{X}(\varepsilon_1,\varepsilon_2)\} \end{align*}
is an open subset of $\mathbb{C}^2$, where the function $d_{X}$ denotes the distance from $x$-axis. Note that $d_{X}(\varepsilon_1,\varepsilon_2)=|\varepsilon_2|$ and this distance is achieved by taking the distance from the point $(\varepsilon_1,\varepsilon_2)$ to the point $(\varepsilon_1,0)$ on $x$-axis. \\ \hf \hf Now, we will show that $V \cap \varphi(U) \neq \emptyset.$ Note that
$$\partial U=\{ (x,h)\in \mathbb{C}^2 : |x|^2+|h|^2=r^2\}.$$
Observe that $\partial U$ is compact; so $\varphi(\partial U)$ is compact and hence closed in $\mathbb{C}^2$. Hence, $d_{\varphi(\partial U)}(\varepsilon,0) = 0$ if and only if $(\varepsilon,0) \in \varphi( \partial U)$. We conclude that $(\varepsilon,0) \in V$ if and only if $(\varepsilon,0) \notin \varphi( \partial U).$ Now, let $(\varepsilon,0) \in \varphi( \partial U)$. Let us assume $\varphi(x,h)=(\varepsilon,0)$ with $|x|^2+|h|^2=r^2$ and $\varepsilon \neq 0$. We conclude from $\varphi_2(x,h)=0$ and \cref{ze1} that \begin{align*} x(h) & = h + O(h^2) \qquad \textnormal{and} \qquad h^4\Big(\frac{\mathcal{B}^{f_t}_4}{24} + O(h) \Big) = \varepsilon. \end{align*}
Now using the fact $|x|^2+|h|^2=r^2$, we conclude that $|\varepsilon| = \dfrac{|\mathcal{B}^{f_t}_4|}{96}r^4+O(r^5)$. Hence we get either $\varepsilon=0$ or $|\varepsilon| = \dfrac{|\mathcal{B}^{f_t}_4|}{96}r^4+O(r^5)$.
Note that $\dfrac{|\mathcal{B}^{f_t}_4|}{96}r^4+O(r^5) \neq 0$ as $\mathcal{B}^{f_t}_4 \neq 0$ and $r$ is sufficiently small. So, $(\varepsilon,0) \in V$ for all nonzero $\varepsilon$ with
$|\varepsilon| < \dfrac{|\mathcal{B}^{f_t}_4|}{96}r^4+O(r^5)$ (i.e. for all $|\varepsilon|$ sufficiently small).
From \cref{h4} we concluded that the system $\varphi(x,h)=(\varepsilon,0)$ has solutions in $U$ where $\varepsilon$ is small but nonzero. Hence $(\varepsilon,0) \in V \cap \varphi(U)$ for some nonzero $\varepsilon$ with $|\varepsilon| < \dfrac{\mathcal{B}^{f_t}_4}{96}r^4+O(r^5)$. Hence, $V \cap \varphi(U)$ is non empty. \\ \hf \hf Next, we note that since $\varphi:U \longrightarrow \mathbb{C}^2$ is a non constant holomorphic map, $\varphi(U)$ is an open subset of $\mathbb{C}^2$. Hence, $V \cap \varphi(U)$ is a non empty open subset of $\mathbb{C}^2$ and has nonzero measure.
Using Sard's Theorem (applied to the function $\varphi:U \longrightarrow \mathbb{C}^2$), we conclude that
$V \cap \varphi(U)$ contains regular values of $\varphi$. Let $(\varepsilon_1,\varepsilon_2) \in V \cap \varphi(U)$ be a regular value of $\varphi$. Therefore by definition of $V$, $d_{\varphi(\partial U)}(\varepsilon_1,\varepsilon_2) > |\varepsilon_2| \geq 0.$ Now, $\varphi(\partial U)$ is a closed subset of $\mathbb{C}^2$ and $d_{\varphi(\partial U)}(\varepsilon_1,\varepsilon_2) >0$ together implies that $(\varepsilon_1,\varepsilon_2) \notin \varphi( \partial U).$ Hence all the hypothesis of \cref{local_degree_kes} are satisfied. We conclude from the proposition that $\textnormal{deg}(\varphi,U,(\varepsilon_1,\varepsilon_2))=\textnormal{deg}(\varphi,U,(\varepsilon_1,0))$, i.e. the number of solutions in $U$ to both the equations $\varphi(x,h)=(\varepsilon_1,\varepsilon_2)$ and $\varphi(x,h)=(\varepsilon_1,0)$ are same. This justifies our claim in \cref{ep2}. \\
\hf \hf Let us now justify the assertion for \cref{epp12}. The argument is similar to the previous argument. We just need to prove the following claim: \begin{claim} Let $U\subseteq \mathbb{C}^3$ be a small open neighborhood of $(0,0,0)$ and $\varphi: U \longrightarrow \mathbb{C}^3$ be given by \begin{align*} \varphi(x,y,u)&:= \Big(f_t(x,y),f_{t_x}(x,y),f_{t_y}(x,y)\Big), \end{align*} where
$f_t$ is as given in \cref{eqf_t} and $f_{t_{20}},f_{t_{11}},f_{t_{02}}$ are as given in \cref{d4_three_nodes_soln}. Let $\widehat{U} \subseteq \mathbb{C}$ be a small open neighborhood of $0$. If $\varepsilon$ is a generic point of $\widehat{U}$, then $(\varepsilon,0,0)$ is a regular value of $\varphi$. \end{claim} \begin{proof} Let $(x,y,u) \in U$ such that $\varphi(x,y,u)=(\varepsilon,0,0)$. We note that \begin{align} \det \begin{pmatrix} f_{t_x}(x,y) & f_{t_{xx}}(x,y) & f_{t_{yx}}(x,y) \\ f_{t_y}(x,y) & f_{t_{xy}}(x,y) & f_{t_{yy}}(x,y) \\ f_{t_u}(x,y) & f_{t_{x_u}}(x,y) & f_{t_{y_u}}(x,y) \end{pmatrix} & = f_{t_u}(x,y) \cdot \det \begin{pmatrix} f_{t_{xx}}(x,y) & f_{t_{yx}}(x,y) \\ f_{t_{xy}}(x,y) & f_{t_{yy}}(x,y) \end{pmatrix}. \label{y4} \end{align} This is because $f_{t_x}(x,y)$ and $f_{t_y}(x,y)$ are both equal to zero.
We now note that $f_t$ has an $A_1$ singularity at $(0,0)$; hence determinant of Hessian of $f_t$ does not vanish at $(0,0)$. Since $(x,y)$ is small, we conclude that the determinant of the Hessian of $f_t$ at $(x,y)$ is nonzero. Hence, if the right hand side of \cref{y4} is zero, then $f_{t_u}(x,y)$ has to be zero. We claim this is not possible for a generic $\varepsilon$. To see why this is so, note that the solution to the equation $\varphi(x,y,u)=(\varepsilon,0,0)$ with $\varepsilon \in \widehat{U}$ is given by \cref{sol3}.
After plugging the value of $(x,y)$ obtained in \cref{sol3} to the expression of $f_{t}(x,y)$, we conclude from \cref{g_3_mult} that \begin{align*} f_{t_u}(x,y)& = -\Big(\frac{3(A_2-A_1)^2 (A_3-A_1)^2}{324 (A_2 - A_3)}\Big) u^2 + O(u^3) \end{align*} Note that $f_{t_u}(x,y)$ is a power series of $u$ which is not identically zero in a small open subset of $\mathbb{C}$ containing the origin and hence it has only finitely many zeros. We conclude that $(\varepsilon,0,0)$ is a regular value of $\varphi$ for all but a finite set of $\varepsilon$; in particular for a generic $\varepsilon$, $(\varepsilon,0,0)$ is a regular value of $\varphi$. \end{proof}
\subsection{Proof of \Cref{npa1}: computation of $N(A_1^{\delta} \mathcal{P} A_1)$ when $0\leq \delta \leq 2$} We will now justify our formula for $N(A_1^{\delta} \mathcal{P} A_1,r, s, n_1, n_2, n_3, \theta)$, when $0 \leq \delta \leq 2$. If $\theta =0$, then the formula follows from \cref{deg_to_one_up_to_down}.\\ \hf \hf Let us now assume $\theta>0$. Recall that (as per the definition in section \ref{notation}) \begin{align*} \A^{\delta}_1 \circ \overline{\hat{\A}}_1 := \{ ([f], [\eta], q_1, \ldots, q_{\delta}, l_{q_{\delta+1}}) \in \mathcal{S}_{\mathcal{D}_{\delta}}\times_{\mathcal{D}} \mathbb{P} W_{\mathcal{D}}: &\textnormal{$f$ has a singularity of type $\A_1$ at $q_1, \ldots, q_{\delta}$}, \\
& ([f], [\eta], l_{q_{\delta+1}}) \in \overline{\hat{A}}_1, ~~\textnormal{$q_1, \ldots, q_{\delta+1}$ all distinct}\}. \end{align*} Let $\mu$ be a generic cycle, representing the class \begin{align*} [\mu] = \mathcal{H}_L^r \cdot \mathcal{H}_p^s \cdot a^{n_1} \lambda^{n_2} (\pi_{\delta+1}^*H)^{n_3} (\pi_{\delta+1}^*\lambda_{W})^{\theta}. \end{align*}
We now define a section of the following bundle \begin{align*} \Psi_{\PP \A_1}: \A^{\delta}_1 \circ \overline{\hat{\A}}_1 \longrightarrow \mathcal{L}_{\PP \A_1} & := \gamma_{\DD}^\ast\otimes \gamma_W^{\ast 2}\otimes \gamma_{\mathbb{P}^3}^{* d}, \qquad \textnormal{given by}\\
\{\Psi_{\PP \A_1}([f], q_1,\ldots,q_\delta,l_{q_{\delta+1}})\}(f\otimes v^{\otimes 2}) &:= \nabla^2 f|_{q_{\delta+1}}(v, v). \nonumber
\end{align*} We will show shortly that this section is transverse to zero. Next, let us define \begin{align*} \mathcal{B} &:= \overline{\A^{\delta}_1 \circ \overline{\hat{\A}}}_1- \A^{\delta}_1 \circ \overline{\hat{\A}}_1. \end{align*} Hence \begin{align} \langle e(\mathcal{L}_{\PP \A_1}), ~~[\overline{\A^{\delta}_1 \circ \overline{\hat{\A}}}_1] \cap [\mu] \rangle & = \N(\A_1^{\delta}\PP \A_1, r, s, n_1, n_2, n_3, \theta) + \mathcal{C}_{\mathcal{B}\cap \mu}, \label{npa1_euler_bdry_eqn} \end{align} where as before, $\mathcal{C}_{\mathcal{B} \cap \mu}$ denotes the contribution of the section to the Euler class from $\mathcal{B} \cap \mu$. When $\delta=0$, the boundary $\mathcal{B}$ is empty. Hence, plugging in $\mathcal{C}_{\mathcal{B} \cap \mu}=0$ and unwinding the left hand side of \cref{npa1_euler_bdry_eqn} gives us the formula of \Cref{npa1} for $\delta=0$. \\ \hf \hf Let us now assume $\delta>0$. Given $k$ distinct integers $i_1, i_2, \ldots, i_k \in [1, \delta+1]$, let $\Delta_{i_1, \ldots, i_{k}}$ be as defined in the proof of \Cref{na1_delta}. Let us define \begin{align*} \hat{\Delta}_{i_1, \ldots, i_{k}}&:= \pi^{-1}(\Delta_{i_1, \ldots, i_{k}}), \end{align*} where $\pi:\mathcal{S}_{\mathcal{D}_{\delta}}\times_{\mathcal{D}} \mathbb{P} W_{\mathcal{D}} \longrightarrow \mathcal{S}_{\mathcal{D}_{\delta+1}}$ is the projection map. Let us define \begin{align*}
\mathcal{B}(q_{i_1}, \ldots, q_{i_{k-1}}, l_{q_{\delta+1}}) & := \mathcal{B}\cap \hat{\Delta}_{i_1, \ldots, i_{k-1}, \delta+1}. \end{align*} Let us now consider $\mathcal{B}(q_i, l_{q_{\delta+1}})$. We claim that, \begin{align} \B (q_i, l_{q_{\delta +1 }}) & \approx \overline{\A_1^{\delta-1} \circ \hat{\A}}_3, \label{two_nodes_Hat_A3} \end{align} where $\B (q_i, l_{q_{\delta +1 }})$ is identified as a subset of $\mathcal{S}_{\mathcal{D}_{\delta-1}}\times_{\widehat{\mathbb{P}}^3} \mathbb{P} W_{\mathcal{D}}$ in the obvious way (namely via the inclusion map where the $(\delta+1)^{\textnormal{th}}$ point is equal to the $i^{\textnormal{th}}$ point). We will justify that shortly. Let us now intersect $\overline{\A_1^{\delta-1} \circ \hat{\A}}_3$ with $\mu$. This will be an isolated set of finite points. Hence, the section $\psi_{\PP \A_1}$ will not vanish on $\overline{\A_1^{\delta-1} \circ \hat{\A}}_3 \cap \mu$. Hence it does not contribute to the Euler class. \\ \hf \hf Next, let us consider $\B (q_{i_1}, q_{i_2}, l_{q_{\delta +1 }})$.
We claim that \begin{align} \B (q_{i_1}, q_{i_2}, l_{q_{\delta +1 }}) & \approx \overline{\A_1^{\delta-2} \circ \hat{A}}_5 \cup \overline{\A_1^{\delta-2} \circ \widehat{D}}_4 . \label{three_nodes_Hat_D4} \end{align} The set $\overline{\A_1^{\delta-2} \circ \hat{A}}_5 \cap \mu$ is empty since the sum total of the dimensions of these two varieties is one less than the dimension of the ambient space. Next, we note that the section $\Psi_{\PP \A_1}$ vanishes everywhere on $\overline{\A_1^{\delta-2} \circ \widehat{D}}_4$; hence it also vanishes on $\overline{\A_1^{\delta-2} \circ \widehat{D}}_4 \cap \mu$.
We claim that the contribution from each of the points of $\B (q_{i_1}, q_{i_2}, l_{q_{\delta +1 }}) \cap \mu$ is $6$. Hence the total contribution from all the components of type $\B (q_{i_1}, q_{i_2}, l_{q_{\delta +1 }})$ is
\begin{align*} 6 \binom{\delta}{2} \N(\A_1^{\delta-2} \widehat{D}_4, n_1, n_2,n_3, \theta). \end{align*} Plugging this in \cref{npa1_euler_bdry_eqn} gives us the formula of \cref{npa1}. \\ \hf \hf Let us now justify the transversality, closure and multiplicity claims. We will follow the setup of \cref{na1_delta}. Suppose \begin{align*} \Psi_{\PP A_1}([f], [\eta], q_1, \ldots, q_{\delta}, l_{q_{\delta+1}}) & =0. \end{align*} As before we assume $\eta$ determines the plane where the last component is zero and $q_{\delta+1}:= [0,0,1,0]$.
Let us consider $T\mathbb{P}^2_{\eta}|_{[q_{\delta+1}]}$. Let $\partial_x$ and $\partial_y$ be the standard basis
vectors for $T\mathbb{P}^2_{\eta}|_{[q_{\delta+1}]}$ (corresponding to the first two coordinates). Hence \begin{align*}
l_{q_{\delta+1}} & = [a \partial_x + b \partial_y] \in \mathbb{P}T\mathbb{P}^2_{\eta}|_{[q_{\delta+1}]} \end{align*} for some complex numbers $a,b$ not both of which are zero. Without loss of generality, we can assume $l_{q_{\delta+1}} = [\partial_x]$. Let us now consider the polynomial \begin{align*} \rho_{20}&:= (X-X_1)^2(X-X_2)^2\ldots \cdot (X-X_{\delta})^2 X^2 Z^{d-2\delta-2} \end{align*} and consider the corresponding curve $\gamma_{20}(t)$. We now note \begin{align*}
\{\{\nabla \Psi_{\PP A_1}|_{([f],[\eta], q_1, \ldots, q_{\delta}, l_{q_{\delta+1}})}\}(\gamma_{20}^{\prime}(0))\}(f \otimes
\partial_x \otimes \partial_x)&= \lambda Z^{d-2\delta-2} \nabla^2 X|_{[0,0,1,0]}(\partial_x, \partial_x). \end{align*}
Since $\nabla^2 X|_{[0,0,1,0]}(\partial_x, \partial_x)$ is nonzero, we conclude that the section is transverse to zero. \\ \hf \hf Next, let us justify the closure claims. Let us start with \cref{two_nodes_Hat_A3}. This statement is saying that when two nodes collide, we get a tacnode. Hence, the proof of \cref{two_nodes_Hat_A3} is same as the proof of \cref{a1_a1_is_a3}. \\ \hf \hf Next, let us consider \cref{three_nodes_Hat_D4}. Again, this statement is saying what happens what happens when three nodes collide. Hence, the proof of \cref{three_nodes_Hat_D4} is same as the proof of \cref{three_node_triple_point}. \\ \hf \hf It remains to justify the contribution from the points of $\overline{\A_1^{\delta-2} \circ \widehat{D}}_4 \cap \mu$. We will use the solutions constructed in \cref{d4_three_nodes_soln}. Using the expression for $f_{t_{20}}$, we note that the multiplicity from each branch is the number of small solutions $u$ to the equation \begin{align*} -\frac{f_{t_{30}}}{3} (A_1 - A_3) u + E_6(u) & = \varepsilon. \end{align*} This is clearly $1$. Since there are $6$ branches, the total multiplicity is $6$. \qed
\subsection{Proof of \Cref{npa2}: computation of $N(A_1^{\delta} \mathcal{P} A_2)$ when $0\leq \delta \leq 2$}
We will justify our formula for $N(A_1^{\delta} \mathcal{P} A_2,r, s, n_1, n_2, n_3, \theta)$, when $0 \leq \delta \leq 2$. Recall that \begin{align*} \A^{\delta}_1 \circ \overline{\PP \A}_1 := \{ ([f], [\eta], q_1, \ldots, q_{\delta}, l_{q_{\delta+1}}) \in \mathcal{S}_{\mathcal{D}_{\delta}}\times_{\mathcal{D}} \mathbb{P} W_{\mathcal{D}}: &\textnormal{$f$ has a singularity of type $\A_1$ at $q_1, \ldots, q_{\delta}$}, \\
& ([f], [\eta], l_{q_{\delta+1}}) \in \overline{\PP \A}_1, ~~\textnormal{$q_1, \ldots, q_{\delta+1}$ all distinct}\}. \end{align*} Let $\mu$ be a generic cycle, representing the class \begin{align*} [\mu] = \mathcal{H}_L^r \cdot \mathcal{H}_p^s \cdot a^{n_1} \lambda^{n_2} (\pi_{\delta+1}^*H)^{n_3} (\pi_{\delta+1}^*\lambda_{W})^{\theta}. \end{align*} Recall that as per the hypothesis of the Theorem, if $\delta=2$ then $\theta=0$. We now define a section of the following line bundle \begin{align} \Psi_{\PP \A_2}: \A^{\delta}_1 \circ \overline{\PP \A}_1 \longrightarrow \mathbb{L}_{\PP \A_2} & := \gamma_{\DD}^\ast\otimes \gamma_W ^\ast \otimes (W/\gamma_{W})^\ast\otimes \gamma_{\mathbb{P}^3}^{* d}, \qquad \textnormal{given by} \nonumber \\ \{\Psi_{\PP \A_2}([f], q_1,\ldots,q_\delta,l_{q_{\delta+1}})\}(f\otimes v \otimes w) &:=
\nabla^2 f|_{q_{\delta+1}}(v,w). \nonumber
\end{align}
We will show shortly that this section is transverse to zero.
Next, let us define \begin{align*} \mathcal{B} &:= \overline{\A^{\delta}_1 \circ \overline{\PP \A}}_1- \A^{\delta}_1 \circ \overline{\PP\A}_1. \end{align*} Hence \begin{align} \langle e(\mathbb{L}_{\PP \A_2}), ~~[\overline{\A^{\delta}_1 \circ \overline{\PP\A}}_1] \cap [\mu] \rangle & = \N(\A_1^{\delta}\PP \A_2, r, s, n_1, n_2, n_3, \theta) + \mathcal{C}_{\mathcal{B}\cap \mu}. \label{npa2_Euler_class_formula} \end{align} Define $\mathcal{B}(q_{i_1}, \ldots q_{i_k}, l_{q_{\delta+1}})$ as before. For simplicity, let us set $(i_1, i_2, \ldots, i_k):= (\delta-k, \ldots, \delta-1, \delta)$. Before we describe $\B (q_{i_1},q_{i_2}, l_{q_{\delta +1 }})$, let us define a few things.
Let $v$ be a fixed nonzero vector that belongs to $l_{q_{\delta+1}}$.
Let us define $W_1, W_2$, $W_3$ and $W_4$ as \begin{align} W_1 &:= \{ ([f], [\eta], q_1, \ldots,q_{\delta}, l_{q_{\delta+1}}) \in
\overline{\A^{\delta}_1 \circ \overline{\PP\A}}_1: \nabla^2f|_{q_{\delta+1}} \not\equiv 0\}, \nonumber \\
W_2 &:= \{ ([f], [\eta], q_1, \ldots,q_{\delta}, l_{q_{\delta+1}}) \in \overline{\A^{\delta}_1 \circ \overline{\PP\A}}_1:
\nabla^2f|_{q_{\delta+1}} \equiv 0 \}, \nonumber \\ W_3 &:= \{ ([f], [\eta], q_1, \ldots,q_{\delta}, l_{q_{\delta+1}}) \in \overline{\A^{\delta}_1 \circ \overline{\PP\A}}_1:
\nabla^3f|_{q_{\delta+1}}(v,v,v) \neq 0 \} \qquad \textnormal{and} \nonumber \\ W_4 &:= \{ ([f], [\eta], q_1, \ldots,q_{\delta}, l_{q_{\delta+1}}) \in \overline{\A^{\delta}_1 \circ \overline{\PP\A}}_1:
\nabla^3f|_{q_{\delta+1}}(v,v,v) = 0 \}. \end{align}
We claim that \begin{align}
\B (q_{\delta}, l_{q_{\delta +1 }}) \cap W_1 & \approx \overline{\A_1^{\delta-1} \circ \PP \A}_3 \cap W_1, \label{hh1} \\ \B (q_{\delta}, l_{q_{\delta +1 }}) \cap W_2 & \approx \overline{\A_1^{\delta-1} \circ \widehat{\D}}_4, \label{hh2} \\
\B (q_{\delta-1}, q_{\delta}, l_{q_{\delta +1 }}) \cap W_1 & \subset \overline{\A_1^{\delta-2} \circ \PP \A}_5 \cap W_1, \label{hh3} \\ \B (q_{\delta-1}, q_{\delta}, l_{q_{\delta +1 }}) \cap (W_2 \cap W_4) & \approx \overline{\A_1^{\delta-2} \circ \PP \D}_4 \cap W_1 \qquad \textnormal{and} \label{hh4} \\ \B (q_{\delta-1}, q_{\delta}, l_{q_{\delta +1 }}) \cap (W_2 \cap W_3) & \subset \overline{\A_1^{\delta-2} \circ \widehat{D}}_5. \label{hh5} \end{align} Notice that equations \eqref{hh3} and \eqref{hh5} say that the left hand side is a subset of the right hand side (unlike the other three equations, which assert equality of sets). We now note that equations \eqref{hh1} and \eqref{hh2}, imply that \begin{align}
\B (q_{i_1}, l_{q_{\delta +1 }}) & \approx \overline{\A_1^{\delta-1} \circ \PP \A}_3\cup \overline{\A_1^{\delta-1} \circ \widehat{\D}}_4, \label{a1_Pa1_clsr}
\end{align} while equations \eqref{hh3}, \eqref{hh4} and \eqref{hh5} imply that \begin{align} \B (q_{i_1},q_{i_2}, l_{q_{\delta +1 }}) & \subset \overline{\A_1^{\delta-2} \circ \PP \A}_5\cup \overline{\A_1^{\delta-2} \circ \PP D}_4 \cup \overline{\A_1^{\delta-2} \circ \widehat{\D}}_5. \label{a1_a1_Pa1_clsr} \end{align}
We claim that the contribution to the Euler class from each of the points of $\overline{\A_1^{\delta-1} \circ \PP \A}_3 \cap \mu$, $\overline{\A_1^{\delta-1} \circ \widehat{\D}}_4 \cap \mu$ and $\overline{\A_1^{\delta-2} \circ \PP \D}_4 \cap \mu$ are
$2,3$ and $4$ respectively. \\ \hf \hf Next, we note that for dimensional reasons, the intersection of $\overline{\A_1^{\delta-1} \circ \PP \A}_5$ with $\mu$ is empty. Hence, by \cref{hh3}, the intersection of $\B (q_{\delta-1}, q_{\delta}, l_{q_{\delta +1 }}) \cap W_1$ with $\mu$ is also empty and hence does not contribute to the Euler class. Finally, let us consider the component corresponding to the left hand side of \cref{hh5}; this is where we will use $\theta=0$. Let us consider the projection map \begin{align*} \pi:\mathcal{S}_{\mathcal{D}_{\delta}} \times_{\widehat{\mathbb{P}}^3} \mathbb{P}W_D \longrightarrow \mathcal{S}_{\mathcal{D}_{\delta+1}}. \end{align*}
We recall that \begin{align*} \overline{\A_1^{\delta-2} \circ \widehat{D}}_5 & = \pi^{-1}(\overline{\A_1^{\delta-2} \circ D}_5). \end{align*}
Since $\theta=0$, we note that $\mu$ is the pullback of a class $\nu$, i.e. \begin{align*} \mu &= \pi^*(\nu). \end{align*} Hence, the intersection of $\mu$ with $\overline{\A_1^{\delta-2} \circ \widehat{D}}_5$ is in one to one correspondence with the intersection of $\nu$ with $\overline{\A_1^{\delta-2} \circ D}_5$. But the degree of the cohomology class $\nu$ is one more than the dimension of the cycle $\overline{\A_1^{\delta-2} \circ D}_5$. Hence, the intersection of $\overline{\A_1^{\delta-2} \circ D}_5$ with $\nu$ is empty and hence, the intersection of $\mu$ with $\overline{\A_1^{\delta-2} \circ \widehat{D}}_5$ is empty.
As a result, by \cref{hh5}, the intersection of $\B (q_{\delta-1}, q_{\delta}, l_{q_{\delta +1 }}) \cap (W_2 \cap W_3)$ with $\mu$ is also empty. Hence the total contribution from all the components of type $\B (q_{i_1}, l_{q_{\delta +1 }})$ equals \begin{align*} 2\binom{\delta}{1}N(\A_1^{\delta-1}\PP\A_3, r, s, n_1, n_2, n_3, \theta) +3\binom{\delta}{1}N(\A_1^{\delta-1}\widehat{\D}_4, r, s, n_1, n_2, n_3, \theta), \end{align*} while the total contribution from all the components of type $\B (q_{i_1},q_{i_2}, l_{q_{\delta +1 }})$ equals \begin{align*} 4\binom{\delta}{2}N(\A_1^{\delta-2}\PP\D_4, r, s, n_1, n_2, n_3, \theta).
\end{align*} Plugging this in \cref{npa2_Euler_class_formula} gives us the formula of \cref{npa2}. \\ \hf \hf Let us now prove the claim about transversality. This follows from following the setup of proof of transversality in Theorem \cref{npa1}. We consider the polynomial \begin{align*} \rho_{11}&:= (X-X_1)^2(X-X_2)^2\ldots \cdot (X-X_{\delta})^2 XY Z^{d-2\delta-2} \end{align*} and the corresponding curve $\gamma_{11}(t)$. Transversality follows by computing the derivative of the section $\Psi_{\PP A_2}$ along the curve $\gamma_{11}(t)$ as before.\\ \hf \hf Next, let us justify the closure and multiplicity claims. We will start by justifying \cref{a1_Pa1_clsr}. It suffices to justify \cref{hh1} and \cref{hh2}. Let us rewrite these two equations explicitly, namely
\begin{align} \{([f], [\eta], q_1, \ldots,q_{\delta}, l_{q_{\delta+1}}) \in \overline{A_1^{\delta}\circ \PP A}_1: q_{\delta} = q_{\delta+1}\} \cap W_1 & = \overline{A_1^{\delta-1}\circ \PP A}_3 \cap W_1 \qquad \textnormal{and} \label{eq1_w1}\\ \{([f], [\eta], q_1, \ldots,q_{\delta}, l_{q_{\delta+1}}) \in \overline{A_1^{\delta}\circ \PP A}_1: q_{\delta} = q_{\delta+1}\} \cap W_2 & = \overline{A_1^{\delta-1}\circ \widehat{D}}_4. \label{eq2_w2} \end{align} Since $\widehat{D}_4$ is a subset of $W_2$, we did not write $\cap W_2$ on the right hand side of \cref{eq2_w2}.\\ \hf \hf Let us now start the proof of \cref{eq1_w1}. Let us first explain why the left hand side of \cref{eq1_w1} is a subset of its right hand side. To see that, first we note that $\PP A_1$ is a subset of $\overline{\hat{A}}_1$. Since we have shown while proving \cref{a1_a1_is_a3} and \cref{a1_a1_a3_exp} that when two nodes collide we get a tacnode in \cref{a1_a1_a3_exp}, we conclude that \begin{align*} \{([f], [\eta], q_1, \ldots,q_{\delta}, l_{q_{\delta+1}}) \in \overline{A_1^{\delta}\circ \hat{A}}_1: q_{\delta} = q_{\delta+1}\} & = \overline{A_1^{\delta-1}\circ \hat{A}}_3. \end{align*} Hence, we conclude that \begin{align} \{([f], [\eta], q_1, \ldots,q_{\delta}, l_{q_{\delta+1}}) \in \overline{A_1^{\delta}\circ \PP A}_1: q_{\delta} = q_{\delta+1}\} & \subset \overline{A_1^{\delta-1}\circ \hat{A}}_3 \nonumber \\ \implies \{([f], [\eta], q_1, \ldots,q_{\delta}, l_{q_{\delta+1}}) \in \overline{A_1^{\delta}\circ \PP A}_1: q_{\delta} = q_{\delta+1}\} \cap W_1 & \subset \overline{A_1^{\delta-1}\circ \hat{A}}_3 \cap W_1. \label{a1_PA1_hat_A3} \end{align} Suppose $([f], [\eta], q_1, \ldots,q_{\delta}, l_{q_{\delta+1}})$ belongs to the left hand side of \cref{a1_PA1_hat_A3}. Since $([f], [\eta], l_{q_{\delta+1}})$ belongs to $\overline{\PP A}_1$, we conclude that \begin{align*}
\nabla^2f|_{q_{\delta+1}}(v,v) & = 0 \qquad \forall ~~v \in l_{q_{\delta+1}}. \end{align*} Since $([f], [\eta], q_1, \ldots,q_{\delta}, l_{q_{\delta+1}})$ is a subset of the right hand side of
\cref{a1_PA1_hat_A3}, we conclude that the Hessian $\nabla^2f|_{q_{\delta+1}}$ is not identically zero, but it has a non trivial Kernel. We claim that $v$ is in the Kernel of the Hessian. To see why, let us assume that the nonzero vector $\tilde{v}$ is in the Kernel of the Hessian, i.e.
$\nabla^2f|_{q_{\delta+1}}(\tilde{v}, \cdot)=0$. Let $w$ be any other vector, linearly independent from $\tilde{v}$. Since the Hessian is not identically zero and the vector space is two dimensional,
we conclude that $\nabla^2f|_{q_{\delta+1}}(w, w) \neq 0$. Hence, writing the vector $v:= \lambda_1 \tilde{v} + \lambda_2 w$
and using $\nabla^2f|_{q_{\delta+1}}(v,v)=0$, we conclude that $\lambda_2 =0$. Hence, $v$ belongs to the Kernel of the Hessian.
But we also note that if $([f], [\eta], l_q) \in \PP A_3$ and $\nabla^2 f|_q(v, \cdot)=0$, then $\nabla^3 f|_q(v,v,v) =0$. Hence, we can improve \cref{a1_PA1_hat_A3} and conclude that
the left hand side of \cref{eq1_w1} is a subset of its right hand side.\\ \hf \hf Let us now prove the converse. We will simultaneously prove the following two statements \begin{align} \{([f], [\eta], q_1, \ldots,q_{\delta}, l_{q_{\delta+1}}) \in \overline{A_1^{\delta}\circ \PP A}_1: q_{\delta} = q_{\delta+1}\} & \supset A_1^{\delta-1}\circ \PP A_3, \label{k1} \\ \Big(\{([f], [\eta], q_1, \ldots,q_{\delta}, l_{q_{\delta+1}}) \in \overline{A_1^{\delta}\circ \PP A}_2: q_{\delta} = q_{\delta+1}\}\Big) \cap \Big(A_1^{\delta-1}\circ \PP A_3\Big) & = \emptyset \qquad \textnormal{and} \label{k2} \end{align} We will prove the following claim:
\begin{claim} \label{one_node_andone_PA1_claim} Let $([f], [\eta], q_1, \ldots, \ldots, q_{\delta-1}, l_{q_{\delta}}) \in A_1^{\delta-1}\circ \PP A_3$. Then there exists points \begin{align} \big([f_t], [\eta_t], q_1(t), \ldots, q_{\delta-2}(t); q_{\delta-1}(t), q_{\delta}(t), l_{q_{\delta+1}(t)}\big) \in A_1^{\delta}\circ \PP A_1 \label{sol_const} \end{align} sufficiently close to $([f], [\eta], q_1, \ldots, \ldots, q_{\delta-1}; q_{\delta}, l_{q_{\delta}})$.
Furthermore, \textit{every} such solution satisfies the condition \begin{align}
\nabla^2f|_{q_{\delta+1}}(v,w) &\neq 0, \label{one_node_one_PA2_int_PA3_empty} \end{align} if $v$ is a nonzero vector that belongs to $l_{q_{\delta+1}(t)}$ and $w$ is a nonzero vector that belongs to
$T\mathbb{P}^2_{\eta}|_{q_{\delta+1}(t)}/l_{q_{\delta+1}(t)}$. In other words, \begin{align*} \big([f_t], [\eta_t], q_1(t), \ldots, q_{\delta-1}(t), q_{\delta}(t), l_{q_{\delta+1}(t)}\big) & \not\in A_1^{\delta} \circ \PP A_2. \end{align*}
\end{claim} \begin{rem} We note that \cref{one_node_andone_PA1_claim} simultaneously proves \cref{k1} and \cref{k2}.
\end{rem}
\noindent \textbf{Proof:} Following the setup of the proofs of claims \ref{two_and_three_node_claim} and \ref{three_node_claim_d4}, we will now work in an affine chart, where we send the plane $\mathbb{P}^2_{\eta_t}$ to $\mathbb{C}^2_{z}$ and the point $q_{\delta}(t) \in \mathbb{P}^2_{\eta_t}$ to $(0,0,0) \in \mathbb{C}^2_{z}$. We also choose coordinates, such that $ \partial_x \in l_{q_{\delta+1}(t)}$. Using this chart, let us write down the Taylor expansion of $f_t$ around the point $(0,0)$, namely \begin{align*} f_t(x,y)&= f_{t_{11}}xy + \frac{f_{t_{02}}}{2} y^2 + \frac{f_{t_{30}}}{6} x^3 + \frac{f_{t_{21}}}{2} x^2y + \frac{f_{t_{12}}}{2} x y^2 + \frac{f_{t_{03}}}{6} y^3+\ldots \end{align*} Since $([f_t], [\eta_t], l_{q_{\delta}(t)}) \in \PP A_1$, we conclude that $f_{t_{20}}$ is zero. Next, let us consider the Taylor expansion of $f$ (not $f_t$). We note that $([f], [\eta], l_{q_{\delta}}) \in \PP A_3$. This means that $f_{11}$ and $f_{02}$ can not both be zero (since that would mean the Hessian is identically zero). If $f_{02} =0$ and $f_{11} \neq 0$, then it implies that $([f], [\eta], l_{q_{\delta}}) \in \hat{A}_1$ (and hence does not belong to $\PP A_3$). Hence, $f_{02} \neq 0$ and hence we conclude that $f_{t_{02}} \neq 0$. Finally, since $([f], [\eta], l_{q_{\delta}}) \in \PP A_3$, we conclude that $f_{{11}}$ and $f_{{30}}$ are zero; hence $f_{t_{11}}$ and $f_{t_{30}}$ are small (close to zero). We will mainly follow the Proof of \cref{two_and_three_node_claim}. Since $f_{t_{02}} \neq 0$ we can make the same change of coordinates $ \hat{y}:= y + B(x)$ as in the Proof of \cref{two_and_three_node_claim} and write $f_t$ as \begin{align*} f_t (x, y(x, \hat{y})) &= \varphi(x,\hat{y})\hat{y}^2 + \frac{\mathcal{B}_2^{f_t}}{2!} x^2 + \frac{\mathcal{B}_3^{f_t}}{3!} x^3 + \frac{\mathcal{B}_4^{f_t}}{4!} x^4 + \mathcal{R}(x)x^5, \end{align*}
where \begin{align} \B^{f_t}_2 & := - \frac{f_{t_{11}}^2}{f_{t_{02}}}, \qquad \B^{f_t}_3 := f_{t_{30}} -\frac{3f_{t_{11}} f_{t_{21}}}{f_{t_{02}}}+\frac{3f_{t_{11}}^2 f_{t_{12}}}{f_{t_{02}}^2} -\frac{3f_{t_{11}}^3 f_{t_{03}}}{f_{t_{02}}^3}, \ldots,
\qquad \varphi(0,0) \neq 0 \label{Bf_k_formula} \end{align} and $\mathcal{R}(x)$ is a holomorphic function defined in a neighborhood of the origin. Since $([f], [\eta], q_{\delta}) \in \PP A_3$, we conclude that $\B^{f_t}_2$ and $\B^{f_t}_3$ are small (close to zero) and $\B^{f_t}_4$ is nonzero. Let us make a further change of coordinates and denote \begin{align*} \hat{\hat{y}}&:= \sqrt{\varphi(x, \hat{y})} \hat{y}. \end{align*} Note that we can choose a branch of the square root since $\varphi(0,0) \neq 0$. Next, for notational convenience, let us now define \begin{align} \hat{f}_t(x, \hat{\hat{y}}) &:= f_t (x, y(x, \hat{y}(\hat{\hat{y}})))), \end{align} i.e. $\hat{f}_t$ is basically $f_t$ written in the new coordinates (namely $x$ and $\hat{\hat{y}}$). Hence, \begin{align*} \hat{f}_t(x, \hat{\hat{y}}) & = \hat{\hat{y}}^2 + \frac{\mathcal{B}_2^{f_t}}{2!} x^2 + \frac{\mathcal{B}_3^{f_t}}{3!} x^3 + \frac{\mathcal{B}_4^{f_t}}{4!} x^4 + \mathcal{R}(x)x^5. \end{align*} We now note that constructing the points on the left hand side of \cref{sol_const} amounts to solving the set of equations \begin{align} \hat{f}_t & = 0, \qquad \hat{f}_{t_x} =0 \qquad \textnormal{and} \qquad \hat{f}_{t_{\hat{\hat{y}}}} =0, \label{eq_k4} \end{align} where $(x, \hat{\hat{y}})$ is small but not equal to $(0,0)$.\\ \hf \hf We will now construct solutions to \cref{eq_k4}. The solutions to \cref{eq_k4} are given by \begin{align}
\hat{\hat{y}} =0, \qquad \mathcal{B}^{f_t}_2 & = \frac{\mathcal{B}_4^{f_t}}{12} x^2 + O(|x|^3) \qquad \textnormal{and} \qquad
\mathcal{B}^{f_t}_3 = -\frac{\mathcal{B}_4^{f_t}}{2} x + O(|x|^2). \label{f11+f30_soln} \end{align} Now we use the expression of $\mathcal{B}^{f_t}_2,\mathcal{B}^{f_t}_3$ and conclude from \cref{f11+f30_soln} that \begin{align}
\frac{f_{t_{11}}^2}{f_{t_{02}}}& = -\frac{\mathcal{B}_4^{f_t}}{12} x^2 + O(|x|^3) \qquad \textnormal{and} \label{f11} \\
f_{t_{30}} & = -3\mathcal{B}_4^{f_t} x+O(|x|^2). \label{f30_soln} \end{align}
Hence, there are two solutions to \cref{f11}, given by \begin{align}
f_{t_{11}} & = \Big(\sqrt{\frac{-f_{t_{02}}\mathcal{B}^{f_t}_4}{12}}\Big) x + O(|x|^2) \qquad \textnormal{or} \qquad
f_{t_{11}} = -\Big(\sqrt{\frac{-f_{t_{02}}\mathcal{B}^{f_t}_4}{12}}\Big) x + O(|x|^2), \label{f11_mult} \end{align} where $\sqrt{}$ denotes a branch of the square root.
Hence, there are \textbf{exactly} two solutions to \cref{eq_k4}, given by \begin{align}
x&= u, \qquad f_{t_{11}} = \pm \Big(\sqrt{\frac{-f_{t_{02}}\mathcal{B}^{f_t}_4}{12}}\Big) u + O(|u|^2) \label{x_and_f11} \end{align} and $\hat{\hat{y}}=0$ and $f_{t_{30}}$ as given by
\cref{f30_soln}, where we plug in the expressions for $x$ and $f_{t_{11}}$ as given by \cref{x_and_f11} to express them in terms of $u$ (the exact expressions in terms of $u$ are not so important, hence we have not written that out explicitly). This proves \cref{one_node_andone_PA1_claim}. Since \cref{x_and_f11} are the \textbf{only} solutions and $\mathcal{B}^{f_t}_4 \neq 0$, we also conclude that \cref{one_node_one_PA2_int_PA3_empty} is true. \qed \\
\hf \hf It remains to compute the multiplicity. We claim the each point of $(A_1^{\delta-1}\circ \PP A_3)\cap \mu$ contributes $2$ to the Euler class in \cref{npa2_Euler_class_formula}.
Using \cref{x_and_f11} we conclude that the multiplicity from each branch is the number of small solutions $u$ to the equation \begin{align*}
\Big(\sqrt{\frac{-f_{t_{02}}\mathcal{B}^{f_t}_4}{12}}\Big) u + O(|u|^2)& = \varepsilon
\qquad \textnormal{and} \qquad -\Big(\sqrt{\frac{-f_{t_{02}}\mathcal{B}^{f_t}_4}{12}}\Big) u + O(|u|^2) = \varepsilon. \end{align*} This number is $1$ in each case and hence, the total multiplicity is $2$. \qed \\ \hf \hf Next, let us justify \cref{eq2_w2}. Let us first explain why the left hand side of \cref{eq2_w2} is a subset of its right hand side. If
$([f], [\eta], q_1, \ldots,q_{\delta}, l_{q_{\delta+1}}) \in W_2$, then it means that $\nabla^2f|_{q_{\delta+1}} = 0$. Hence, it means that $([f], [\eta], l_{q_{\delta+1}}) \in \overline{\widehat{D}}_4$. Hence, the left hand side of \cref{eq2_w2} is a subset of its right hand side.\\
\hf \hf Let us now prove \cref{eq2_w2}. Before that, let us introduce a new space. Let us define \begin{align*}
\widehat{D}_4^{\#}&:= \{([f], [\eta], l_{q}) \in \widehat{D}_4: \nabla^3f|_q(v,v,v) \neq 0 ~~\textnormal{if} ~~v \in l_q-0\}. \end{align*} Note that $\overline{\widehat{D}^{\#}_4} = \overline{\widehat{D}}_4$. We will now simultaneously prove the following two statements: \begin{align} \{([f], [\eta], q_1, \ldots,q_{\delta}, l_{q_{\delta+1}}) \in \overline{A_1^{\delta}\circ \PP A}_1: q_{\delta} = q_{\delta+1}\} & \supset A_1^{\delta-1}\circ \widehat{D}_4^{\#} \qquad \textnormal{and} \label{kk1} \\ \Big(\{([f], [\eta], q_1, \ldots,q_{\delta}, l_{q_{\delta+1}}) \in \overline{A_1^{\delta}\circ \PP A}_2: q_{\delta} = q_{\delta+1}\}\Big) \cap \Big(A_1^{\delta-1}\circ \widehat{D}_4^{\#} \Big) & = \emptyset. \label{kk2} \end{align} We will prove the following claim: \begin{claim} \label{one_node_and_one_PA1_claim_D4} Let $([f], [\eta], q_1, \ldots, \ldots, q_{\delta-1}, l_{q_{\delta}}) \in A_1^{\delta-1}\circ \widehat{D}_4^{\#}$. Then there exists points \begin{align} \big([f_t], [\eta_t], q_1(t), \ldots, q_{\delta-2}(t); q_{\delta-1}(t), q_{\delta}(t), l_{q_{\delta+1}(t)}\big) \in A_1^{\delta}\circ \PP A_1 \label{sol_const_2} \end{align} sufficiently close to $([f], [\eta], q_1, \ldots, \ldots, q_{\delta-1}; q_{\delta}, l_{q_{\delta}})$.
Furthermore, \textit{every} such solution satisfies the condition \begin{align}
\nabla^2f|_{q_{\delta+1}}(v,w) &\neq 0, \label{one_node_one_PA2_int_PA3_empty_2} \end{align} if $v$ is a nonzero vector that belongs to $l_{q_{\delta+1}(t)}$ and $w$ is a nonzero vector that belongs to
$T\mathbb{P}^2_{\eta}|_{q_{\delta+1}(t)}/l_{q_{\delta+1}(t)}$. In other words, \begin{align*} \big([f_t], [\eta_t], q_1(t), \ldots, q_{\delta-1}(t), q_{\delta}(t), l_{q_{\delta+1}(t)}\big) & \not\in A_1^{\delta} \circ \PP A_2. \end{align*} \end{claim} \begin{rem} We note that \cref{one_node_and_one_PA1_claim_D4} proves \cref{kk1} and \cref{kk2} simultaneously (since $\overline{\widehat{D}^{\#}_4} = \overline{\widehat{D}}_4$). \end{rem}
\noindent \textbf{Proof:} Following the setup of the proofs of claims \ref{two_and_three_node_claim}, \ref{three_node_claim_d4} and \ref{one_node_andone_PA1_claim}, we will now work in an affine chart, where we send the plane $\mathbb{P}^2_{\eta_t}$ to $\mathbb{C}^2_{z}$ and the point $q_{\delta}(t) \in \mathbb{P}^2_{\eta_t}$ to $(0,0,0) \in \mathbb{C}^2_{z}$. We also choose coordinates, such that $ \partial_x \in l_{q_{\delta+1}(t)}$. Using this chart, let us write down the Taylor expansion of $f_t$ around the point $(0,0)$, namely \begin{align*} f_t(x,y)&= f_{t_{11}}xy + \frac{f_{t_{02}}}{2} y^2 + \frac{f_{t_{30}}}{6} x^3 + \frac{f_{t_{21}}}{2} x^2y + \frac{f_{t_{12}}}{2} x y^2 + \frac{f_{t_{03}}}{6} y^3+\ldots \end{align*} Since $([f_t], [\eta_t], l_{q_{\delta}(t)}) \in \PP A_1$, we conclude that $f_{t_{20}}$ is zero. Next, since $([f], [\eta], l_{q_{\delta}}) \in \widehat{D}_4$, we conclude that $f_{20}, \, f_{11}$ and $f_{02}$ are zero; hence $f_{t_{11}}$ and $f_{t_{02}}$ are small (close to zero). Hence, constructing points on the right hand side of \cref{sol_const_2} amounts to finding solutions to the set of equations \begin{align} f_t & = 0, \qquad f_{t_x} =0 \qquad \textnormal{and} \qquad f_{t_{y}} =0, \label{eq_s1} \end{align} where $(x,y)$ is small but not equal to $(0,0)$. Let us define \begin{align*} g_t(x,y) & = -2f_t(x,y)+x f_{t_x}(x,y) +y f_{t_y}(x,y). \end{align*} We note that $f_t(x,y)$ and $g_t(x,y)$ have the same cubic term in the Taylor expansion. Furthermore, $g_t(x,y)$ does not contain any quadratic term. Since $([f], [\eta], l_{q_{\delta}}) \in \widehat{D}_4$, we conclude that
$f_{t_{30}} \neq 0$.
Let \begin{align*} x&:= \hat{x} + E_1(\hat{x}, \hat{y}) \qquad \textnormal{and} \qquad y:= \hat{y} + E_2(\hat{x}, \hat{y}) \end{align*} be changes change of coordinates (where $E_1$ and $E_2$ are second order terms), such that \begin{align*} g_t &= \frac{f_{t_{30}}}{6}(\hat{x}- A_1 \hat{y})(\hat{x}- A_2 \hat{y}) (\hat{x}- A_3 \hat{y}) \end{align*} There are three solutions to $g_t = 0$, given by $\hat{y}=u$ and $\hat{x}= A_i \hat{u}$, for $i=1,2$ and $3$. Converting back in terms of $x$ and $y$, we conclude that the solutions to $g_t =0$ are given by \begin{align*}
y& = u \qquad \textnormal{and} \qquad x = A_i u + O(|u|^2). \end{align*}
Let us consider the solution $x = A_1 u + O(|u|^2)$; the other two cases can be dealt with similarly. We plug this solution into the equations $f_{t_x} =0$ and $f_{t_y} =0$ and solve for $f_{t_{11}}$ and $f_{t_{02}}$ in terms of $u$. Doing that, we get the solutions to \cref{eq_s1} are given by \begin{align}
y &= u, \qquad x = A_1 u + O(|u|^2), \nonumber \\
f_{t_{11}} & = -\frac{f_{t_{30}}}{6}(A_1-A_2)(A_1-A_3) u + O(|u|^2) \qquad \textnormal{and} \qquad
f_{t_{02}} = \frac{f_{t_{30}}}{3} A_1 (A_1-A_2) u + O(|u|^2) \label{f11_sol_8} \end{align}
and two more similar solutions corresponding to $x = A_2 u + O(|u|^2)$ and $x = A_3 u + O(|u|^2)$. This proves the first assertion of \cref{one_node_and_one_PA1_claim_D4}. Furthermore, since $f_{t_{30}} \neq 0$ and $A_1, A_2$ and $A_3$ are distinct, we conclude using \cref{f11_sol_8} that $f_{t_{11}} \neq 0$; this proves \cref{one_node_one_PA2_int_PA3_empty_2}. \qed \\ \hf \hf It remains to compute the multiplicity. We claim the each point of $(A_1^{\delta-1}\circ \widehat{D}_4^{\#})\cap \mu$ contributes $3$ to the Euler class in \cref{npa2_Euler_class_formula}.
Using \cref{f11_sol_8} we conclude that the multiplicity from each branch is the number of small solutions $u$ to the equation \begin{align*}
-\frac{f_{t_{30}}}{6}(A_1-A_2)(A_1-A_3) u + O(|u|^2) & = \varepsilon.
\end{align*} This number is $1$ and hence, the total multiplicity is $3$. Finally, we note that since $\mu$ is a generic cycle all points of $(A_1^{\delta-1}\circ \widehat{D}_4)\cap \mu$ will actually belong to $(A_1^{\delta-1}\circ \widehat{D}_4^{\#})\cap \mu$.\qed \\ \hf \hf Before proceeding further, note that we have proved \begin{align} \Big(\{([f], [\eta], q_1, \ldots,q_{\delta}, l_{q_{\delta+1}}) \in \overline{A_1^{\delta}\circ \PP A}_1: q_{\delta-1}=q_{\delta} = q_{\delta+1}\}\Big) \cap \Big(A_1^{\delta-1}\circ \widehat{D}_4^{\#}\Big) & = \emptyset. \label{k4} \end{align} To see why that is so, our proof of the claim shows that the family we constructed can not have a third node. \\ \hf \hf Next, let us prove equations \eqref{hh3}, \eqref{hh4} and \eqref{hh5} (i.e. we will analyze what happens when three points come together). Let us start with the proof of \eqref{hh3}. Let us show that \begin{align} \Big(\{([f], [\eta], q_1, \ldots,q_{\delta}, l_{q_{\delta+1}}) \in \overline{A_1^{\delta}\circ \PP A}_1: q_{\delta-1}=q_{\delta} = q_{\delta+1}\}\Big) \cap \Big(A_1^{\delta-1}\circ \PP A_4\Big) & = \emptyset. \label{k6} \end{align} We note that \cref{k6} immediately implies \cref{hh3}. In order to prove \cref{k6}, it suffices to prove the following claim:
\begin{claim} \label{two_nodes_and_one_PA1_claim_not_PA4} Let $([f], [\eta], q_1, \ldots, \ldots, q_{\delta-2}, l_{q_{\delta-1}}) \in A_1^{\delta-2}\circ \PP A_4$. Then there does not exist any point \begin{align} \big([f_t], [\eta_t], q_1(t), \ldots, q_{\delta-2}(t); q_{\delta-1}(t), q_{\delta}(t), l_{q_{\delta+1}(t)}\big) \in A_1^{\delta}\circ \PP A_1 \label{sol_const_ag3} \end{align} sufficiently close to $([f], [\eta], q_1, \ldots, \ldots, q_{\delta-1}; q_{\delta}, l_{q_{\delta}})$.
\end{claim}
\noindent \textbf{Proof:} Let us continue with the setup of \cref{one_node_andone_PA1_claim}.
As before, since $f_{t_{02}} \neq 0$, we can make a change of coordinates $\hat{y} :=y+ B(x)$ and write $f_t$ as \begin{align*} f_t (x, y(x, \hat{y})) &= \varphi(x,\hat{y})\hat{y}^2 + \frac{\mathcal{B}_2^{f_t}}{2!} x^2 + \frac{\mathcal{B}_3^{f_t}}{3!} x^3 + \frac{\mathcal{B}_4^{f_t}}{4!} x^4 +\frac{\mathcal{B}_5^{f_t}}{5!} x^5+\frac{\mathcal{B}_6^{f_t}}{6!} x^6+ \mathcal{R}(x)x^7 \end{align*} where $\mathcal{B}^{f_t}_k$ are as defined in \cref{Bf_k_formula}, $\varphi(0,0) \neq 0$ and
$\mathcal{R}(x)$ is a holomorphic function defined in a neighborhood of the origin.
Let us make a further change of coordinates and denote \begin{align*} \hat{\hat{y}}&:= \sqrt{\varphi(x, \hat{y})} \hat{y}. \end{align*} as in the Proof of \cref{one_node_andone_PA1_claim}. Let us denote the polynomial $f_t$ by $\hat{f}_t$ which is a polynomial in two variables $x$ and $\hat{\hat{y}}$. Hence, \begin{align*} \hat{f}_t (x, \hat{\hat{y}}) &= \hat{\hat{y}}^2 + \frac{\mathcal{B}_2^{f_t}}{2!} x^2 + \frac{\mathcal{B}_3^{f_t}}{3!} x^3 + \frac{\mathcal{B}_4^{f_t}}{4!} x^4 +\frac{\mathcal{B}_5^{f_t}}{5!} x^5+\frac{\mathcal{B}_6^{f_t}}{6!} x^6+ \mathcal{R}(x)x^7. \end{align*} We claim that there does not exist any solutions to the set of equations \begin{align} \hat{f}_t(u_1, v_1) & = 0, ~~\hat{f}_x(u_1, v_1) =0, ~~\hat{f}_{\hat{\hat{y}}}(u_1, v_1) =0 \qquad \textnormal{and} \label{b1}\\ \hat{f}_t(u_2, v_2) & = 0, ~~\hat{f}_x(u_2, v_2) =0, ~~\hat{f}_{\hat{\hat{y}}}(u_2, v_2) =0, \label{b2} \end{align} where $(u_1, v_1)$ and $(u_2, v_2)$ and $(0,0)$ are all distinct, but close to each other.\\ \hf \hf We now note that the only solutions to the set of equation \cref{b1} and \cref{b2} is given by
\begin{align} v_1, v_2 & =0, \nonumber \\
\B^{f_t}_2 & = \frac{1}{360} \B^{f_t}_6 u_1^2 u_2^2 + O(|(u_1, u_2)|^5), \quad
\B^{f_t}_3 = -\frac{1}{60} \B^{f_t}_6(u_1^2 u_2 + u_1 u_2^2)+ O(|(u_1, u_2)|^4), \nonumber \\
\B^{f_t}_4 &= \frac{1}{30}\B^{f_t}_6(u_1^2+ 4 u_1 u_2 + u_2^2)+ O(|(u_1, u_2)|^3) \quad \textnormal{and} \quad
\B^{f_t}_5 = -\frac{1}{3}\B^{f_t}_6 (u_1+u_2)+ O(|(u_1, u_2)|^2). \label{A6_nhbd_PA1} \end{align} To see why this is so, we simply note that \cref{b1} and \cref{b2} are the same as \cref{three_node_uv1} and \cref{three_node_uv2}; hence, the argument is exactly the same as how we justified
\cref{A6_nhbd} is the solution to \cref{b1} and \cref{b2}. \\ \hf \hf We now note that $v_1,v_2$ are both zero; hence $u_1$ and $u_2$ are both nonzero, but small. Hence, $\B^{f_t}_5$ is close to zero. This is a contradiction, since $([f], [\eta], l_{q_{\delta}}) \in \PP A_4$.
\hf \hf Next, let us prove \eqref{hh4}. We will prove the following claim: \begin{claim} \label{two_nodes_and_one_PA1_claim_PD4} Let $([f], [\eta], q_1, \ldots, \ldots, q_{\delta-2}, l_{q_{\delta-1}}) \in A_1^{\delta-2}\circ \PP D_4$. Then there exists points \begin{align} \big([f_t], [\eta_t], q_1(t), \ldots, q_{\delta-3}(t); q_{\delta-2}(t), q_{\delta-1}(t), q_{\delta}(t), l_{q_{\delta+1}(t)}\big) \in A_1^{\delta}\circ \PP A_1 \label{sol_const_2_ag} \end{align} sufficiently close to $([f], [\eta], q_1, \ldots, \ldots, q_{\delta-2}; q_{\delta-1}, q_{\delta-1}, l_{q_{\delta-1}})$.
Furthermore, \textit{every} such solution satisfies the condition \begin{align}
\nabla^2f|_{q_{\delta+1}}(v,w) &\neq 0, \label{one_node_one_PA2_int_PA3_empty_2_ag} \end{align} if $v$ is a nonzero vector that belongs to $l_{q_{\delta+1}(t)}$ and $w$ is a nonzero vector that belongs to
$T\mathbb{P}^2_{\eta}|_{q_{\delta+1}(t)}/l_{q_{\delta+1}(t)}$. In other words, \begin{align*} \big([f_t], [\eta_t], q_1(t), \ldots, q_{\delta-3}(t); q_{\delta-2}(t), q_{\delta-1}(t), q_{\delta}(t), l_{q_{\delta+1}(t)}\big) & \not\in A_1^{\delta} \circ \PP A_2. \end{align*} \end{claim}
\noindent \textbf{Proof:} Following the setup of the proof of claim \ref{one_node_and_one_PA1_claim_D4},
let us write down the Taylor expansion of $f_t$ around the point $(0,0)$, namely \begin{align*} f_t(x,y)&= f_{t_{11}}xy + \frac{f_{t_{02}}}{2} y^2 + \frac{f_{t_{30}}}{6} x^3 + \frac{f_{t_{21}}}{2} x^2y + \frac{f_{t_{12}}}{2} x y^2 + \frac{f_{t_{03}}}{6} y^3+\ldots \end{align*} Since $([f_t], [\eta_t], l_{q_{\delta}(t)}) \in \PP A_1$, we conclude that $f_{t_{20}}$ is zero. Next, since $([f], [\eta], l_{q_{\delta}}) \in \PP D_4$, we conclude that $f_{11}$, $f_{02}$ and $f_{30}$ are zero; hence $f_{t_{11}}$, $f_{t_{02}}$ and $f_{t_{30}}$ are small (close to zero). Constructing points on the right hand side of \cref{sol_const_2_ag} amounts to finding solutions to the set of equations \begin{align} f_t(x_1, y_1)&=0, \qquad f_{t_x}(x_1, y_1)=0, \qquad f_{t_{y}}(x_1, y_1) =0 \qquad \textnormal{and} \label{eq1_ag} \\ f_t(x_2, y_2)&=0, \qquad f_{t_x}(x_2, y_2)=0, \qquad f_{t_{y}}(x_2, y_2) =0, \label{eq2_ag} \end{align} where $(0,0), (x_1, y_1)$ and $(x_2, y_2)$ are all distinct (but close to each other). As before, we define \[g_t(x,y):= xf_{t_{x}}(x,y) + yf_{t_{y}}(x,y) - 2f_t(x,y).\] We note that $g_t$ has no quadratic term and has the same cubic term as $f_t$. The cubic term of $f$ can be written as either $\frac{f_{{03}}}{6}(y-A_1(0) x)(y-A_2(0) x)y$ (if $f_{{03}} \neq 0$) or it can be written as $\frac{xy}{2}(f_{21}x + f_{12} y)$ (if $f_{{03}} = 0$). We will assume the former case; the latter case can be dealt with similarly. Hence, we can write $g_t$ as \begin{align*} g_t(x,y)& = \dfrac{f_{t_{03}}}{6}(y-A_1x)(y-A_2 x)(y-A_3x) + E(x,y), \end{align*} where $E$ is a fourth order term. Let us assume that $A_3$ is close to zero. We also note that since $f_{t_{21}} \neq 0$, hence $A_1$ and $A_2$ are both nonzero. Using the equation $g_t = 0$, let us consider the solution \begin{align*}
x&= u \qquad \textnormal{and} \qquad y = A_1 u + O(|u|^2). \end{align*} Let us now use $f_{t_x}(x,y)=0$ and solve for $f_{t_{11}}$ in terms of $u$. Doing that, we get \begin{align*}
f_{t_{11}} & = \dfrac{f_{t_{03}}}{6}(A_1^2-A_1A_2-A_1A_3+ A_2A_3)u + O(|u|^2). \end{align*} Plugging in this value of $f_{t_{11}}$ into the equation $f_{t_{y}}$ and solving for $f_{t_{02}}$, we get that \begin{align*}
f_{t_{02}} & = \dfrac{f_{t_{03}}}{6}\Big(-2 A_1 + 2 A_2 + 2 A_3 - \frac{2 A_2 A_3}{A_1}\Big) u + O(|u|^2). \end{align*}
Let us now try to produce a second node. We will justify shortly that $x=v$ and $y=A_2v + O(|v|^2)$ is a not a possible solution.
Hence, let us consider $x=v$ and $y=A_3v + O(|v|^2)$. Plugging this into $f_{t_y}(x,y)=0$ and solving for $u$ in terms of $v$, we conclude that \begin{align*}
u & = \Big(\frac{A_1(A_3-A_2)}{(A_1-A_2)(A_1-2A_3)}\Big)v + O(|v|^2). \end{align*} Plugging in this value for $u$ into $f_{t_x}(x,y)=0$ and solving for $A_3$, we conclude that \begin{align*}
A_3 & = O(|v|). \end{align*} Plugging in the value of $A_3$ into $u$ and then plugging that back into $f_{t_{11}}$ and $f_{t_{02}}$, we conclude that \begin{align*}
u& = \frac{A_2}{A_2-A_1} v + O(|v|^2), \qquad
f_{t_{11}} = -\dfrac{f_{t_{03}}}{6}A_1A_2 v + O(|v|^2) \qquad \textnormal{and} \qquad
f_{t_{02}} = \dfrac{f_{t_{03}}}{3}A_2 v + O(|v|^2). \end{align*} There are four ways to construct such solutions (interchange $(A_1, A_3)$, with $(A_2, A_3)$). Furthermore, we can permute the nodal points. From the expression for $f_{t_{11}}$ we see that the order of vanishing is $1$; hence the total multiplicity is $4$. \\
\hf \hf It remains to show why we reject the solution $x=v$ and $y = A_2 v+ O(|v|^2)$. If we take that solution, then we plug it in $f_{t_x}=0$, then solving for $u$ (in terms of $v$), we conclude that \begin{align*}
u & = \Big(\frac{A_3-A_2}{A_1-A_3}\Big)v + O(|v|^2) \end{align*} Plugging this into $f_{t_{y}}$, we conclude that \begin{align*}
f_{t_y} & = \dfrac{f_{t_{03}}}{3} \Big( \frac{(A_1-A_2)^2(A_3-A_2)}{A_1}\Big) v^2 + O(|v|^2). \end{align*}
This is clearly nonzero, if $v$ is small and nonzero. Hence, we reject the solution corresponding to $x=v$ and $y = A_2 v + O(|v|^2)$. This completes the proof. \\
\hf \hf Finally, let us justify \cref{hh5}. This follows from \cref{k4}. This completes the proof of \Cref{npa2}. \qed
\subsection{Proof of \Cref{npa3}: computation of $N(A_1^{\delta} \mathcal{P} A_3)$ when $0\leq \delta \leq 1$}
We will justify our formula for $N(A_1^{\delta} \mathcal{P} A_3,r, s, n_1, n_2, n_3, \theta)$, when $0 \leq \delta \leq 1$. Recall that \begin{align*} \A^{\delta}_1 \circ \overline{\PP \A}_2 := \{ ([f], [\eta], q_1, \ldots, q_{\delta}, l_{q_{\delta+1}}) \in \mathcal{S}_{\mathcal{D}_{\delta}}\times_{\mathcal{D}} \mathbb{P} W_{\mathcal{D}}: &\textnormal{$f$ has a singularity of type $\A_1$ at $q_1, \ldots, q_{\delta}$}, \\
& ([f], [\eta], l_{q_{\delta+1}}) \in \overline{\PP \A}_2, ~~\textnormal{$q_1, \ldots, q_{\delta+1}$ all distinct}\}. \end{align*} Let $\mu$ be a generic cycle, representing the class \begin{align*} [\mu] = \mathcal{H}_L^r \cdot \mathcal{H}_p^s \cdot a^{n_1} \lambda^{n_2} (\pi_{\delta+1}^*H)^{n_3} (\pi_{\delta+1}^*\lambda_{W})^{\theta}. \end{align*} We now define a section of the following bundle \begin{align} \Psi_{\PP \A_3}: \A^{\delta}_1 \circ \overline{\PP \A}_2 \longrightarrow \mathbb{L}_{\PP \A_3} & := \gamma_{\DD}^\ast\otimes\gamma_{W}^{\ast 3}\otimes \gamma_{\mathbb{P}^3}^{* d}, \qquad \textnormal{given by} \nonumber \\ \{\Psi_{\PP \A_3}([f], [\eta], q_1,\ldots,q_\delta,l_{q_{\delta+1}})\}(f\otimes v^{\otimes 3}) &:=
\nabla^3f|_{q_{\delta+1}}(v,v,v). \nonumber \end{align} Analogous to \cite[Lemma 6.1]{BM13_2pt_published},
we conclude that for $d \geq 4$, \begin{align} \overline{\PP \A}_2 &= \PP \A_2 \cup \overline{\PP \A}_3 \cup \overline{\widehat{D}}_4. \label{pa2_closure} \end{align} Furthermore, analogous to \cite[Lemma 6.3]{BM13_2pt_published}
we conclude that for $d \geq 4$, \begin{align} \overline{\overline{\A^{\delta}_1} \circ \PP \A}_2 &= (\overline{\A^{\delta}_1} \circ \PP \A_2) \cup \overline{\A^{\delta}_1} \circ (\overline{\PP \A}_2- \PP \A_2) \cup \overline{\A^{\delta-1}_1} \circ ( \Delta \overline{\PP \A}_4 \cup \Delta \overline{\widehat{D}}_5). \label{pa2_closure_again} \end{align}
Let us define \begin{align*} \mathcal{B} &:= \overline{A_1^{\delta} \circ \overline{\PP A}}_2 - \A^{\delta}_1 \circ (\PP\A_2\cup \overline{\PP A}_3). \end{align*}
We will show shortly that the section $\Psi_{\PP \A_3}$ vanishes on the points of $\A_1^{\delta}\circ \PP A_3$ transversally.
Hence, \begin{align} \langle e(\mathbb{L}_{\PP \A_3}), ~~[\overline{\A^{\delta}_1 \circ \overline{\PP\A}}_2] \cap [\mu] \rangle & = \N(\A_1^{\delta}\PP \A_3, n_1, n_2,n_3, \theta) + \mathcal{C}_{\mathcal{B}\cap \mu}. \label{npa3_Euler_class_formula} \end{align} We now give an explicit description of $\mathcal{B}$. Let us first define \begin{align*} \mathcal{B}_0 &:= \{ ([f], [\eta], q_1, \ldots q_{\delta}, l_{q_{\delta+1}}) \in \mathcal{B}: q_1, q_2 \ldots q_{\delta+1} ~~\textnormal{are all distinct}\}. \end{align*} In other words, $\mathcal{B}_0$ is that component of the boundary, where all the points are still distinct. By \cref{pa2_closure}, we conclude that \begin{align*} \mathcal{B}_0 &= \overline{A_1^{\delta}} \circ \overline{\widehat{D}}_4. \end{align*} If we intersect $\mathcal{B}_0$ with $\mu$ then we will get a finite set of points. Since the representative $\mu$ is generic, we conclude that the third derivative along $v$ will not vanish, i.e. the section $\Psi_{\PP A_3}$ will not vanish on those points. Hence, $\mathcal{B}_0\cap \mu$ does not contribute to the Euler class. \\ \hf\hf Next, let us consider the components of $\mathcal{B}$ where one (or more) of the $q_i$ become equal to the last point $q_{\delta+1}$. Define $\mathcal{B}(q_{i_1}, \ldots q_{i_k}, l_{q_{\delta}})$ as before. Analogous to the proof of \cite[Lemma 6.3]{BM13_2pt_published},
we conclude that \begin{align*} \B (q_1, l_{q_{\delta +1 }}) & \approx \overline{\A_1^{\delta-1} \circ \PP \A}_4 \cup \overline{\A_1^{\delta-1} \circ \widehat{D}}_5. \end{align*} Furthermore, analogous to the proof of \cite[Corollary 6.13, Page 700]{BM13_2pt_published}, we conclude that
the contribution to the Euler class from each of the points of $\overline{\A_1^{\delta-1} \circ \PP \A}_4 \cap \mu$ is $2$. Finally, we note that the section $\Psi_{\PP A_3}$ does not vanish on $\overline{\A_1^{\delta-1} \circ \widehat{D}}_5 \cap \mu$, since $\mu$ is generic. Hence, the total contribution from all the components of type $\B (q_{i_1}, l_{q_{\delta +1 }})$ equals \begin{displaymath} 2\binom{\delta}{1}N(\A_1^{\delta-1}\PP\A_4,n_1,n_2,n_3, \theta). \end{displaymath} Plugging in this in \cref{npa3_Euler_class_formula} gives us the formula of \cref{npa3}. \\ \hf \hf It just remains to prove the transversality claim. This follows from following the setup of proof of transversality in Theorem \cref{npa2}. We consider the polynomial \begin{align*} \rho_{30}&:= (X-X_1)^2(X-X_2)^2\ldots \cdot (X-X_{\delta})^2 X^3 Z^{d-2\delta-3} \end{align*} and the corresponding curve $\gamma_{30}(t)$. Transversality follows by computing the derivative of the section $\Psi_{\PP A_3}$ along the curve $\gamma_{30}(t)$ as before. \qed
\subsection{Proof of \Cref{npa4}: computation of $N(\mathcal{P} A_4)$} We will now justify our formula for \newline $N(\mathcal{P} A_4,r, s, n_1, n_2, n_3, \theta)$.
Let $\mu$ be a generic cycle, representing the class \begin{align*} [\mu] = \mathcal{H}_L^r \cdot \mathcal{H}_p^s \cdot a^{n_1} \lambda^{n_2} (\pi^*H)^{n_3} (\pi^*\lambda_{W})^{\theta}. \end{align*} Let $v \in \gamma_{W}$ and $w \in \pi^*W/\gamma_{W}$ be two fixed nonzero vectors. Let us introduce the following abbreviation: \begin{align*}
f_{ij} & := \nabla^{i+j} f|_{q} (\underbrace{v,\cdots v}_{\textnormal{$i$ times}}, \underbrace{w,\cdots w}_{\textnormal{$j$ times}}). \end{align*} We now define a section of the following bundle \begin{align} \Psi_{\PP \A_4}: \overline{\PP \A}_3 \longrightarrow \mathbb{L}_{\PP \A_4} & := \gamma_{\DD}^{\ast 2}\otimes\gamma_W^{\ast 4}\otimes(W/\gamma_{W})^{\ast 2}\otimes \gamma_{\mathbb{P}^3}^{\ast 2d}, \label{psiPA_4} \\ \{\Psi_{\PP \A_4}([f], l_{q})\}(f^{\otimes 2} \otimes v^{\otimes 4}\otimes w^{\otimes 2}) &:= f_{02}A_4^f, \qquad \textnormal{where} \qquad \A^{f}_4 := f_{40}-\frac{3 f_{21}^2}{f_{02}}. \label{A^f_4}
\end{align}
Analogous to \cite[Lemma 6.1]{BM13_2pt_published}, we conclude that \begin{align} \overline{\PP A}_3 & = \PP A_3 \cup \overline{\PP A}_4 \cup \overline{\PP D}_4. \label{pa3_closure} \end{align}
Hence, let us define \begin{align*} \mathcal{B} &:= \overline{\PP A}_3 - \PP A_3 \cup \overline{\PP A}_4. \end{align*} We will show shortly that
the section $\Psi_{\PP \A_4}$ vanishes on the points of $\PP A_4$ transversally.
Hence, \begin{align} \langle e(\mathbb{L}_{\PP \A_4}), ~~[\overline{\PP\A}_3] \cap [\mu] \rangle & = \N(\A_1^{\delta}\PP \A_4, r, s, n_1, n_2, n_3, \theta) + \mathcal{C}_{\mathcal{B}\cap \mu}. \label{npa4_Euler_class_formula} \end{align} Let us now study the boundary $\mathcal{B}$.
By \cref{pa3_closure}, we conclude that \begin{align*} \mathcal{B} \cap \mu &= \overline{\PP D}_4 \cap \mu. \end{align*}
Since the representative $\mu$ is generic, we conclude that the directional derivative $f_{21}$ will not vanish on those points. Since $f_{02} =0$ on $\mathcal{B}$, we conclude that \begin{align*} f_{02}A^f_4 &= f_{02}f_{40} - 3f_{21}^2 \neq 0 \end{align*} if $f_{21} \neq 0$.
Hence, the section $\Psi_{\PP A_4}$ will not vanish on $\mathcal{B}\cap \mu$. Hence, the total boundary contribution is zero and \cref{npa4_Euler_class_formula} gives us the formula of \cref{npa4}. \\ \hf \hf It remains to prove the claim regarding transversality. This follows from following the setup of proof of transversality in
\Cref{npa3}. We consider the polynomial \begin{align*} \rho_{40}&:= X^4 Z^{d-4} \end{align*} and the corresponding curve $\gamma_{40}(t)$. Transversality follows by computing the derivative of the section $\Psi_{\PP A_4}$ along the curve $\gamma_{40}(t)$ as before. \qed
\subsection{Proof of \Cref{npd4}: computation of $N(\mathcal{P} D_4)$} We will now justify our formula for $N(\mathcal{P} D_4,r, s, n_1, n_2, n_3, \theta)$. Let $\mu$ be a generic cycle, representing the class \begin{align*} [\mu] & = \mathcal{H}_L^r \cdot \mathcal{H}_p^s \cdot a^{n_1} \lambda^{n_2} (\pi^*H)^{n_3} (\pi^*\lambda_{W})^{\theta}. \end{align*} As before, let $v \in \gamma_{W}$ and $w \in \pi^*W/\gamma_{W}$ be two fixed nonzero vectors. Define a section of the following bundle \begin{align} \Psi_{\PP \D_4}: \overline{\PP \A}_3 \longrightarrow \mathbb{L}_{\PP \D_4} & := \gamma_{\DD}^{\ast}\otimes(W/\gamma_W)^{\ast 2}\otimes \gamma_{\mathbb{P}^3}^{*d}, \qquad \textnormal{given by} \nonumber \\
\{\Psi_{\PP \D_4}([f], l_{q})\}(f\otimes w^{\otimes 2}) &:= \nabla^2f|_q(w,w). \label{psiD_4} \end{align}
We recall \cref{pa3_closure}, namely \begin{align} \overline{\PP A}_3 & = \PP A_3 \cup \overline{\PP A}_4 \cup \overline{\PP D}_4. \label{pa3_closure_again} \end{align} We now define \begin{align*} \mathcal{B} &:= \overline{\PP A}_3 - (\PP\A_3\cup \overline{\PP D}_4). \end{align*} We will show that the section $\Psi_{\PP \D_4}$ vanishes on the points of $\PP D_4$ transversally.
Hence, \begin{align} \langle e(\mathbb{L}_{\PP \D_4}), ~~[\overline{\PP\A}_3] \cap [\mu] \rangle & = \N(\PP \D_4, r,s, n_1, n_2, n_3, \theta) + \mathcal{C}_{\mathcal{B}\cap \mu}. \label{npd4_Euler_class_formula} \end{align} By definitions, the section $\Psi_{\PP D_4}$ does not vanish on $\PP A_4 \cap \mu$. Hence, the total boundary contribution is zero and \cref{npa4_Euler_class_formula} gives us the formula of \cref{npd4}. \\ \hf \hf It remains to prove the claim regarding transversality. This follows from following the setup of proof of transversality in
\Cref{npa4}. We consider the polynomial \begin{align*} \rho_{02}&:= Y^2 Z^{d-2} \end{align*} and the corresponding curve $\gamma_{02}(t)$. Transversality follows by computing the derivative of the section $\Psi_{\PP D_4}$ along the curve $\gamma_{02}(t)$ as before. \qed
\section{Verification with other results and low degree checks} \label{low_degree_checks} Let us make a few low degree checks. We will abbreviate $N(A_1^{\delta+1}, r,s,0,0)$ as $N(A_1^{\delta+1}, r,s)$. \subsection{Verification with S.~Kleiman and R.~Piene's result} \label{KP_check} Let us start by verifying the numbers predicted by the algorithm of S.~Kleiman and R.~Piene in \cite{KP2}. Let us explain how to obtain the formula for $N(A_1^{\delta+1}, r,s)$ using \cite[Algorithm 2.3, Page 5]{KP2}. Let us first define four polynomials (called Bell polynomials), given by \begin{align*} P_1(a_1):= a_1, \qquad P_2(a_1, a_2)&:= a_1^2 + a_2, \qquad P_3(a_1, a_2, a_3):= a_1^3+3a_1 a_2 + a_3 \qquad \textnormal{and} \\ P_4(a_1, a_2, a_3, a_4)&:= a_1^4 + 6 a_1^2 a_2 + 3 a_2^2 + 4 a_1 a_3 + a_4. \end{align*} We define the following cycles in $\mathcal{S}_{\mathcal{D}_{1}}$, namely \begin{align} v&:= \lambda + dH, \qquad w_1:= a-3H \qquad \textnormal{and} \qquad w_2:= a^2-2aH+3aH^2. \label{vw} \end{align} Note that $v= c_1(\mathcal{L}_{A_0})$ and $w_i= c_i(T^*W)$, where $\mathcal{L}_{A_0}$ and $W$ are the bundles defined in section \ref{na1_delta_proof}. The algorithm \cite[Algorithm 2.3, Page 5]{KP2} produces polynomials $b_i(v,w_1, w_2)$ of degree $i+2$ (from $i=1$ to $8$). Let us write down the expressions explicitly, \begin{align} b_1(v, w_1, w_2)&= v^3 + v^2w_1 + vw_2, \qquad b_2(v, w_1, w_2) = -7 v^4-13 v^3 w_1-6 v^2 w_1^2-7 v^2 w_2-6 v w_1 w_2, \nonumber \\ b_3(v, w_1, w_2)& = 138 v^5+394 v^4 w_1+376 v^3 w_1^2+138 v^3 w_2 +120 v^2 w_1^3+256 v^2 w_1 w_2+120 v w_1^2 w_2 \qquad \textnormal{and} \nonumber \\ b_4(v, w_1, w_2)& = -4824 v^6-19134 v^5 w_1-28842 v^4 w_1^2-3888 v^4 w_2-19572 v^3 w_1^3 \nonumber \\
& ~~ -12438 v^3 w_1 w_2
-5040 v^2 w_1^4-13596 v^2 w_1^2 w_2 \nonumber \\
& ~~ +936 v^2 w_2^2-5040 v w_1^3
w_2+936 v w_1 w_2^2. \label{b1_defn} \end{align}
The numbers $N(A_1^{\delta+1}, r,s)$ will be computed from the polynomials $P_{\delta+1}$ by intersecting cycles in $\mathcal{S}_{\mathcal{D}_{\delta+1}}$. Let $\pi_i:\mathcal{S}_{\mathcal{D}_{\delta+1}} \longrightarrow \mathcal{S}_{\mathcal{D}_{1}}$ denote
the $i^{\textnormal{th}}$ projection map. Then \begin{align*} N(A_1, r,s)&= [b_1]\cdot \mathcal{H}_L^r \cdot \mathcal{H}_p^s, \end{align*} where the right hand side is an intersection number on $\mathcal{S}_{\mathcal{D}_{1}}$. Note that we plug in the values for $v, w_1$ and $w_2$ from \cref{vw} in \cref{b1_defn}, use \cref{HL_Hp_class} for $\mathcal{H}_L$ and $\mathcal{H}_p$ and the ring structure as given by \cref{ring_str} to compute the intersection number. Next, let us explain how to compute $N(A_1^2, r,s)$. This is given by \begin{align} N(A_1^2, r,s)&= (\pi_1^*b_1)\cdot (\pi_2^*b_1) \cdot \mathcal{H}_L^r \cdot \mathcal{H}_p^s + b_2 \cdot \mathcal{H}_L^r \cdot \mathcal{H}_p^s. \label{two_nodes_kp} \end{align} The first number on the right hand side of \cref{two_nodes_kp} is an intersection number on $\mathcal{S}_{\mathcal{D}_{2}}$, while the second one is an intersection number on $\mathcal{S}_{\mathcal{D}_{1}}$. Similarly, \begin{align*} N(A_1^3, r,s)&= \Big((\pi_1^*b_1)\cdot (\pi_2^*b_1) \cdot (\pi_3^*b_1) + 3 (\pi_1^*b_1)\cdot (\pi_2^*b_1) + b_3\Big) \cdot \mathcal{H}_L^r \cdot \mathcal{H}_p^s \qquad \textnormal{and} \\ N(A_1^4, r,s)&= \Big((\pi_1^*b_1)\cdot (\pi_2^*b_1)\cdot (\pi_3^*b_1) \cdot (\pi_4^*b_1) + 6 (\pi_1^*b_1) \cdot (\pi_2^*b_1) \cdot (\pi_3^*b_1) \\ & \qquad + 3 (\pi_1^*b_2)\cdot (\pi_2^*b_2) + 4 (\pi_1^*b_1)\cdot (\pi_2^*b_3) + b_4 \Big)\cdot \mathcal{H}_L^r \cdot \mathcal{H}_p^s. \end{align*} We have written a mathematica program to implement this formula and verified that the answers agree with our formula.
\subsection{Verification with T.~Laraakker's result} Next we note that in \cite[Appendix A, Page 32]{TL}, T.~Laraakker has explicitly written down the formulas for $N(A_1^{\delta+1},0,0)$. We have verified that our formulas agree with his.
\subsection{Verification with the second author and R.~Singh's result} We now verify some of the numbers obtained by R.~Mukherjee and R.~Singh in \cite{RS}. In \cite{RS}, the authors compute $C_d^{\textnormal{Planar}, \mathbb{P}^3}(r,s)$, the number of planar genus zero degree $d$ curves in $\mathbb{P}^3$ intersecting $r$ lines and passing through $s$ points having a cusp (where $r+2s = 3d+1$). Let us compare this with
$N_d(A_1^{\delta} A_2, r, s)$, the number of planar degree $d$ curves in $\mathbb{P}^3$, passing through $r$ lines and passing through $s$ points, that have $\delta$ (ordered) nodes and one cusp
(where $r + 2s =\dfrac{d(d+3)}{2} +1-\delta$). For $d =3$, and $\delta =0$, this number should be the same as the characteristic number of genus zero planar cubics in $\mathbb{P}^3$ with a cusp, i.e. $C_d(r, s)$. We have verified that is indeed the case. We tabulate the numbers for the readers convenience: \begin{align*} C_3(10, 0)& = 17760, \quad C_3(8, 1) = 2064, \quad C_3(6, 2) = 240 \quad \textnormal{and} \qquad C_3(4, 3) = 24.
\end{align*}
These numbers are the same as $N_d(A_1^{\delta} A_2, r, s)$ for $d=3$ and $\delta =0$. \\ \hf \hf Next, we note that when $d=4$ and $\delta =2$, the number $\frac{1}{\delta !}N_d(A_1^{\delta} A_2, r, s)$ is same as the characteristic number of genus zero planar quartics in $\mathbb{P}^3$ with a cusp, i.e. $C_d(r, s)$. We have verified that fact. The numbers are
\begin{align*} C_4(13, 0)& = 10613184, \quad C_4(11, 1) = 760368, \quad C_4(9, 2) = 49152 \quad \textnormal{and} \quad C_4(7, 3) = 2304.
\end{align*}
These numbers are the same as $\frac{1}{2!}N_d(A_1^{\delta} A_2, r, s)$ for $d=4$ and $\delta =2$. We have to divide out by a factor of $\delta!$ because in the definition of $N_d(A_1^{\delta} A_2, r, s)$, the nodes are ordered.
\subsection{Enumerativity of BPS numbers computed by R.~Pandharipande} We will now verify some of the numbers predicted by the conjecture made by Pandharipande in \cite{RPDeg}, regarding the enumerativity of the BPS numbers for $\mathbb{P}^3$. Let $N^d_g(r,s)$ denote the genus $g$ Gromov-Witten invariant of $\mathbb{P}^3$ (corresponding to the insertion of $r$ lines and $s$ points) and let $E^d_g(r,s)$ denote the corresponding BPS invariant as given by \cite[Equations 5 and 9, Pages 493 and 494]{RPDeg}. The numbers $E^d_g(r,s)$ are conjectured to be integers. Even if the conjecture is true,
it is not always clear if the the BPS numbers have an enumerative significance. We will now give some evidence for the enumerativity of some of the BPS number. \\ \hf \hf Let us consider the case $g=2$ and $d=4$. It is far from clear that $E^d_2(r,s)$ is enumerative when $d=4$, because the moduli space of curves has more than the expected dimension (see the remark in \cite{RPDeg} just after Theorem 3, Page 494). We claim that $E^d_2(r,s)$ is enumerative when $d=4$. To see how, we first note that every degree $4$, genus $2$ curve lies inside some $\mathbb{P}^2$ (this follows from the Castelnuovo bound, \cite[Page 527]{GH}). Since the genus of a smooth degree $4$ curve is $3$, we conclude that the corresponding enumerative invariant is equal to the characteristic number of planar degree $4$ curves in $\mathbb{P}^3$ with one node. We have verified that $E^d_2(r,s)$ is indeed equal to $N_d(A_1, r,s)$ for all $r$ and $s$ when $d=4$. We tabulate the numbers for the readers convenience \begin{align} N_4(A_1, 16, 0)&= 258300, \quad N_4(A_1, 14, 1)= 15498, \quad N_4(A_1, 12, 2)= 792 \quad \textnormal{and} \quad N_4(A_1, 10, 3) = 27. \label{BPS_g_2} \end{align}
The degree four, genus two BPS numbers are directly tabulated in \cite[Page 43]{AGath} and are seen to be equal to the above numbers listed in \cref{BPS_g_2}.
\section{Explicit Formulas} \label{expfor} \noindent For the convenience of the reader, we write down some explicit formulas.
\begin{align*} N(r,s, 0, 0) &= \begin{cases} \frac{1}{324} d(d^2-1)(d+2)\left(d^2 +4d +6 \right) \left(2d^3 +6d^2+13d+3 \right) & \mbox{if} ~~ s =0, \\ \frac{1}{36}d(d^2-1)(d+2) \left( 2d^2 +8d +3 \right) & \mbox{if} ~~s=1,\\ \frac{1}{3}d(d-1)(d+4) & \mbox{if} ~~s= 2, \\ 1 & \mbox{if} ~~s= 3.
\end{cases} \\ N(A_1, r,s, 0, 0) &= \begin{cases}
\frac{1}{108}d(d^2 -1)^2 (d+2)(d+3) \left(2d^4 +4d^3 +d^2 -10d -6 \right) & \mbox{if} ~~ s =0, \\
\frac{1}{12} d(d-1)^2(d+3) \left( 2d^4 +6d^3 -9d^2 -3d -2 \right) & \mbox{if} ~~s=1,\\ d(d-1)^2 \left( d^2 +3d-6 \right) & \mbox{if} ~~s= 2, \\
3(d-1)^2 & \mbox{if} ~~s= 3.
\end{cases}\\ N(A_2, r,s, 0, 0) &= \begin{cases}
\frac{1}{27} d(d^2 -1)(d^2 -4) \left( 2 d^6+12 d^5+11 d^4 -30 d^3-49 d^2-18 \right) & \mbox{if} ~~ s =0, \\
\frac{1}{3} d(d-1)(d-2) \left(2 d^5+12 d^4+d^3-54 d^2+9 d+6 \right) & \mbox{if} ~~s=1,\\ 4d(d-1)(d-2) \left(d^2+3 d-8\right) & \mbox{if} ~~s= 2, \\
12(d-1)(d-2) & \mbox{if} ~~s= 3.
\end{cases} \\ N(A_3, r,s, 0, 0) &= \begin{cases}
\frac{1}{162} d(d-1)(d-2)\big( 50 d^8+408 d^7+539 d^6-2556 d^5-6625 d^4 \\
\qquad \qquad \qquad \qquad +762 d^3+10050 d^2-11232 d+8208\big) & \mbox{if} ~~ s =0, \\
\frac{1}{18} (d-2) (d-1) \big(50 d^6+258 d^5-485 d^4-2241 d^3 \\
\qquad \qquad \qquad \qquad \qquad \qquad \qquad +2172 d^2+1512 d-648\big) & \mbox{if} ~~s=1,\\ \frac{2}{3} d(d-2)(d+5) \left(25 d^2-96 d+84\right) & \mbox{if} ~~s= 2, \\ 2 \left(25 d^2-96 d+84\right) & \mbox{if} ~~s= 3.
\end{cases}\\ N(A_4, r,s, 0, 0) &= \begin{cases}
\frac{5}{27} (d-1)(d-3) \big(6 d^9+50 d^8+41 d^7-445 d^6-715 d^5 \\
\qquad \qquad \qquad +1529 d^4+2720 d^3-7902 d^2+7164 d-2160\big) & \mbox{if} ~~ s =0, \\
\frac{5}{3} (d-3) \big(6 d^7+26 d^6-105 d^5-231 d^4 \\
\qquad \qquad \qquad \qquad +765 d^3-107 d^2-762 d+360\big) & \mbox{if} ~~s=1,\\
20 d(d-3) (3 d-5) \left(d^2+3 d-12\right) & \mbox{if} ~~s= 2, \\
60(d-3)(3d -5) & \mbox{if} ~~s= 3.
\end{cases}\\ N(D_4, r,s, 0, 0) &= \begin{cases} \frac{5}{36} (d-1)(d-2)^2 (d+4) \big(2 d^7+12 d^6-d^5-66 d^4 -91 d^3 \\ \qquad \qquad \qquad \qquad \qquad +234 d^2-270 d+108\big) & \mbox{if} ~~ s =0, \\ \frac{5}{4} (d-2)^2 \left(2 d^6+12 d^5-15 d^4-102 d^3+85 d^2+90 d-48\right) & \mbox{if} ~~s=1,\\ 15 d(d-2)^2 \left(d^2+3 d-12\right) & \mbox{if} ~~s= 2, \\
45 (d-2)^2 & \mbox{if} ~~s= 3.
\end{cases}\\ N(A_1^2, r,s, 0, 0) &= \begin{cases} \frac{1}{108} d(d^2-1)(d^2-4) \big(6 d^8+30 d^7-25 d^6-255 d^5-142 d^4 \\ \qquad \qquad \qquad+333 d^3+629 d^2+18 d+198\big) & \mbox{if} ~~ s =0, \\ \frac{1}{12} d(d-1)(d-2) \big(6 d^7+30 d^6-55 d^5-297 d^4+190 d^3 \\ \qquad \qquad \qquad \qquad +537 d^2-69 d-78\big) & \mbox{if} ~~s=1,\\
d(d-1)(d-2) \left(d^2+3 d-8\right) \left(3 d^2-3 d-11\right) & \mbox{if} ~~s= 2, \\ 3 (d-1)(d-2) \left(3 d^2-3 d-11\right) & \mbox{if} ~~s= 3.
\end{cases}\\ \end{align*}
\begin{align*} N(A_1 A_2, r,s, 0, 0) &= \begin{cases}
\frac{1}{27} d(d-1)(d-2)(d-3) \big(6 d^9+60 d^8+155 d^7-186 d^6 \\
\qquad -1288 d^5 -1422 d^4+641 d^3+1512 d^2-2034 d+1836\big) & \mbox{if} ~~ s =0, \\
\frac{1}{3} (d^2 -1)(d-2)(d-3) \big(6 d^6+36 d^5-37 d^4-338 d^3 \\
\qquad \qquad \qquad +123 d^2+438 d-144\big) & \mbox{if} ~~s=1,\\
4 d(d-2)(d-3)(d+5) \left(3 d^3-6 d^2-11 d+18\right) & \mbox{if} ~~s= 2, \\
12 (d-3) \left(3 d^3-6 d^2-11 d+18\right) & \mbox{if} ~~s= 3.
\end{cases}\\ N(A_1 A_3, r,s, 0, 0) &= \begin{cases}
\frac{1}{54} (d-1)(d-3) \Big(50 d^{11}+358 d^{10}-489 d^9-6967 d^8 \\ -3139 d^7 +40955 d^6+40482 d^5-112250 d^4-131080 d^3 \\
\qquad \qquad +436176 d^2-402480 d+120960\Big) & \mbox{if} ~~ s =0, \\ \frac{1}{6} (d-3) \big(50 d^9+158 d^8-1471 d^7-2389 d^6+14857 d^5 \\ \qquad \qquad +2359 d^4 -41156 d^3+7912 d^2+41808 d-19440\big) & \mbox{if} ~~s=1,\\
2 d(d-3) \left(d^2+3 d-12\right) \left(25 d^3-71 d^2-122 d+280\right) & \mbox{if} ~~s= 2, \\
6 (d-3) \left(25 d^3-71 d^2-122 d+280\right) & \mbox{if} ~~s= 3.
\end{cases}\\ N(A_1^3, r,s, 0, 0) &= \begin{cases} \frac{1}{108} d(d-1)(d-2) \Big(18 d^{12}+108 d^{11}-315 d^{10}-2664 d^9 \\ +470 d^8+21919 d^7+19103 d^6-58136 d^5-106948 d^4 \\ \qquad \qquad +7039 d^3+129360 d^2-165798 d+110700\Big) & \mbox{if} ~~ s =0, \\ \frac{1}{12} (d-1)(d-2) \Big(18 d^{10}+54 d^9-567 d^8-1179 d^7+6383 d^6 \\
+7774 d^5-25775 d^4-20197 d^3+26955 d^2+20802 d-8640\Big) & \mbox{if} ~~s=1,\\
d(d-2)(d+5) \Big(9 d^6-54 d^5+9 d^4+423 d^3 \\
\qquad \qquad \qquad \qquad \qquad \qquad -458 d^2-829 d+1050\Big) & \mbox{if} ~~s= 2, \\
3 \left(9 d^6-54 d^5+9 d^4+423 d^3-458 d^2-829 d+1050\right) & \mbox{if} ~~s= 3.
\end{cases}\\ N(A_1^2 A_2, r,s, 0, 0) &= \begin{cases}
\frac{1}{9} (d-1)(d-3) \Big(6 d^{13}+36 d^{12}-159 d^{11}-1124 d^{10}+1209 d^9 \\
\qquad +12169 d^8+664 d^7-52991 d^6-39896 d^5+127254 d^4 \\
\qquad \qquad +129112 d^3-452904 d^2+413280 d-120960\Big) & \mbox{if} ~~ s =0, \\
(d-3) \Big(6 d^{11}+12 d^{10}-249 d^9-236 d^8+3653 d^7+367 d^6 \\
-20186 d^5+6389 d^4+38600 d^3-7828 d^2-42896 d+19680\Big) & \mbox{if} ~~s=1,\\
12 d(d-3) \left(d^2+3 d-12\right) \big(3 d^5-12 d^4-30 d^3 \\
\qquad \qquad \qquad \qquad \qquad \qquad \qquad +125 d^2+82 d-280\big) & \mbox{if} ~~s= 2, \\
36 (d-3) \left(3 d^5-12 d^4-30 d^3+125 d^2+82 d-280\right) & \mbox{if} ~~s= 3.
\end{cases} \end{align*} \begin{align*} N(A_1^4, r,s, 0, 0) &= \begin{cases}
\frac{1}{36} (d-1)(d-3) \Big(18 d^{15}+90 d^{14}-747 d^{13}-3843 d^{12}+11660 d^{11} \\
+63140 d^{10}-75352 d^9-486678 d^8+73143 d^7+1773729 d^6+1150606 d^5 \\
\qquad -4123550 d^4-3282032 d^3+12893256 d^2-11795040 d+3404160\Big) & \mbox{if} ~~ s =0, \\
\frac{1}{4} (d-3) \Big(18 d^{13}+18 d^{12}-945 d^{11}-261 d^{10}+18590 d^9-4254 d^8 \\
\qquad -164328 d^7+80206 d^6+653953 d^5-362481 d^4-1051128 d^3 \\
\qquad \qquad \qquad +245636 d^2+1215312 d-554880\Big) & \mbox{if} ~~s=1,\\
3d(d-3) \left(d^2+3 d-12\right) \Big(9 d^7-45 d^6-135 d^5+801 d^4 \\
\qquad \qquad \qquad +691 d^3-4671 d^2-1386 d+7880\Big) & \mbox{if} ~~s= 2, \\
9 (d-3) \big(9 d^7-45 d^6-135 d^5+801 d^4+691 d^3 \\
\qquad \qquad \qquad -4671 d^2-1386 d+7880\big) & \mbox{if} ~~s= 3.
\end{cases} \end{align*}
\section{Acknowledgment} The ideas of this paper originated while the second author had discussions with Martijn Kool and Ties Laarakker regarding the papers \cite{BM8} and \cite{KST}. During the discussion we wondered if one can count planar curves in $\mathbb{P}^3$ with singularities. As shown by Ties Laarakker in \cite{TL}, one can adapt the techniques in \cite{KST} to count $\delta$-nodal planar curves in $\mathbb{P}^3$. On the other hand, the discussions also led us to conclude that by adapting the methods of \cite{R.M}, \cite{BM13_2pt_published} and \cite{BM8}, we can enumerate planar curves in $\mathbb{P}^3$ with singularities that are more degenerate than nodes. The result is this present paper. The second author is therefore very grateful to Martijn Kool and Ties Laarakker for the discussions and fruitful exchange of ideas that resulted in this paper. The second author would also like to thank Steven Kleiman for his comments and pointing out that the ideas of \cite{KP1} and \cite{KP2} combined can be used to compute the numbers obtained in this paper (see \Cref{KP_remark}). The second author would also like to thank Center for Quantum Geometry of Moduli Space at Aarhus, Denmark (DNRF95) for giving him a chance to spend six weeks there, when the author got the initial idea for this project; the visit was was mainly paid by the grant ``EU-IRSES Fellowship within FP7/2007-2013 under grant agreement number 612534, project MODULI - Indo European Collaboration on Moduli Spaces." Finally, the second author would like to acknowledge the External Grant he has obtained, namely MATRICS (File number: MTR/2017/000439) that has been sanctioned by the Science and Research Board (SERB).
Both the authors are grateful to Anantadulal Paul and Rahul Singh for several fruitful discussions.
\end{document} | arXiv |
Semi-nonparametric modeling of topological domain formation from epigenetic data
Emre Sefer ORCID: orcid.org/0000-0002-9186-02701 &
Carl Kingsford2
Algorithms for Molecular Biology volume 14, Article number: 4 (2019) Cite this article
Hi-C experiments capturing the 3D genome architecture have led to the discovery of topologically-associated domains (TADs) that form an important part of the 3D genome organization and appear to play a role in gene regulation and other functions. Several histone modifications have been independently associated with TAD formation, but their combinatorial effects on domain formation remain poorly understood at a global scale.
We propose a convex semi-nonparametric approach called nTDP based on Bernstein polynomials to explore the joint effects of histone markers on TAD formation as well as predict TADs solely from the histone data. We find a small subset of modifications to be predictive of TADs across species. By inferring TADs using our trained model, we are able to predict TADs across different species and cell types, without the use of Hi-C data, suggesting their effect is conserved. This work provides the first comprehensive joint model of the effect of histone markers on domain formation.
Our approach, nTDP, can form the basis of a unified, explanatory model of the relationship between epigenetic marks and topological domain structures. It can be used to predict domain boundaries for cell types, species, and conditions for which no Hi-C data is available. The model may also be of use for improving Hi-C-based domain finders.
The emerging evidence suggests that 3D nuclear architecture is important for the regulation of gene expression and it is tightly linked to the function of the genome. For instance, expression in the beta-globin locus is mediated by folding to bring an enhancer and associated transcription factors within close proximity of a gene [1, 2]. Similarly, loci of mutations that affect expression of genomically far-away genes (eQTLs) interact significantly more frequently within a contact range in 3D to their regulated genes [3], indicating that 3D genome structure plays a wide-spread role in gene regulation. Lastly, spatial regions that interact with nuclear lamina are generally inactive [4]. Measuring and modeling the 3D shape of a genome is thus essential to obtain a more complete understanding of how cells function.
Chromatin interactions obtained from a variety of recent chromosome conformation capture experimental techniques such as Hi-C [5] have resulted in significant advances in our understanding of the geometry of chromatin structure [6, 7]. These experiments yield matrices of counts that represent the frequency of cross-linking between restriction fragments of DNA at a certain resolution. Analysis of the resulting matrix by Dixon et al. [8] led to the discovery of topologically-associated domains (TADs) which correspond to consecutive, highly-interacting matrix regions typically a few megabases in size that are close in densely packed chromatin. TADs have been identified across different cell cycle phases and in prokaryotes [9]. Several lines of evidence suggest that TADs are a building block of genomic regulatory architecture [10, 11]. Segmental packaging of genome via TADs likely have critical roles in cell dynamics such as long-range transcriptional regulation and cell differentiation [12, 13].
The mechanism by which these TADs form and are demarcated is still largely unknown. A plethora of epigenetic modifications have been identified in metazoan genomes that are associated with 3D genome shape [14], and thus TADs. Several modifications have been found to be specifically correlated with TAD boundaries [8]. For instance, histone modifications H3K4me3m H3K27ac and insulator proteins are enriched within TAD boundaries [15], although the causal direction of these associations is still unknown [12]. Despite these analyses, the complete picture of how histone modifications are related to TAD formation is missing. This is partially because previous analyses relating histone marks to domain boundaries have often considered each histone mark independently, without accounting for their combined affects. It is unknown to what extent relationships between the histone markers are important or whether there is a small set of markers that are of primary importance.
Here, we develop and train a joint model, which we call nTDP (Nonparameteric Topological Domain Partitioner), of how histone modifications are associated with domain boundaries and interiors. We show that we are able to train this model optimally in polynomial time because its likelihood function is convex. The model does not make any assumptions about the effect of each histone mark on domain formation, and instead fits the histone-domain relationship nonparametrically. Using this model, we systematically identify a small set of histone markers that in combination appear to explain TAD boundaries. We find a small number of epigenetic elements account for a large proportion of the accuracy of TAD prediction. All of these identified marks fail to predict domain boundaries when considered independently. We show that these markers are conserved across species and cell types in a very strong way: models trained on mouse continue to work well on human, and models trained on IMR90 cells continue to work on embryonic stem cells.
Our approach, nTDP, can form the basis of a unified, explanatory model of the relationship between epigenetic marks and topological domain structures. It can be used to predict domain boundaries for cell types, species, and conditions for which no Hi-C data is available. The model may also be of use for improving Hi-C-based domain finders. The rest of the paper is organized as follows: We start by formally defining nTDP model. Then, we present provably optimal methods to train our model from markers and TADs, as well as to predict TADs over trained model. Lastly, we present results on prediction of domains on the same species as well as across species and cell types.
Previous work mainly focused on analyzing epigenetic data in an unsupervised way. Segway [16] and ChromHMM [17] take as input a collection of genomics datasets and learn chromatin states that exhibit similar epigenetic activity patterns which then have different interpretations such as transcriptionally active, Polycomb-repressed. Libbrecht et al. [18] improve Segway predictions by integrating Hi-C data which is not as abundant as histone data, whereas [19] jointly infers chromatin state maps in multiple genomes by a hierarchical model. However, none of these methods deal directly with TADs. Even though a subset of their chromatin states overlap with TADs, predicting TADs from them heuristically does not perform well. Additionally, they either ignore the histone densities, or make parametric distribution assumptions such as geometric or normal which are not always reflected in the true data. When modified to run in a supervised setting, they cannot capture the most informative subset of epigenetic elements.
The recent approach [20] proposes a supervised learning method based on random forests to predict TAD boundaries from histone modifications and chromatin proteins. In general, this approach is reported to perform quite accurately in predicting boundaries. However, it does not model interior TAD segments and it treats each segment independently ignoring the fact that TADs form as a result of the joint effects of multiple segments.
The nTDP model
The likelihood function
Let V be the ordered set of genome restriction fragments (bins), where each bin v represents the interval \([vr-r+1\), vr], where r is the Hi-C resolution. Let M be the set of histone modifications (markers) over V. The marker data \(H = (h_{vm})\) is a \(|V| \times |M|\)-matrix where its (v, m)'th entry \(h_{vm}\) is the count of the occurrences of marker m inside segment v. Let \(d=[s, e]\) be a domain (interval) where s and e are its start and end boundaries respectively, \(\{s+1, \ldots , e-1\}\) are the segments inside d, and let \(D = \{[s_{1}, e_{1}], [s_{2}, e_{2}], \ldots , [s_{i}, e_{i}] \}\) be a partition of V where none of the domains overlap.
We propose a supervised, semi-nonparametric, high-dimensional model nTDP that uses H to model and predict D. Our model can be seen as a generalization of Conditional Random Field [21, 22] where we have continuous weights instead of binary features and where we model the marker effects semi-nonparametrically.
Specifically, we assume there are 3 types of segments in V that are relevant for modeling: those that are at the domain boundaries (\(V_{\mathbf {b}}\)), those that are in the interior of domains (\(V_{\mathbf {i}}\)), and those that are not part of a domain (\(V_{\mathbf {e}}\)), and we have \(V = V_{\mathbf {b}}\cup V_{\mathbf {i}} \cup V_{\mathbf {e}}\). For each marker type m, we have 3 types of effect functions, \(f^b_m(c, \mathbf {w_m^b}), f_m^i(c, \mathbf {w_m^i}), f_m^e(c, \mathbf {w_m^e})\), that will describe the relationship between marker count c and the fragment type (b, i, e) for marker type m. Here, \(\bf {w_m^b},\bf {w_m^i},\bf {w_m^e}\) are parameters that we will fit to determine the shape of the effect function. Thus, for example, \(f_m^i(c, \mathbf {w_m^i})\) will describe how a count of c for marker m influences whether the fragment is in the interior (i) of a domain.
We assume that these effect functions combine linearly. Therefore, let
$$\begin{aligned} E^{b}_{vq} = \sum _{m \in M} f^{b}_{m}(c^{q}_{vm}, \mathbf {w^{b}_{m}}) \end{aligned}$$
be the total effect of all the markers on fragment v for boundary formation (b). Summations \(E^{i}_{vq}\) and \(E^{e}_{vq}\) are defined analogously for interior (i) and inter-domain fragments (e).
Let W be the union of model parameters \(\mathbf {w_m^b}, \mathbf {w_m^i}, \mathbf {w_m^e}\), and let \(D^\text {train} = \{D^q : q = 1,\dots ,Q\}\) be several domain decompositions (in different sequences or conditions) and let \(H^\text {train} = \{H^q : q = 1,\dots , Q\}\) be a set of corresponding histone markers. Under the assumption that the training pairs are independent, the log-likelihood of parameters W given \(D^\text {train}\) is
$$\begin{aligned} \log \left (P(D^{\text {train}} | W, H^{\text {train}})\right ) = \sum _{q}\log (P(D^{q} | W, H^{q}) ). \end{aligned}$$
We define the probability \(P(D^{q}, W, H^{q}) = \frac{\exp ^{F(D^{q}, W, H^{q})}}{\sum _{F^{'}} \exp ^{F^{'}}}\) where \(F(D^{q}, W, H^{q})\) is the total quality of partition \(D^{q}\) and marker data \(H^{q}\) under model parameters W. Let \(V^{q}\) be the set of segments in pair q. Due to independence of segments:
$$\begin{aligned}\log \left (P(D^{q} | W, H^{q})\right) = \overbrace{\sum _{d=[s, e] \in D^{q}} \left( \sum _{v \in \{s, e\}} \overline{c}_{b} E^{b}_{vq} + \sum _{v =s+1}^{e-1}\overline{c}_{i} E^{i}_{vq} \right) \, + \sum _{v \in V^{q}_{e}} \overline{c}_{e} E^{e}_{vq}}^{\log \left( P(D^{q} , W, H^{q})\right) = F(D^{q}, W, H^{q})} \nonumber & -\log (Z^{q}_{|V^{q}|}) \end{aligned}$$
where \(Z^{q}_{|V^{q}|} = \sum _{D'}\,P(D' , W, H^{q})\) is the partition function defined over all possible nonoverlapping partitions \(D'\), \(\overline{c}_{b}, \overline{c}_{i}, \overline{c}_{e}\) are relative weights of different types of fragments to account for unbalanced training set, and \(V_e^q\) is the set of fragments that do not belong to any domain in \(D^q\).
Nonparametric form of the effect functions
Because the shape of the marker effect function is unknown, we choose the f functions from the nonparametric family of Bernstein basis polynomials. Bernstein polynomials can approximate any effect function and additionally can handle imposed shape constraints such as monotonicity and concavity.
Let A be the chosen dimension of these polynomials; larger A results in a more expressive family, but more parameters to fit. Let \(m_{\max }\) be the maximum possible density of marker m. This is is used to transform the input \(c^{q}_{vm}\) to the range [0, 1]; therefore define \(p^{q}_{vm} = {c^{q}_{vm}}/{m_{\max }}\). We model \(f^{b}_{m}(c^{q}_{vm}, \mathbf {w^{b}_{m}})\) for segment v by a Bernstein polynomial \(B_{A}(p^{q}_{vm}, \mathbf {w^{b}_{m}})\) as in:
$$\begin{aligned} f^{b}_{m}(c^{q}_{vm}, \mathbf {w^{b}_{m}}) = B_{A}\left( p^{q}_{vm}, \mathbf {w^{b}_{m}} \right) = \sum _{i=0}^{A} w^{b}_{m}[i] \overbrace{ {A \atopwithdelims ()i} \left( p^{q}_{vm}\right) ^{i} \left( 1-p^{q}_{vm}\right) ^{A-i}}^{b_{i, A} (p^{q}_{vm})} \end{aligned}$$
where \(b_{i, A} (p^{q}_{vm})\) are the base Bernstein kernels.
Optimal algorithms for training and inference
We must train the parameters W for the above model using data of the form \(D^\text {train}, H^\text {train}\). We will examine these trained parameters (and several good solutions for them) for insights into which markers are most informative for describing \(D^\text {train}\) and thus topological domains.
Training: Given a set of marker data \(H^{\text {train}}\), likely from several chromosomes and cell conditions, and corresponding set of TAD decompositions \(D^{\text {train}}\), we estimate the most likely parameters W according to Eq. 2.
Inference: Given marker data H model parameters W, we estimate the best domain partition D of the track.
A nice feature of the objective (3) is that it is convex in its arguments, \(\{\mathbf {w^{b}_{m}}, \mathbf {w^{i}_{m}},\) \(\mathbf {w^{e}_{m}}\}_{m \in M}\), which follows from linearity, composition rules for convexity, and convexity of the negative logarithm. However, training involves several challenges: (a) computing the partition function \(Z^{q}_{|V^{q}|}\) in (3), and (b) estimating W so that the weights are sparse. We solve each of these challenges next.
Estimating the partition function
We estimate \(Z^{q}_{|V^{q}|}\) in (3) recursively in polynomial time since each segment can belong to one of 4 states: domain start (sb), inside a domain (i), domain end (eb), non-domain (e), and state of each segment depends only on the previous segment's state. Let \(Y = \{sb, i, eb, e\}\), and \(Z_{|V^{q}|}^{q} = Z^{q}_{|V^{q}|, eb}\,+\,Z^{q}_{|V^{q}|, e}\) which components can be estimated by:
$$\begin{aligned} Z^{q}_{v, x} = \sum _{y \in Y} Z^{q}_{v-1, y} T_{y, x} \exp ^{E^{x}_{vq}} \end{aligned}$$
where \(Z^{q}_{v,sb}\), \(Z^{q}_{v,i}\), \(Z^{q}_{v,eb}\), \(Z^{q}_{v,e}\) represent the partition function up to segment v ending with sb, i, eb and non-domain respectively. T is a \(4 \times 4\) binary state transition matrix where \(T_{y, x} = 1\) if a segment can be assigned to x given previous segment is assigned to state y such as \((y, x) \in \{(sb, i), (sb, eb), (i, i), (i, eb), (eb, sb)\), \((eb, e), (e, sb), (e, e)\}\), otherwise 0. Initial conditions are \(Z^{q}_{0,sb} = Z^{q}_{0,i} = Z^{q}_{0,eb} = 0\), \(Z^{q}_{0,e} = 1\). To avoid overflow in estimating \(Z^{q}_{|V^{q}|}\) and speed it up, we estimate \(\log (Z^{q}_{|V^{q}|})\) by expressing it in terms of log of the sum of exponentials and forward and backward variables (\(\alpha\), \(\beta\)) similar to Hidden Markov Model [21].
Estimating a sparse set of good histone effect parameters
We would like to augment objective function (2) so that we select a sparse subset of markers from the data and avoid overfitting. If the coefficients \(\mathbf {w^{b}_{m}} = 0\), then there is no influence of marker m. For this purpose, we will impose grouped lasso type of regularization on the coefficients \(w_{mk}\). Grouped lasso regularization has the tendency to select a small number of groups of non-zero coefficients but push other groups of coefficients to be zero.
We introduce two types of regularization. First, we require that many of the weights be 0 using an \(L_2\)-norm regularization term. Second, we want the effect functions \(\{f\}\) to be smooth. Let \(P = \{b, i, e\}\). We modify our objective to trade off between these goals:
$$\begin{aligned} \mathop {\text{argmin}}\limits _{W}\;-\sum _{q}\log \left (P(D^{q}| W, H^{q})\right )+\overbrace{\lambda _{1} \sum _{p \in P} \left ( \sum _{m \in M} \left|\left|\mathbf {w^{p}_{m}}\right|\right| \right)^{2}+\lambda _{2}\sum _{p \in P} \sum _{m \in M} R(f^{p}_{m})}^{\text {Regularization}} \end{aligned}$$
where \(\lambda _{1}\), \(\lambda _{2}\) are the regularization parameters, and \(R(f^{p}_{m})\) is the smoothing function for effect of marker m at \(p \in P\):
$$\begin{aligned} R(f^{p}_{m}) = \int _{x} \left( \frac{\partial ^{2} f^{p}_{m}(x,\mathbf {w^{p}_{m}})}{\partial x^{2}}\right) ^{2} dx \end{aligned}$$
Group lasso in (6) uses the square of block \(l_{1}\)-norm instead of \(l_{2}\)-norm group lasso which does not change the regularization properties [23]. Objective function (6) is convex which follows from the convexity of \(R(f^{p}_{m})\) as proven in Theorem 1.
\(R(f^{p}_{m})\) is a convex function of \(\mathbf {w^{p}_{m}}\).
Second-order derivative in (7) can be written more explicitly as in (8) according to [24]:
$$\begin{aligned} \frac{\partial ^{2} f^{p}_{m}(x,\mathbf {w^{p}_{m}})}{\partial x^{2}} = A(A-1) \sum _{i=0}^{A-2} (w^{p}_{m}[i+2]-2w^{p}_{m}[i+1]+w^{p}_{m}[i]) \genfrac(){0.0pt}0{A-2}{i} x^{i}(1-x)^{A-2-i} \end{aligned}$$
which turns \(R(f^{p}_{m})\) into (9):
$$\begin{aligned}&\int _{0}^{1} \left( \frac{\partial ^{2} f^{p}_{m}(x)}{\partial x^{2}}\right) ^{2} dx = A^{2}(A-1)^{2} \sum _{i=0}^{A} \sum _{j=i}^{A} (w^{p}_{m}[i] w^{p}_{m}[j])\nonumber \\&\quad \left( \sum _{q=\overline{e}_{i}}^{\min (i,2)} \sum _{r=\overline{e}_{j}}^{\min (j,2)} (-1)^{q+r} \genfrac(){0.0pt}0{2}{q} \genfrac(){0.0pt}0{2}{r} T^{i-q}_{j-r}(x) \right) \end{aligned}$$
where \(\overline{e}_{p} = \max (0,2-A+p)\), \(T^{i-q}_{j-r}(x)\) is defined below and \(\beta (i+j-q-r+1,2A-3-i-j+q+r)\) is the beta function:
$$\begin{aligned} T^{i-q}_{j-r}(x) = \genfrac(){0.0pt}0{A-2}{i-q} \genfrac(){0.0pt}0{A-2}{j-r} \underbrace{\int _{0}^{1} x^{i-q}(1-x)^{A-2-i+q} x^{j-r}(1-x)^{A-2-j+r} dx}_{\beta (i+j-q-r+1, 2A-3-i-j+q+r)} \end{aligned}$$
\(R(f^{p}_{m})\) is quadratic function of \(\mathbf {w^{p}_{m}}\). Its convexity follows from semidefiniteness of the resulting polynomial. \(\square\)
We note that (6) is convex, but it is a nonsmooth optimization problem because of the regularizer. We solve it efficiently by using an iterative algorithm from multiple kernel learning [23]. By Cauchy-Schwarz inequality:
$$\begin{aligned} \sum _{p \in P} \left ( \sum _{m \in M} \left|\left|\mathbf {w^{p}_{m}}\right|\right| \right )^{2} \le \sum _{p \in P} \sum _{m \in M} \frac{\left|\left|\mathbf {w^{p}_{m}}\right|\right|^{2}}{\gamma _{mp}} \end{aligned}$$
where \(\gamma _{mp} \ge 0\), \(\sum _{m \in M} \gamma _{mp} = 1, \;\, p \in P\), and the equality in (11) holds when
$$\begin{aligned} \gamma _{mp} = \frac{\left|\left|\mathbf {w^{p}_{m}}\right|\right|}{\sum _{m \in M} \left|\left|\mathbf {w^{p}_{m}}\right|\right|}, \quad p \in P \end{aligned}$$
This modification turns the objective into the following which is jointly convex in both \(\mathbf {w^{p}_{m}}\) and \(\gamma _{mp}\):
$$\begin{aligned}\mathop {\text{argmin}}\limits _{W}\;-\sum _{q}\log \left (P(D^{q} | W, H^{q})\right )\,+\,\sum _{p \in P} \sum _{m \in M} \left (\lambda _{1}\frac{\left|\left|\mathbf {w^{p}_{m}}\right|\right|^{2}}{\gamma _{mp}}\,+\,\lambda _{2} R(f^{p}_{m})\right ) \end{aligned}$$
$$\begin{aligned}&\,\text {s.t.} \;\; \sum _{m \in M} \gamma _{mp} = 1.0,\quad p \in P \end{aligned}$$
$$\begin{aligned}&\;\;\;\;\quad \gamma _{mp} \ge 0,\quad m \in M, p \in P \end{aligned}$$
We solve this by alternating between the optimization of \(\mathbf {w^{p}_{m}}\) and \(\gamma _{mp}\). When we fix \(\gamma _{mp}\), we can find the optimal \(\mathbf {w^{p}_{m}}\) by any quasi-newton solver such as L-BFGS [25] which runs faster than the other solvers such as iterative scaling or conjugate gradient. When we fixed \(\mathbf {w^{p}_{m}}\), we can obtain the best \(\gamma _{mp}\) by the closed form equation (12). Both steps iterate until convergence.
Training extensions
We can model a variety of shape-restricted effect functions by Bernstein polynomials that cannot be easily achieved by other nonparametric approaches such as smoothing splines [24]. We add the following constraints to ensure monotonicity:
$$\begin{aligned} w^{b}_{m}[i] \le w^{b}_{m}[i+1], \quad i = 0, \ldots , A-1 \end{aligned}$$
which is a realistic assumption since increasing the marker density should not decrease its effect. We can also ensure concavity of the effect function by:
$$\begin{aligned} w^{b}_{m}[i-1] - 2w^{b}_{m}[i] + w^{b}_{m}[i+1] \le 0, \quad i = 1, \ldots , A-1 \end{aligned}$$
which has a natural diminishing returns property: the increase in the value of the effect function generated by an increase in the marker density is smaller when output is large than when it is small. Our problem is different than smoothing splines since our loss function is more complicated than traditional spline loss functions due to partition function estimation in (5) which makes it hard to directly apply the smoothing spline methods [26]. In addition, these nonnegativity and other shape constraints can be naturally enforced in our method.
We can also extend the problem to modeling multiple domain subclasses instead of a single class where domains are categorized into subclasses according to their gene-expression profiles such as TADs with highly-active genes, TADs with repressive genes, etc.
Inferring domains using the trained model
Given marker data H over a single track and W, the inference log-likelihood is:
$$\begin{aligned} \mathop {\text{argmax}}\limits _{D}\;\log \left (P(D | W, H)\right ) = \sum _{d=[s, e] \in \overline{D}} r_{se} x_{se}\;+\;\sum _{v \in V} E^{e}_{v} y_{v} \end{aligned}$$
where \(\overline{D} = \{[s, e]\,|\,s,e \in V, \, e-s \ge 1\}\) is the set of all potential domains of length at least 2 and \(r_{se} = E^{b}_{s} + E^{b}_{e} + \sum _{v =s+1}^{e-1} E^{i}_{v}\). The intuition is that variable \(x_{se}=1\) when the solution contains interval [s, e], and variable \(y_{v}=1\) if v is not assigned to any domain. The \(\log (Z_{|V|})\) term is removed during inference since it is same for all D. We solve (19)–(20) to find the best partition D:
$$\begin{aligned}&\mathop {\text{argmax}}\limits _{D}\; \sum _{d=[s, e] \in \overline{D}} r_{se} x_{se}\;+\;\sum _{v \in V} E^{e}_{v} \left( 1 - \sum _{[s, e] \in M[v]} x_{se} \right) \end{aligned}$$
$$\begin{aligned}\text {s.t.} \;\; x_{se} + x_{s'e'} \le 1\qquad \forall \; \text {domains } [s,e], [s',e'] \text { that overlap} \end{aligned}$$
where M[v] is the set of intervals that span fragment v. We replace \(y_{v}\) in (18) with \(1 - \sum _{[s, e] \in M[v]} x_{se}\) since each segment can be assigned to at most a single domain. (20) ensures that inferred domains do not overlap. Problem (19)–(20) is Maximum Weight Independent Set in interval graph defined over domains which can be solved optimally by dynamic programming in \(O(|V|^{2})\) time.
Experimental setup
We binned ChIP-Seq histone modification and DNase-seq data at \(40~\hbox {kb}\) resolution, estimate RPKM (Reads Per Kilobase per Million) measure for each bin, and transform values x in each bin by \(\log (x+1)\), which reduces the distorting effects of high values. In the case of 2 or more replicates, the RPKM-level for each bin is averaged to get a single histone modification file, in order to minimize batch-related differences. We convert any data mapped to hg19 (mm8) to hg18 (mm9) using UCSC liftOver tool. We define TADs over human IMR90, human embryonic stem (ES), and mouse ES cells Hi-C data [8] at 40 kb resolution after normalization by [27]. We use consensus domains from Armatus [28] as the true TAD partition by selecting threshold \(\gamma\) where maximum Armatus domain size is closest to the maximum Dixon et al. [8] domain size (\(\gamma =0.5\) for IMR90, \(\gamma =0.6\) for human ES, and \(\gamma =0.2\) for mouse ES cells).
We solved the training optimization problem by L-BFGS [25]. We use the public implementation of Armatus [28], and obtain histone modifications from NIH Roadmap Epigenomics [29] and UCSC Encode [30]. Code and datasets can be found at http://www.cs.cmu.edu/~ckingsf/research/ntdp. nTDP is reasonably fast: we train on all human IMR90 chromosomes in less than 3 h on a MacBook Pro with 2.5Ghz processor and \(8\hbox {Gb}\) Ram. The iterative procedure in general converges in fewer than 10 iterations.
We prevent overfitting by following a two-step nested cross-validation which has inner and outer steps. The outer K-fold cross-validation, for example, trains on all autosomal human chromosomes except the one to be predicted. Within each loop of outer cross-validation, we perform \((K-1)\)-fold inner cross-validation to estimate the regularization parameters.
nTDP finds a small subset of modifications predictive of TADs
We identified a minimal set of histone marks that can model TADs as follows: we run nTDP independently on each chromosome of human IMR90 to obtain 21 sets of marks. These sets overlap significantly across all chromosome pairs (hypergeometric \(p < 0.05\) for all pair-wise comparisons), and a total of 16 modifications cover all chromosomes. Despite the regularization, the weights of several of these marks are still very close to 0, so we identify a non-redundant subset of the modifications by Bayesian information criterion (BIC) [21] which penalizes model complexity more strongly.
As we increase the number of included modifications from 1 to 16, the BIC decrease nearly stops after 4 modifications, with some additional small reduction up to 6 modifications. The sets of 4 and 6 modifications that were most informative are: {H3K36me3, H3K4me1, H3K4me3, H3K9me3} and {H3K4me3, H3K79me2, H3K27ac, H3K9me3, H3K36me3, H4K20me1}. These non-redundant set of elements are preserved when we repeat this procedure between species. We find that only these 4–6 modifications are needed to accurately predict TADs.
These marks are common in good models
The 4 modifications {H3K36me3, H3K4me1, H3K4me3, H3K9me3} are also enriched among a collection of high quality training solutions. We measure the agreement between estimated and true partitions by normalized variation of information \(NVI=\frac{VI}{\log |V|}\) [31] where VI measures the similarity between two partitions and lower score means better performance. We analyze the fraction of models with 4 histone modifications for which NVI score is at least \(95\%\) of optimum NVI score obtained by running nTDP over all modifications as in Fig. 1. We find 161, 139 solutions satisfying this criteria among 1820 candidates for human IMR90 and human ES histone modifications respectively. We find the 4 histone modifications above to be significantly overrepresented in the set of models for both human IMR90 and ES cells (hypergeometric \(p < 0.0001\)). As a boundary case, restricting the effect function to be a linear function of model parameters in human IMR90 cells does not significantly change the results as in Fig. 2. In another species, mouse ES cells, these 4 histone modifications are also the most informative predictors of TADs as in Fig. 3. These significance values combined with the results above suggest the importance of the identified modifications in TADs.
Fraction of histone modifications appearing in a best scoring four-modification model in a human IMR90, b human ES. Best scoring is defined as reaching at least \(95\%\) of NVI score of the model with all modifications
Fraction of histone modifications appearing in a best scoring four-modification parametric model in human IMR90. Best scoring is defined similar to Fig. 1
Fraction of histone modifications appearing in a best scoring four-modification model in mouse ES. Best scoring is defined similar to Fig. 1
These marks have nearly optimal coherence score
We assess the performance of various subset of modifications by the coherence score which is the exponential of the negative mean log-likelihood of each chromosome on the test set, and it is normalized by the best model coherence score as in Table 1. As such it is a relative measure of the quality of various models. The coherence score using only the set {H3K36me3, H3K4me1, H3K4me3, H3K9me3} is almost as high as the score for all 28 histone modifications in human IMR90. Restricting the effect function shape to be nonnegative and concave slightly improves the score. Similar ordering of models according to coherence scores is also observed in human ES cells as in Table 2. Our analysis indicates that the remaining modifications carry either redundant information or are less important for TADs.
Table 1 Normalized coherence scores of various marker subsets in human IMR90 cells
Table 2 Normalized coherence scores of various marker subsets in human ES cells
Predicting TADs from histone marks in human
nTDP is able to predict domain boundaries accurately using 4 histone marks alone in both human IMR90 and human ES cells. We compare TAD prediction performance of nTDP with the chromatin state partition predicted by Segway [16] in terms of NVI. Even though Segway does not predict TADs directly, its chromatin state partition can still be used as a baseline. Training with all 28 histone modifications instead of with the identified 4 modifications does not lead to a major performance increase as shown in Fig. 4a even though it increases the training time approximately 4 times for human IMR90 cells. Restricting the effect function to be monotonic and concave only slightly increases the performance. Chromatin states inferred by Segway do not directly correspond to TADs which leads to a lower TAD prediction performance even though they have other meaningful interpretations.
We find combinatorial effects of histone modifications to be important for accurate domain prediction since none of the modifications can achieve NVI score better than 0.2 when considered independently. To verify that there are not inherent structures in the data that can lead to an easy prediction, we randomly shuffle domains in the training set by preserving their lengths without shuffling modifications, which NVI score is never better than 0.3 in all chromosomes showing the importance of histone modification distributions in TADs.
nTDP also predicts TADs accurately across different species as well as across different cell types as in Fig. 4b–d. For example, if we train on human IMR90 data, the model we obtain is still able to recover domains in human ES cells (Fig. 4b). Using the identified 4 histone modifications achieves NVI score of 0.13 in human ES whereas using all modifications achieves slightly lower 0.11 on average. This performance difference is not significant except chromosomes 7 and 21 in human ES. This holds true across species as well: training on human ES data, for example, produces a model that can work well on mouse ES data. Accurate prediction of TADs by training with the identified 4 histone modifications across different species and cell types suggests the consistency of the identified modifications across species and cell types.
TAD prediction performance on different chromomes. a human IMR90 to human IMR90: infer each human IMR90 chromosome by training with all IMR90 chromosomes except the one to be inferred, b human IMR90 to human ES, c human ES to human ES, d human ES to mouse ES are defined similarly
Multiscale analysis of the predicted TADs. Performance over true TAD partitions at different scales obtained via different Armatus \(\gamma\) in human IMR90
Multiscale analysis of the predicted TADs
We find that our predicted TADs match true TADs more accurately at different scales defined by different Armatus \(\gamma\)'s as in Fig. 5. We observe a slight performance improvement if we define true TAD partition at lower Armatus \(\gamma\) values in human IMR90 which correspond to longer TADs. This figure suggests that some of our wrong TAD predictions may actually correspond to longer TAD blocks which we erroneously interpret as incorrect due to a scale mismatch.
We formulate semi-nonparametric modeling of TADs in terms of histone modifications, and propose an efficient provably optimal solution nTDP for training and inference. Experimental results on human and mouse cells show that a common subset of histone modifications can accurately predict TADs across cell types and species. Via our trained model, we also accurately predict TADs without using any Hi-C data which is especially useful for understanding the 3D genome conformation on species with limited Hi-C data. We expect our method to become increasingly useful with faster accumulation of epigenomic datasets than Hi-C interaction data. Additionally, some of our mispredictions may actually correspond to TADs at different scales suggesting a possibility of better inference performance than presented here.
Tolhuis B, Palstra R-J, Splinter E, Grosveld F, de Laat W. Looping and interaction between hypersensitive sites in the active \(\beta\)-globin locus. Mol Cell. 2002;10(6):1453–65.
Baù D, Marti-Renom MA. Structure determination of genomic domains by satisfaction of spatial restraints. Chromosome Res. 2011;19(1):25–35. https://doi.org/10.1007/s10577-010-9167-2.
Duggal G, Wang H, Kingsford C. Higher-order chromatin domains link eQTLs with the expression of far-away genes. Nucleic Acids Res. 2014;42(1), 87–96. https://doi.org/10.1093/nar/gkt857. http://nar.oxfordjournals.org/content/42/1/87.full.pdf+html
Guelen L, et al. Domain organization of human chromosomes revealed by mapping of nuclear lamina interactions. Nature. 2008;453(7197):948–51.
Lieberman-Aiden E, et al. Comprehensive mapping of long-range interactions reveals folding principles of the human genome. Science. 2009;326(5950), 289–293. https://doi.org/10.1126/science.1181369. http://www.sciencemag.org/cgi/reprint/326/5950/289.pdf
Gibcus JH, Dekker J. The hierarchy of the 3D genome. Mol Cell. 2013;49(5):773–82.
Rao SS, et al. A 3D map of the human genome at kilobase resolution reveals principles of chromatin looping. Cell. 2014;159(7):1665–80.
Dixon JR, Selvaraj S, Yue F, Kim A, Li Y, Shen Y, Hu M, Liu JS, Ren B. Topological domains in mammalian genomes identified by analysis of chromatin interactions. Nature. 2012;485(7398):376–80. https://doi.org/10.1038/nature11082.
Le TBK, et al. High-resolution mapping of the spatial organization of a bacterial chromosome. Science. 2013;342(6159), 731–734. https://doi.org/10.1126/science.1242059. http://www.sciencemag.org/content/342/6159/731.full.pdf
Hou C, Li L, Qin ZS, Corces VG. Gene density, transcription, and insulators contribute to the partition of the drosophila genome into physical domains. Mol Cell. 2012;48(3):471–84. https://doi.org/10.1016/j.molcel.2012.08.031.
Sexton T, Yaffe E, Kenigsberg E, Bantignies F, Leblanc B, Hoichman M, Parrinello H, Tanay A, Cavalli G. Three-dimensional folding and functional organization principles of the Drosophila genome. Cell. 2012;148(3):458–72.
Nora EP, et al. Segmental folding of chromosomes: a basis for structural and regulatory chromosomal neighborhoods? BioEssays. 2013;35(9):818–28. https://doi.org/10.1002/bies.201300040.
Phillips-Cremins JE, Sauria MEG, Sanyal A, Gerasimova TI, Lajoie BR, Bell JSK, Ong C-T, Hookway TA, Guo C, Sun Y, Bland MJ, Wagstaff W, Dalton S, McDevitt TC, Sen R, Dekker J, Taylor J, Corces VG. Architectural protein subclasses shape 3D organization of genomes during lineage commitment. Cell. 2013;153(6):1281–95. https://doi.org/10.1016/j.cell.2013.04.053.
Bickmore WA, van Steensel B. Genome architecture: domain organization of interphase chromosomes. Cell. 2013;152(6):1270–84. https://doi.org/10.1016/j.cell.2013.02.001.
Emre S, Geet D, Carl K. Deconvolution of ensemble chromatin interaction data reveals the latent mixing structures in cell subpopulations. J Comput Biol. 2016;23(6):425–38.
Hoffman MM, Buske OJ, Wang J, Weng Z, Bilmes JA, Noble WS. Unsupervised pattern discovery in human chromatin structure through genomic segmentation. Nat Methods. 2012;9:473–6. https://doi.org/10.1038/nmeth.1937.
Ernst J, Kellis M. ChromHMM: automating chromatin-state discovery and characterization. Nat Methods. 2012;9(3):215–6.
Libbrecht MW, Ay F, Hoffman MM, Gilbert DM, Bilmes JA, Noble WS. Joint annotation of chromatin state and chromatin conformation reveals relationships among domain types and identifies domains of cell type-specific expression. Genome Res. 2015; 25, 544–557. https://doi.org/10.1101/gr.184341.114. http://genome.cshlp.org/content/early/2015/02/12/gr.184341.114.full.pdf+html
Ho JW, et al. Comparative analysis of metazoan chromatin organization. Nature. 2014;512(7515):449–52.
Bednarz P, Wilczyński B. Supervised learning method for predicting chromatin boundary associated insulator elements. J Bioinf Comput Biol. 2014;12(06):1442006.
Murphy KP. Machine learning: a probabilistic perspective. Adaptive computation and machine learning series. Cambridge: Mit Press; 2012.
Zhou J, Troyanskaya OG. Global quantitative modeling of chromatin factor interactions. PLoS Comput Biol. 2014;10(3):1003525. https://doi.org/10.1371/journal.pcbi.1003525.
Bach FR. Exploring large feature spaces with hierarchical multiple kernel learning. In: Advances in neural information processing systems. 2009; p. 105–112.
McKay Curtis S, Ghosh SK. A variable selection approach to monotonic regression with Bernstein polynomials. J Appl Stat. 2011;38(5):961–76.
Liu DC, Nocedal J. On the limited memory BFGS method for large scale optimization. Math Program. 1989;45(1–3):503–28.
Wahba G. Spline models for observational data. Philadelphia: Society for Industrial and Applied Mathematics; 1990.
Yaffe E, Tanay A. Probabilistic modeling of Hi-C contact maps eliminates systematic biases to characterize global chromosomal architecture. Nat Genet. 2011;43(11):1059–65. https://doi.org/10.1038/ng.947.
Filippova D, Patro R, Duggal G, Kingsford C. Identification of alternative topological domains in chromatin. Alg Mol Biol. 2014;9(1):14. https://doi.org/10.1186/1748-7188-9-14.
Bernstein BE, et al. The NIH roadmap epigenomics mapping consortium. Nat Biotechnol. 2010;28(10):1045–8.
ENCODE Project Consortium. An integrated encyclopedia of DNA elements in the human genome. Nature. 2012;489(7414):57–74.
Meilă M. Comparing clusterings—an information based distance. J Multivariate Anal. 2007;98(5):873–95.
Both authors contributed to this paper. Both authors read and approved the final manuscript.
An early version of this paper was published in WABI 2015 conference proceedings. Authors would like to thank the reviewers for their helpful comments.
These results can be reproduced using the data and scripts provided at http://www.cs.cmu.edu/~ckingsf/research/ntdp
This research is funded in part by the Gordon and Betty Moore Foundation's Data-Driven Discovery Initiative through Grant GBMF4554 to Carl Kingsford, by the US NSF (1256087, 1319998), and by the US NIH (HG006913, HG007104). Carl Kingsford received support as an Alfred P. Sloan Research Fellow.
Machine Learning Department, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, 15213, USA
Emre Sefer
Computational Biology Department, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, 15213, USA
Carl Kingsford
Correspondence to Emre Sefer.
Sefer, E., Kingsford, C. Semi-nonparametric modeling of topological domain formation from epigenetic data. Algorithms Mol Biol 14, 4 (2019). https://doi.org/10.1186/s13015-019-0142-y
Accepted: 26 February 2019
Topological domains
Epigenetic modifications | CommonCrawl |
\begin{document}
\baselineskip=18pt
\begin{center}\Large\bf Efficiency of Reversible MCMC Methods:\ \ Elementary
\\[4pt]
Derivations and Applications to Composite Methods \end{center}
\centerline{\large
\ Radford M.\ Neal \ \ \ \ \ \ \ \ \ \ Je{f}frey S.\ Rosenthal} \centerline{\small\texttt{[email protected]}\hspace{24pt}
\texttt{[email protected]~~~}}
\centerline{\sl Department of Statistical Sciences, University of Toronto}
\centerline{(May 29, 2023)}
\begin{quotation}\noindent \textbf{Abstract.}\ \ We review criteria for comparing the efficiency of Markov chain Monte Carlo (MCMC) methods with respect to the asymptotic variance of estimates of expectations of functions of state, and show how such criteria can justify ways of combining improvements to MCMC methods. We say that a chain on a finite state space with transition matrix $P$ efficiency-dominates one with transition matrix $Q$ if for every function of state it has lower (or equal) asymptotic variance. We give elementary proofs of some previous results regarding efficiency dominance, leading to a self-contained demonstration that a reversible chain with transition matrix $P$ efficiency-dominates a reversible chain with transition matrix $Q$ if and only if none of the eigenvalues of $Q-P$ are negative. This allows us to conclude that modifying a reversible MCMC method to improve its efficiency will also improve the efficiency of a method that randomly chooses either this or some other reversible method, and to conclude that improving the efficiency of a reversible update for one component of state (as in Gibbs sampling) will improve the overall efficiency of a reversible method that combines this and other updates. We also establish conditions that can guarantee that a method is not efficiency-dominated by any other method. \end{quotation}
\section{Introduction}
Markov chain Monte Carlo (MCMC) algorithms (e.g.~\cite{mcmchandbook}) estimate the expected value of a function $f:S\to{\bf R}$ with respect to a probability distribution $\pi$ on a state space $S$, which in this paper we assume to be finite, using an estimator such as $$ \widehat{f}_N \ = \ {1 \over N} \sum_{k=1}^N f(X_k) \, , $$ where $X_1,X_2,X_3,\ldots$ is a time-homogeneous Markov chain with stationary distribution $\pi$, having transition probabilities $P(x,y)$ from state $x$ to state $y$ (often viewed as a matrix $P$).
An important measure of the efficiency of this estimator is its \underbar{asymptotic variance}: \begin{equation}\label{vdef} \!\!\!v(f,P) \ := \ \lim_{N\to\infty} N \, {\rm Var}\Big[ \widehat{f}_N \Big] \ = \ \lim_{N\to\infty} N \, {\rm Var}\Big[ {1 \over N} \sum_{i=1}^N f(X_i) \Big] \ = \ \lim_{N\to\infty} {1 \over N} \, {\rm Var}\Big[ \sum_{i=1}^N f(X_i) \Big] \, .\ \ \ \end{equation} For the irreducible Markov chains used for MCMC, the initial state of the chain does not affect the asymptotic variance, and the bias of the estimator converges to zero at rate $1/N$ regardless of initial state. (In practice, an initial portion of the chain is usually simulated but not used for estimation, in order to reduce the bias in a finite-length run.)
If we run the chain for a large number of iterations, $N$, we therefore expect that $v(f,P)/N$ will be an indication of the likely squared error of the estimate obtained. Indeed, when $v(f,P)$ is finite, one can show (e.g., \cite[Theorem 5]{tierney}) that a Central Limit Theorem applies, with the distribution of $(\widehat{f}-E_{\pi}(f))\,/\,\sqrt{v(f,P)/N}$ converging to $N(0,1)$.
We are therefore motivated to try to modify the chain to reduce $v(f,P)$, ideally for all functions $f$. We say that one transition matrix, $P$, \underbar{efficiency-dominates} another one, $Q$, if $$ v(f,P) \ \le \ v(f,Q) \qquad \textrm{for \ all} \ f:S\to{\bf R} \, . $$ Various conditions are known \cite{tierney, geyerstatsci, radfordnonrev} which ensure that $P$ efficiency-dominates $Q$. One of these, for reversible chains, is the \underbar{Peskun-dominance} condition \cite{peskun, tierney2}
which on a finite state space is that $P(x,y) \ge Q(x,y)$ for all $x\not=y$. But this is a very strong condition, and $P$ might well efficiency-dominate $Q$ even if it does not Peskun-dominate it.
In this paper, we focus on reversible chains with a finite state space. We present several known equivalences of efficiency dominance, whose proofs were previously scattered in the literature, sometimes only hinted at, and sometimes based on very technical mathematical arguments. We provide complete \underbar{elementary} proofs of them in Sections~\ref{sec-varform}, \ref{sec-equiv}, and~\ref{sec-effequivproof}, using little more than simple linear algebra techniques.
In Section~\ref{sec-newresults}, we use these equivalences to derive new results, which can show efficiency dominance for some chains constructed by composing multiple component transition matrices, as is done for the Gibbs Sampler. We also show how one can sometimes prove that a reversible chain cannot be efficiency-dominated by any other reversible chain. These results are applied to methods for improving Gibbs sampling in a companion paper~\cite{radfordnew}. These equivalences also allow an easy re-derivation, in Section~\ref{sec-peskun}, of the fact that Peskun dominance implies efficiency dominance.
\section{Background Preliminaries} \label{sec-prelim}
We assume that the state space $S$ is finite, with $|S|=n$, and let $\pi$ be a probability distribution on $S$, with $\pi(x)>0$ for all $x\in S$, and $\sum_{x\in S}\! \pi(x) = 1$. For functionals \mbox{$f,g:S\to{\bf R}$}, define the $L^2(\pi)$ inner product by $$ \inn{f}{g} \ = \ \sum_{x\in S} f(x) \, g(x) \, \pi(x) \, . $$ That is, $\inn{f}{g} = {\bf E}_\pi[f(X) \, g(X)]$. Equivalently, if we let $S=\{1,2,\ldots,n\}$, represent a function $f$ by the column vector $f=\big[ f(1),\ldots,f(n) \big]^T$, and let $D={\rm diag}(\pi)$ be the $n \times n$ diagonal matrix with $\pi(1),\,\ldots,\pi(n)$ on the diagonal, then $\inn{f}{g}$ equals the matrix product~$f^T\! D g$.
We aim to estimate expectations with respect to $\pi$ by using a time-homogeneous Markov chain $X_1,X_2,X_3,\ldots$ on $S$, with transition probabilities
\mbox{$P(x,y) = {\bf P}(X_{t+1}\!=\!y \, | \, X_t\!=\!x)$,} often written as a matrix $P$, for which $\pi$ is a \underbar{stationary distribution} (or \underbar{invariant distribution}): $$
\pi(y)\ =\ \sum_{x\in S} \pi(x) P(x,y) $$ Usually, $\pi$ is the only stationary distribution, though we sometimes consider transition matrices that are not irreducible (see below), for which this is not true, as building-blocks for other chains.
For $f:S\to{\bf R}$, let $(Pf):S\to{\bf R}$ be the function defined by $$ (Pf)(x) \ = \ \sum_{y\in S} P(x,y) \, f(y) \, . $$ Equivalently, if we represent $f$ as a vector of its values for elements of $S$, then $Pf$ is the product of the matrix $P$ with the vector $f$. Another interpretation is that
$(Pf)(x) = E_P[f(X_{t+1})|X_t=x]$, where $E_P$ is expectation with respect to the transitions defined by $P$. We can see that $$ \inn{f}{Pg} \ = \ \sum_{x\in S} \sum_{y\in S} f(x) \, P(x,y) \, g(y) \, \pi(x) \, . $$ Equivalently, $\inn{f}{Pg}$ is the matrix product $f^T\! D P g$. Also, $\inn{f}{Pg} = {\bf E}_{\pi,P}[f(X_t) \, g(X_{t+1})]$, where ${\bf E}_{\pi,P}$ means expectation with respect to the Markov chain with initial state drawn from the stationary distribution $\pi$ and proceeding according to $P$.
A transition matrix $P$ is called \underbar{reversible} with respect to $\pi$ if $\pi(x) \, P(x,y) = \pi(y) \, P(y,x)$ for all $x,y\in S$. This implies that $\pi$ is a stationary distribution for $P$, since $\sum_x \pi(x) P(x,y) = \sum_x \pi(y) P(y,x) = \pi(y) \sum_x P(y,x) = \pi(y)$.
If $P$ is reversible, $\inn{f}{Pg} = \inn{Pf}{g}$ for all $f$ and $g$ --- i.e., $P$ is \underbar{self-adjoint} (or, \underbar{Hermitian}) with respect to $\inn{\cdot}{\cdot}$. Equivalently, $P$ is reversible with respect to $\pi$ if and only if $DP$ is a symmetric matrix --- i.e., $DP$ is self-adjoint with respect to the classical dot-product. This allows us to easily verify some well-known facts about reversible Markov chains:
\begin{lemma} If $P$ is reversible with respect to $\pi$ then: (a) the eigenvalues of $P$ are real; (b) these eigenvalues can be associated with real eigenvectors; (c) if $\lambda_i$ and $\lambda_j$ are eigenvalues of $P$ with $\lambda_i \ne \lambda_j$, and $v_i$ and $v_j$ are real eigenvectors associated with $\lambda_i$ and $\lambda_j$, then $v_i^T D v_j = 0$, where $D$ is the diagonal matrix with $\pi$ on the diagonal (i.e., $\inn{v_i}{v_j}=0$); (d) all the eigenvalues of $P$ are in $[-1,1]$. \end{lemma}
\newpartitle{Proof} Since $DP$ is symmetric and $D$ is diagonal, $DP=(DP)^T=P^TD$. (a) If $Pv\,=\,\lambda v$ with $v$ non-zero, then $\overline\lambda\, \overline v^T \,=\, \overline v^T\! P^T$, hence $\overline \lambda\,(\overline v^T\! D v)\,=\, \overline v^T\! P^T D v \,=\, \overline v^T\! DP v\,=\,\lambda\, (\overline v^T\! D v)$. Since $\overline v^T\! D v$ is non-zero (because $D$ has positive diagonal elements), it follows that $\overline \lambda=\lambda$, and hence $\lambda$ is real. (b) If $Pv\,=\,\lambda v$, with $P$ and $\lambda$ real and $v$ non-zero, then at least one of
$\mbox{Re}(v)$ and $\mbox{Im}(v)$ is non-zero and is a real eigenvector associated with~$\lambda$. (c) $\lambda_i(v_i^T D v_j)\,=\, v_i^T P^T D v_j\, =\, v_i^T DP v_j\, =\, \lambda_j (v_i^T D v_j)$, which when $\lambda_i\ne\lambda_j$ implies that $v_i^T D v_j=0$. (d) Since rows of $P$ are non-negative and sum to one, the absolute value of an element of the vector $Pv$ can be no larger than the largest absolute value of an element of $v$. If $Pv=\lambda v$, this implies that $|\lambda| \le 1$, hence $\lambda\in[-1,1]$. \qed
The self-adjoint property implies that $P$ is a ``normal operator'', which guarantees (e.g., \cite[Theorem~2.5.3]{horn}) the existence of an \underbar{orthonormal basis}, $v_1,v_2,\ldots,v_n$, of eigenvectors for $P$, with $Pv_i = \lambda_i v_i$ for each $i$, and $\inn{v_i}{v_j}=\delta_{ij}$. (In particular, this property implies that $P$ is \underbar{diagonalisable} or \underbar{non-defective}, but it is stronger than that.) Without loss of generality, we can take $\lambda_1=1$, and $v_1 = {\bf 1} := \big[ 1,1,\ldots,1 \big]^T$, so that $v_1(x) = {\bf 1}(x) = 1$ for all $x\in S$, since $P {\bf 1} = {\bf 1}$ due to the transition probabilities in $P$ summing to one. We can assume for convenience that all of $P$'s eigenvalues (counting multiplicity) satisfy $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_n$.
In terms of orthonormal eigenvectors of $P$, any functions $f,g:S\to{\bf R}$ can be written as linear combinations $f = \sum_{i=1}^n a_i v_i$ and $g = \sum_{j=1}^n b_j v_j$. It then follows from orthonormality of these eigenvectors that $$ \inn{f}{g} = \sum_i a_i b_i,\ \ \
\inn{f}{f} = \sum_i (a_i)^2,\ \ \
\inn{f}{Pg} = \sum_i a_i b_i \lambda_i,\ \ \
\inn{f}{Pf} = \sum_i (a_i)^2 \lambda_i. $$
A matrix $A$ that is self-adjoint with respect to $\inn{\cdot}{\cdot}$ has a \underbar{spectral representation} in terms of its eigenvalues and eigenvectors as $A = \sum_{i=1}^n \lambda_i v_i v_i^T D$. If $h:{\bf R}\to{\bf R}$ we can define $h(A) := \sum_{i=1}^n h(\lambda_i) v_i v_i^T D$, which is easily seen to be self-adjoint. Using $h(\lambda)=1$ gives the identity matrix. One can also easily show that $h_1(A)+h_2(A)=(h_1+h_2)(A)$ and $h_1(A)h_2(A)=(h_1h_2)(A)$, and hence (when all $\lambda_i\ne0$) that $A^{-1}=\sum_{i=1}^n \lambda_i^{-1} v_i v_i^T D$ and (when all $\lambda_i\ge0$) that $A^{1/2}=\sum_{i=1}^n \lambda_i^{1/2} v_i v_i^T D$, so both of these are self-adjoint. Finally, note that if $A$ and $B$ are self-adjoint, so is $ABA$.
We say that $P$ is \underbar{irreducible} if movement from any $x$ to any $y$ in $S$ is possible via some number of transitions that have positive probability under $P$. An irreducible chain will have only one stationary distribution. A reversible irreducible $P$ will have $\lambda_i<1$ for $i\ge2$. (As an aside, this implies that $P$ is \underbar{variance bounding}, which in turn implies that $v(f,P)$ from~\eqref{vdef} must be finite \cite[Theorem~14]{varbound}.) For MCMC estimation, we want our chain to be irreducible, but irreducible chains are sometimes built using transition matrices that are not irreducible --- for example, by letting $P=(1/2)P_1+(1/2)P_2$, where $P_1$ and/or $P_2$ are not irreducible, but $P$ is irreducible.
An irreducible $P$ is \underbar{periodic} with period $p$ if $S$ can be partitioned into $p>1$ subsets $S_0,\ldots,S_{p-1}$ such that $P(x,y)=0$ if $x\in S_a$ and $y \notin S_b$, where $b=a\!+\!1\ \mbox{mod}\ p$ (and this is not true for any larger $p$). Otherwise, $P$ is \underbar{aperiodic}. An irreducible periodic chain that is reversible must have period 2, and will have $\lambda_n=-1$ and $\lambda_i>-1$ for $i \ne n$. A reversible aperiodic chain will have all $\lambda_i>-1$.
Since $v(f,P)$ as defined in~\eqref{vdef} only involves variance, we can subtract off the mean of $f$ without affecting the result. Hence, we can always assume without loss of generality that $\pi(f)=0$, where $\pi(f) := {\bf E}_\pi(f) = \sum_{x\in S} f(x) \, \pi(x) = \inn{f}{{\bf 1}}$. In other words, we can assume that $f\in L^2_0(\pi) := \{ f: \pi(f)=0, \ \pi(f^2)<\infty\}$, where the condition that $\pi(f^2) = {\bf E}_\pi(f^2)$ be finite is automatically satisfied when $S$ is finite, and hence can be ignored. Also, if $\pi(f) = 0$, then $\inn{f}{{\bf 1}} = \inn{f}{v_1} = 0$, so $f$ is orthogonal to $v_1$, and hence its coefficient $a_1$ is zero.
Next, note that $$ \inn{f}{P^kg} \ = \ \sum_{x\in S} f(x) \, (P^kg)(x) \, \pi(x) \ = \ \sum_{x\in S} \sum_{y\in S} f(x) P^k(x,y) g(y) \pi(x) \ = \ {\bf E}_{\pi,P}[ f(X_t) \, g(X_{t+k}) ]\,. $$ were $P^k(x,y)$ is the $k$-step transition probability from $x$ to $y$. If $f\in L^2_0(\pi)$ (i.e., the mean of $f$ is zero), this is the \underbar{covariance} of $f(X_t)$ and $g(X_{t+k})$, when the chain is started in stationarity (and hence is the same for all $t$). We define this as the \underbar{lag-$k$ autocovariance}, $\gamma_k$: $$ \gamma_k \ := \ {\rm Cov}_{\pi,P}[f(X_t),f(X_{t+k})] \ := \ {\bf E}_{\pi,P}[f(X_t) f(X_{t+k})] \ = \ \inn{f}{P^kf}, \ \ \ \mbox{for $f\in L^2_0(\pi)$}. $$ If $f = \sum_{i=1}^n a_i v_i$ as above (with $a_1=0$ since the mean of $f$ is zero), then using orthonormality of the eigenvectors $v_i$, $$ \gamma_k \ = \ \inn{f}{P^kf} \ = \ \sum_{i=2}^n \sum_{j=2}^n \inn{a_iv_i}{P^k(a_jv_j)} \ = \ \sum_{i=2}^n \sum_{j=2}^n \inn{a_iv_i}{(\lambda_j)^k a_jv_j)} \ = \ \sum_{i=2}^n (a_i)^2 (\lambda_i)^k \, . $$
In particular, $\gamma_0 = \inn{f}{f} := \|f\|_{L^2(\pi)} = \sum_i (a_i)^2$. (If the state space $S$ were not finite, we would need to require $f \in L^2(\pi)$, but finite variance is guaranteed with a finite state space.)
One particular example of a transition matrix $P$, useful for comparative purposes, is $\Pi$, the operator corresponding to i.i.d.\ sampling from $\pi$. It is defined by $\Pi(x,y) = \pi(y)$ for all $x\in S$. This operator satisfies $\Pi {\bf 1} = {\bf 1}$, and $\Pi f = 0$ whenever $\pi(f)=0$. Hence, its eigenvalues are $\lambda_1=1$ and $\lambda_i=0$ for $i\ne1$.
\section{Relating Asymptotic Variance to Eigenvalues} \label{sec-varform}
In this section, we consider some expressions for the asymptotic variance, $v(f,P)$, of~\eqref{vdef}, beginning with a result relating the asymptotic variance to the eigenvalues of $P$. This result (as observed by~\cite{geyerstatsci}) can be obtained (at least in the aperiodic case) as a special case of the more technical results of Kipnis and Varadhan \cite[eqn~(1.1)]{kipnis}.
\begin{proposition}\label{vevalform} If $P$ is an irreducible (but possibly periodic) Markov chain on a finite state space $S$, which is reversible with respect to $\pi$, with orthonormal basis $v_1,v_2,\ldots,v_n$ of eigenvectors, and corresponding eigenvalues $\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_n$, and $f\in L^2_0(\pi)$ with $f=\sum_i a_i v_i$, then the limit $v(f,P)$ in~\eqref{vdef} exists, and $$ v(f,P) \ = \ \sum_{i=2}^n (a_i)^2 \, + \, 2 \, \sum_{i=2}^n\, (a_i)^2 \, {\lambda_i \over 1-\lambda_i} \ = \ \sum_{i=2}^n\, (a_i)^2 \, {1+\lambda_i \over 1-\lambda_i} \, . $$ \end{proposition}
\newpartitle{Proof} First, by expanding the square, using stationarity, and collecting like terms, we obtain the well-known result that for $f\in L^2_0(\pi)$, \begin{eqnarray*} {1 \over N} \, {\rm Var}\,\bigg(\sum_{i=1}^N f(X_i) \bigg) & = & {1 \over N} \, {\bf E}_{\pi,P}\left[ \bigg( \sum_{i=1}^N f(X_i) \bigg)^{\!2}\, \right] \\[3pt] & = & {1 \over N} \, \bigg( N \, {\bf E}_{\pi,P}[f(X_j)^2] \,+\, 2 \sum_{k=1}^{N-1} (N\!-\!k)\ {\bf E}_{\pi,P}[f(X_j) \ f(X_{j+k})]\, \bigg)
\\[3pt]
& = & \gamma_0 \,+\, 2 \sum_{k=1}^{N-1} {N-k \over N} \ \gamma_k \, , \end{eqnarray*} where $\gamma_k = {\rm Cov}_{\pi,P}[f(X_j),f(X_{j+k})] = \inn{f}{P^kf}$ is the lag-$k$ autocovariance in stationarity.
Now, $f = \sum_{i=1}^n a_i v_i$, with $a_1=0$ since $\pi(f)=0$, so $\gamma_k \ = \ \inn{f}{P^kf} \ = \ \sum_{i=2}^n (a_i)^2 (\lambda_i)^k$ and $\gamma_0 \ = \ \sum_{i=2}^n (a_i)^2$. The above then gives that \begin{equation}\label{firstvareqn} {1 \over N} \, {\rm Var}\left( \sum_{i=1}^N f(X_i) \right) \ = \ \sum_{i=2}^n (a_i)^2 + 2 \sum_{k=1}^{N-1} {N-k \over N} \ \sum_{i=2}^n (a_i)^2 (\lambda_i)^k \, , \end{equation} i.e.\ $$ {1 \over N} \, {\rm Var}\left( \sum_{i=1}^N f(X_i) \right) \ = \ \sum_{i=2}^n (a_i)^2 + 2 \sum_{k=1}^\infty \sum_{i=2}^n I_{k \le N-1} \ {N-k \over N} \ (a_i)^2 (\lambda_i)^k \, . $$ If $P$ is aperiodic, then
$\Lambda := \max_{i \ge 2} |\lambda_i| \, < \, 1$, hence
$\sum_{k=1}^\infty \Big| \sum_{i=2}^n I_{k \le N-1} \, {N-k \over N} \ (a_i)^2 (\lambda_i)^k \Big| \le \sum_{k=1}^\infty \sum_{i=2}^n
\Big| I_{k \le N-1} \, {N-k \over N} \ (a_i)^2 (\lambda_i)^k \Big| \,\le\, \sum_{k=1}^\infty \sum_{i=2}^n (a_i)^2 (\Lambda)^k \,=\, \gamma_0 \, \Lambda/(1-\Lambda) < \infty$, so the above sum is \underbar{absolutely summable}. This lets us exchange the limit and summations to obtain $$ v(f,P) \ := \ \lim_{N\to\infty} {1 \over N} \, {\rm Var}\left( \sum_{i=1}^N f(X_i) \right) \ = \ \sum_{i=2}^n (a_i)^2 + 2 \sum_{i=2}^n \sum_{k=1}^\infty \lim_{N\to\infty} \left[ I_{k \le N-1} \ {N-k \over N} \ (a_i)^2 (\lambda_i)^k \right] $$ $$ \ \ \ \ \ \ = \ \sum_{i=2}^n (a_i)^2 + 2 \sum_{i=2}^n \sum_{k=1}^\infty (a_i)^2 (\lambda_i)^k \ = \ \sum_{i=2}^n (a_i)^2 + 2 \sum_{i=2}^n\, (a_i)^2 {\lambda_i \over 1-\lambda_i} \ = \ \sum_{i=2}^n\, (a_i)^2 \, {1+\lambda_i \over 1-\lambda_i} \, . $$
If $P$ is periodic, with $\lambda_n=-1$, then the above $\Lambda=1$, and $\sum_{k=1}^\infty (\lambda_n)^k$ is not even defined, so the above argument does not apply. Instead, separate out the $i=n$ term in~\eqref{firstvareqn} to get $$ {1 \over N} \, {\rm Var}\left( \sum_{i=1}^N f(X_i) \right) \ = \ \sum_{i=2}^n (a_i)^2 \, + \, 2 \sum_{k=1}^{N-1} {N-k \over N} \ \sum_{i=2}^{n-1} (a_i)^2 (\lambda_i)^k \, + \, 2 \sum_{k=1}^{N-1} {N-k \over N} (a_n)^2 (-1)^k \, . $$
Since $\Gamma := \max\{|\lambda_2|,|\lambda_3|,\ldots,|\lambda_{n-1}|\} < 1$, the previous argument applies to the middle double-sum term to show that $$ \lim_{N\to\infty} 2 \sum_{k=1}^{N-1} {N-k \over N} \ \sum_{i=2}^{n-1} (a_i)^2 (\lambda_i)^k \ = \ 2 \sum_{i=2}^{n-1}\, (a_i)^2\, {\lambda_i \over 1-\lambda_i} \, . $$ As for the final term, writing values for $k$ as $2m\!-\!1$ or $2m$, we have \defI_{\rm even}(N){I_{\rm even}(N)} \defI_{\rm odd}(N-1){I_{\rm odd}(N-1)} $$ \sum_{k=1}^{N-1} {N-k \over N} (-1)^k \ = \ {1 \over N} \sum_{m=1}^{\lfloor (N-1)/2 \rfloor} \Big[-(N-2m+1)+(N-2m)\,\Big] \ - \ {1 \over N}\, I_{\rm odd}(N-1) $$ $$ \ = \ {1 \over N} \sum_{m=1}^{\lfloor (N-1)/2 \rfloor} \Big[-1\Big] \ - \ {1 \over N}\, I_{\rm odd}(N-1) \ = \ - {\lfloor (N-1)/2 \rfloor \over N} \ - \ {1 \over N}\, I_{\rm odd}(N-1) \, , $$ which converges as $N\to\infty$ to $-{1 \over 2} = {-1 \over 1-(-1)} = {\lambda_n \over 1-\lambda_n}$. So, we again obtain that $$ v(f,P)\ =\ \lim_{N\to\infty} {1 \over N} \, {\rm Var}\left( \sum_{i=1}^N f(X_i) \right) \ = \ \sum_{i=2}^n (a_i)^2 \, + \, 2 \, \sum_{i=2}^{n-1}\, (a_i)^2 {\lambda_i \over 1-\lambda_i} \, + \, 2\, (a_n)^2 {\lambda_n \over 1-\lambda_n}\ \ $$ $$ \ = \ \sum_{i=2}^n (a_i)^2 \, + \, 2 \, \sum_{i=2}^n (a_i)^2 {\lambda_i \over 1-\lambda_i} \ = \ \sum_{i=2}^n\, (a_i)^2\, {1+\lambda_i \over 1-\lambda_i} \, . \eqno\darkbox $$
Note that when $P$ is periodic, $\lambda_n$ will be $-1$, and the final term in the expression for $v(f,P)$ will be zero. Such a periodic $P$ will have zero asymptotic variance when estimating the expectation of a function $f$ for which $a_n$ is the only non-zero coefficient.
When $P$ is aperiodic, we can obtain from Proposition~\ref{vevalform} the more familiar \cite{bartlett,billingsley,changeyer,olle,huang,tierney} expression for $v(f,P)$ in terms of sums of autocovariances, though it is not needed for this paper (and actually holds also without the reversibility condition \cite[Theorem 20.1]{billingsley}):
\begin{proposition}\label{onelagproposition} If $P$ is a reversible, irreducible, aperiodic Markov chain on a finite state space $S$ with stationary distribution $\pi$, and $f\in L^2_0(\pi)$, then \begin{equation}\label{gammasum} v(f,P)\ =\ \lim_{N\to\infty} {1 \over N} \, {\rm Var}_\pi\left( \sum_{i=1}^{N} f(X_i) \right) \ = \ \gamma_0 + 2 \sum_{k=1}^\infty \gamma_k \, , \end{equation} where $\gamma_k = {\rm Cov}_{\pi,P}[f(X_t),f(X_{t+k})]$ is the lag-$k$ autocovariance in stationarity. \end{proposition}
\newpartitle{Proof} Since $\gamma_k = \inn{f}{P^kf}$, and as above in the aperiodic case
$\Lambda := \sup_{i \ge 2} |\lambda_i| < 1$, the double-sum is again absolutely summable, and we compute directly that if $f=\sum_i a_i v_i$ then $$ \gamma_0 + 2 \sum_{k=1}^\infty \gamma_k \ = \ \inn{f}{f} + 2 \sum_{k=1}^\infty \inn{f}{P^kf} \ = \ \sum_i (a_i)^2 + 2 \sum_{k=1}^\infty \sum_i (a_i)^2 (\lambda_i)^k $$ $$ \ = \ \sum_i (a_i)^2 + 2 \sum_i (a_i)^2 \, \sum_{k=1}^\infty (\lambda_i)^k \ = \ \sum_i (a_i)^2 + 2 \sum_i (a_i)^2 \, {\lambda_i \over 1-\lambda_i} \, , $$ so the result follows from Proposition~\ref{vevalform}. \qed
Proposition~\ref{vevalform} gives the following formula for $v(f,P)$ (see also~\cite[Lemma 3.2]{mirageyer}):
\begin{proposition}\label{vprop} The asymptotic variance, $v(f,P)$, for the functional $f \in L^2_0(\pi)$ using an \underbar{irreducible} Markov chain $P$ which is \underbar{reversible} with respect to $\pi$ satisfies the equation $$ v(f,P) \ = \ \inn{f}{f}\ +\ 2\, \inn{f}{P(I\!-\!P)^{-1} f} \, , $$ which we can also write as $v(f,P) = \inn{f}{f}\ +\ 2\, \inn{f}{{P \over I-P} \, f}$, or as $v(f,P) = \inn{f}{{I+P \over I-P} \, f}$. \end{proposition}
\newpartitle{Proof} Let $P$ have an orthonormal basis $v_1,v_2,\ldots,v_n$, with eigenvalues $\lambda_1,\lambda_2,\ldots,\lambda_n$, and using this basis let $f=\sum_i a_i v_i$. Note that $a_1=0$, since $f$ has mean zero, so we can ignore $v_1$ and $\lambda_1$. Define $h(\lambda) := \lambda (1-\lambda)^{-1}$. As discussed in Section~\ref{sec-prelim}, applying $h$ to the eigenvalues of $P$ will produce another self-adjoint matrix, with the same eigenvectors, which will equal $P(I-P)^{-1}$. Using this, we can write \begin{eqnarray*} \inn{f}{f}\ +\ 2\, \inn{f}{P(I\!-\!P)^{-1} f} & = & \sum_i (a_i)^2
\ +\ 2 \sum_i \sum_j\, \inn{a_i v_i}{P(I\!-\!P)^{-1} (a_j v_j)} \\[3pt] & = & \sum_i (a_i)^2
\ +\ 2 \sum_i \sum_j\, \inn{a_i v_i}{\lambda_j (1-\lambda_j)^{-1} (a_j v_j)} \\[3pt] & = & \sum_i (a_i)^2\ +\ 2 \sum_i\, (a_i)^2\, \lambda_i\, (1-\lambda_i)^{-1}\, , \end{eqnarray*} so the result follows from Proposition~\ref{vevalform}. \qed
\newpartitle{Remark} The inverse $(I\!-\!P)^{-1}$ in Proposition~\ref{vprop} is on the restricted space $L^2_0(\pi)$ of functions $f$ with $\pi(f)=0$. Consequently, if we write $P = \sum_{i=1}^n \lambda_i v_i v_i^T D$, so $I-P = \sum_{i=1}^n (1-\lambda_i) v_i v_i^T D$, then on $L^2_0(\pi)$ this becomes $I-P = \sum_{i=2}^n (1-\lambda_i) v_i v_i^T D$, so $(I-P)^{-1} = \sum_{i=2}^n (1-\lambda_i)^{-1} v_i v_i^T D$, By contrast, $I\!-\!P$ will \underbar{not} be invertible on the full space $L^2(\pi)$ of \underbar{all} functions on $S$, since, for example, $(I\!-\!P){\bf 1}=0$.
\section{Efficiency Dominance Equivalences} \label{sec-equiv}
Combining Proposition~\ref{vprop} with the definition of efficiency dominance proves:
\begin{proposition}\label{pinvprop} For reversible irreducible Markov chain transition matrices $P$ and $Q$, $P$ efficiency-dominates $Q$ \underbar{if and only if} $\inn{f}{P(I\!-\!P)^{-1} f} \le \inn{f}{Q(I\!-\!Q)^{-1} f}$ for all $f \in L^2_0(\pi)$, or informally that $\inn{f}{{P \over I-P}f} \le \inn{f}{{Q \over I-Q}f}$ for all $f \in L^2_0(\pi)$. \end{proposition}
Next, we need the following fact:
\begin{lemma}\label{effequivlemma}
If $P$ and $Q$ are reversible and irreducible Markov chain transition matrices, $\inn{f}{P(I\!-\!P)^{-1}f} \le \inn{f}{Q(I\!-\!Q)^{-1}f}$ for all $f \in L^2_0(\pi)$ \underbar{if and only if} $\inn{f}{Pf} \le \inn{f}{Qf}$ for all $f \in L^2_0(\pi)$. \end{lemma}
Lemma~\ref{effequivlemma} follows from the very technical results of Bendat and Sherman \cite{bendat}. In Section~\ref{sec-effequivproof} below, we discuss this, and also present an alternative \underbar{elementary} proof.
Combining Lemma~\ref{effequivlemma} and Proposition~\ref{pinvprop} immediately shows the following, which is also shown by Mira and Geyer~\cite[Theorem 4.2]{mirageyer}:
\begin{theorem}\label{linprop} For Markov chain transition matrices $P$ and $Q$ that are reversible and irreducible, $P$ efficiency-dominates $Q$ \underbar{if and only if} $\inn{f}{Pf} \le \inn{f}{Qf}$ for all $f \in L^2_0(\pi)$, or equivalently, \underbar{if and only if} $\inn{f}{(Q\!-\!P)f} \ge 0$ for all $f \in L^2_0(\pi)$. \end{theorem}
\newpartitle{Remark} Here the restriction that $f \in L^2_0(\pi)$, i.e.\ that $\pi(f)=0$, can be omitted, since if $c := \pi(f) \not= 0$ then $f=f_0+c$ where $\pi(f_0)=0$, and $\inn{f}{Pf} = \inn{f_0+c}{P(f_0+c)} = \inn{f_0}{Pf_0} + c^2$, and similarly for $Q$. But we do not need this fact here.
\newpartitle{Remark} Some authors (e.g.~\cite{mira}) say that $P$ \underbar{covariance-dominates} $Q$ if $\inn{f}{Pf} \le \inn{f}{Qf}$ for all $f\in L^2_0(\pi)$, or equivalently if ${\rm Cov}_{\pi,P}[f(X_t), \, f(X_{t+1})]$ is always smaller under $P$ than under $Q$. The surprising conclusion of Theorem~\ref{linprop} is that this is \underbar{equivalent} to efficiency dominance --- i.e., to $v(f,P) \le v(f,Q)$ for all $f\in L^2_0(\pi)$. So, there is no need to consider the two concepts separately.
To make the condition $\inn{f}{(Q\!-\!P)f} \ge 0$ for all $f$ more concrete, we have the following:
\begin{lemma}\label{poslemma} Any self-adjoint matrix $J$ satisfies $\inn{f}{Jf} \ge 0$ for all $f$ \underbar{if and only if} the eigenvalues of $J$ are all non-negative, which is \underbar{if and only if} the eigenvalues of $DJ$ are all non-negative where $D={\rm diag}(\pi)$. \end{lemma}
\newpartitle{Proof} Let $J$ have orthonormal basis of eigenvectors $v_1,v_2,\ldots,v_n$ as in Section~\ref{sec-prelim}, so any $f$ can be written as $f = \sum_{i=1}^n a_i v_i$. Then $$ \inn{f}{Jf} \ = \ \inn{\sum_i a_i v_i}{\sum_j a_j Jv_j} \ = \ \inn{\sum_i a_i v_i}{\sum_j a_j \lambda_j v_j} \ = \ \sum_i (a_i)^2 \lambda_i \, . $$ If each $\lambda_i \ge 0$, then this expression must be $\ge 0$. Conversely, if some $\lambda_i<0$, then choosing $f=v_i$ gives $\inn{f}{Jf} = \lambda_i < 0$. This proves the first statement.
For the second statement, recall that $DJ$ is self-adjoint with respect to the classical dot-product. Hence, by the above, the matrix product $f^TDJf \ge 0$ for all $f$ \underbar{if and only if} the eigenvalues of $DJ$ are all non-negative. So, since $f^TDJf = \inn{f}{Jf}$, the two statements are equivalent. \qed
Combining Lemma~\ref{poslemma} (with $J$ replaced by $Q\!-\!P$) with Theorem~\ref{linprop} shows:
\begin{theorem}\label{eigenthm} If $P$ and $Q$ are reversible irreducible Markov chain transitions, $P$ efficiency-dominates $Q$ \underbar{if and only if} the operator $Q-P$ (equivalently, the matrix $Q-P$) has all eigenvalues non-negative, which is \underbar{if and only if} the matrix $D\,(Q-P)$ has all eigenvalues non-negative. \end{theorem}
\newpartitle{Remark} By Theorem~\ref{eigenthm}, if $Q-P$ has even a single negative eigenvalue, say $(Q-P)\,z\, =\, -cz$ where $c>0$, then there must be some $f\in L^2_0(\pi)$ such that $v(f,Q) < v(f,P)$. By following through our proof of Lemma~\ref{effequivlemma} in Section~\ref{sec-effequivproof} below, it might be possible to construct such an $f$ \underbar{explicitly} in terms of $z$ and $c$. We leave this as an open problem.
\newpartitle{Remark} It might be possible to give another alternative proof of Theorem~\ref{eigenthm} using the step-wise approach of \cite{radfordnonrev}, by writing $Q\!-\!P = R_1+R_2+\ldots+R_\ell$ where each $R_i$ is of rank one (e.g., $R_i=\lambda_i v_i v_i^T D$ with $\lambda_i$ and $v_i$ an eigenvalue and eigenvector of $Q\!-\!P$). We leave this as another open problem.
Theorem~\ref{eigenthm} allows us to prove the following, which helps justify the phrase ``efficiency-dominates'' (see also \cite[Section 4]{mirageyer}):
\begin{theorem}\label{partialord} Efficiency dominance is a \underbar{partial order} on reversible chains, i.e.: \begin{itemize} \item[(a)] It is \underbar{reflexive}: $P$ always efficiency-dominates $P$; \item[(b)] It is \underbar{antisymmetric}: if $P$ efficiency-dominates $Q$, and $Q$ efficiency-dominates $P$, then $P=Q$; \item[(c)] It is \underbar{transitive}: if $P$ efficiency-dominates $Q$, and $Q$ efficiency-dominates $R$, then $P$ efficiency-dominates $R$. \end{itemize} \end{theorem}
\newpartitle{Proof} Statement~(a) is trivial. Statement~(c) is true because $v(f,P) \le v(f,Q)$ and $v(f,Q) \le v(f,R)$ imply $v(f,P) \le v(f,R)$. For statement~(b), Theorem~\ref{eigenthm} implies that both $Q\!-\!P$ and $P\!-\!Q$ have all eigenvalues non-negative, hence their eigenvalues must all be zero, which implies (since $Q\!-\!P$ is self-adjoint) that $Q\!-\!P=0$, and hence $P=Q$. \qed
\newpartitle{Remark} Statement (b) of Theorem~\ref{partialord} does not hold if we do not assume reversibility. For example, if $S=\{1,2,3\}$, and $\pi={\rm Uniform}(S)$, and $P(1,2)=P(2,3)=P(3,1)=1$, and $Q(1,3)=Q(3,2)=Q(2,1)=1$, then $v(f,P)=v(f,Q)=0$ for all $f:S\to{\bf R}$, so they each (weakly) efficiency-dominate the other, but $P\not=Q$.
\section{New Efficiency Dominance Results} \label{sec-newresults}
Using Theorems~\ref{linprop} and~\ref{eigenthm}, we can now prove some new results about efficiency dominance that are useful when Markov chains are constructed by combining two or more chains.
We first consider the situation where we randomly choose to apply transitions defined either by $P$ or by $Q$. For example, $P$ might move about one region of the state space well, while $Q$ moves about a different region well. Randomly choosing either $P$ or $Q$ may produce a chain that moves well over the entire state space. The follow theorem says that if in this situation we can improve $P$ to $P'$, then the random combination will also be improved:
\begin{theorem} Let $P$, $P'$, and $Q$ be reversible with respect to $\pi$, with $P$ and $P'$ irreducible, and let $0<a<1$. Then $P'$ efficiency-dominates $P$ \underbar{if and only if} $aP'+(1\!-\!a)Q$ efficiency-dominates $aP+(1\!-\!a)Q$. \end{theorem}
\newpartitle{Proof} Since $P$ and $P'$ are irreducible, so are $aP'+(1\!-\!a)Q$ and $aP+(1\!-\!a)Q$. So by Theorem~\ref{eigenthm}, $P'$ efficiency-dominates $P$ \underbar{if and only if} $P-P'$ has all non-negative eigenvalues, which is clearly \underbar{if and only if} $a(P\!-\!P') = [aP'+(1\!-\!a)Q] - [aP+(1\!-\!a)Q]$ has all non-negative eigenvalues, which is \underbar{if and only if} $aP'+(1\!-\!a)Q$ efficiency-dominates $aP+(1\!-\!a)Q$. \qed
The next result applies to Markov chains built using component transition matrices that are not necessarily irreducible, such as single-variable updates in a random-scan Gibbs sampler, again showing that improving one of the components will improve the combination, assuming the combination is irreducible:
\begin{theorem}\label{gibbsthm} Let $P_1,\ldots,P_{\ell}$ and $P'_1,\ldots,P'_{\ell}$ be reversible with respect to $\pi$ (though not necessarily irreducible). Let $a_1,\ldots,a_{\ell}$ be mixing probabilities, with $a_k>0$ and $\sum_k a_k = 1$, and let $P = a_1 P_1 + ... + a_{\ell} P_{\ell}$ and $P' = a_1 P'_1 + ... + a_{\ell} P'_{\ell}$. Then if $P$ and $P'$ are irreducible, and for each $k$ the eigenvalues of $P_k-P'_k$ (or of $D(P_k-P'_k)$ where $D={\rm diag}(\pi)$) are all non-negative, then $P'$ efficiency-dominates $P$. \end{theorem}
\newpartitle{Proof} Choose any $f\in L^2_0(\pi)$. Since $P_k$ and $P'_k$ are self-adjoint, we have from Lemma~\ref{poslemma} that
$\inn{f}{(P_k-P'_k)f} \ge 0$ for each $k$.
Then, by linearity, $$ \inn{f}{(P-P')f} \ = \ \inn{f}{\sum_k a_k \, (P_k-P'_k)f} \ = \ \sum_k a_k \, \inn{f}{(P_k-P'_k)f} \ \ge \ 0 \, , $$ too. Hence, by Theorem~\ref{linprop}, $P'$ efficiency-dominates $P$. \qed
In the Gibbs sampling application, the state is composed of $\ell$ components, so that $S = S_1 \times S_2 \times \cdots \times S_{\ell}$, and $P_k$ is the transition that samples a value for component $k$, independent of its current value, from its conditional distribution given the values of other components, while leaving the values of these other components unchanged. Since it leaves other components unchanged, such a $P_k$ will not be irreducible. $P_k$ will be a block-diagonal matrix, in a suitable ordering of states (different for each $k$), with
$B=|S|/|S_k|$ blocks, each of size $K=|S_k|$.
For example, suppose $\ell=2$, $S_1=\{1,2\}$, $S_2=\{1,2,3\}$, and $\pi(x)=1/9$ except that $\pi((1,2))=4/9$. With lexicographic ordering, the Gibbs sampling transition matrix for the second component, $P_2$, will be $$
P_2\ =\
\pmatrix{ 1/6 & 4/6 & 1/6 & 0 & 0 & 0
\cr
1/6 & 4/6 & 1/6 & 0 & 0 & 0
\cr
1/6 & 4/6 & 1/6 & 0 & 0 & 0
\cr
0 & 0 & 0 & 1/3 & 1/3 & 1/3
\cr
0 & 0 & 0 & 1/3 & 1/3 & 1/3
\cr
0 & 0 & 0 & 1/3 & 1/3 & 1/3}\\[-1pt] $$
Each block of $P_k$ can be regarded as the $K\times K$ transition matrix for a Markov chain having $S_k$ as its state space, which is reversible with respect to the conditional distribution on $S_k$ given the values for other components associated with this block. For each block, the eigenvalues and eigenvectors of this transition matrix give rise to corresponding eigenvalues and eigenvectors of $P_k$, after prepending and appending zeros to the eigenvector according to how many blocks precede and follow this block. If the transition matrix for each block is irreducible, there will be $B$ eigenvalues of $P_k$ equal to one, with eigenvectors of the form $[ 0, \ldots, 0, 1, \ldots, 1, 0 \ldots, 0]^T$, which are zero except for a series of $K$ ones corresponding to one of the blocks.
We can try to improve the efficiency of $P$ by modifying one or more of the $P_k$. An improvement to $P_k$ can take the form of an improvement to one of its blocks --- corresponding to particular values of components of the state other than component $k$. For the example above, we could try to improve $P$ by improving $P_2$, with the improvement to $P_2$ taking the form of an improvement to how the second component is changed when the first component has the value 2, as follows: $$
P'_2\ =\
\pmatrix{ 1/6 & 4/6 & 1/6 & 0 & 0 & 0
\cr
1/6 & 4/6 & 1/6 & 0 & 0 & 0
\cr
1/6 & 4/6 & 1/6 & 0 & 0 & 0
\cr
0 & 0 & 0 & 0 & 1/2 & 1/2
\cr
0 & 0 & 0 & 1/2 & 0 & 1/2
\cr
0 & 0 & 0 & 1/2 & 1/2 & 0 }\\[-1pt] $$ The change to the $3\times3$ lower-right block still leaves it reversible with respect to the conditional distribution for the second component given the value 2 for the first component (which is uniform), but introduces an antithetic aspect to the sampling. This new $3\times3$ block efficiency-dominates the original Gibbs sampling block.
Theorem~\ref{gibbsthm} can be used to show that the $P'$ built with this modified $P'_2$ (and with $P'_1=P_1$) efficiency-dominates $P$ built with the original $P_1$ and $P_2$. The difference $P_2-P'_2$ will also be block diagonal, and its eigenvalues will be those of the differences in the individual blocks (which are zero for blocks that have not been changed). In the example above, the one block that changed has difference: $$
\pmatrix{1/3 & 1/3 & 1/3
\cr 1/3 & 1/3 & 1/3
\cr 1/3 & 1/3 & 1/3}
\ -\
\pmatrix{ 0 & 1/2 & 1/2
\cr 1/2 & 0 & 1/2
\cr 1/2 & 1/2 & 0 }
\ =\
\pmatrix{+1/3 & -1/6 & -1/6
\cr -1/6 & +1/3 & -1/6
\cr -1/6 & -1/6 & +1/3 } $$ The eigenvalues of this difference matrix are $1/2$, $1/2$, and zero. The eigenvalues of $P_2-P'_2$ will be these plus more zeros. If $P'_1=P_1$, Theorem~\ref{gibbsthm} then guarantees that $P'$ efficiency-dominates $P$, the original Gibbs sampling chain.
More generally, suppose a Gibbs sampling chain is changed by modifying one or more of the blocks of one or more of the $P_k$, with the new blocks efficiency-dominating the old Gibbs sampling blocks (seen as transition matrices reversible with respect to the conditional distribution for that block). Then by Theorem~\ref{eigenthm}, the eigenvalues of the differences between the old and new blocks are all non-negative, which implies that the eigenvalues of $P_k-P'_k$ all non-negative for each $k$, which by Theorem~\ref{gibbsthm} implies that the modified chain efficiency-dominates the original Gibbs sampling chain. The practical applications of this are developed further in the companion paper~\cite{radfordnew}.
We will now present some results relating eigenvalues of transition matrices to efficiency dominance, which can sometimes be used to show that a reversible transition matrix \underbar{cannot} be efficiency-dominated by any other reversible transition matrix.
Say that $P$ \underbar{eigen-dominates} $Q$ if both $P$ and $Q$ are reversible and the eigenvalues of $P$ are no greater than the corresponding eigenvalues of $Q$ --- that is, when the eigenvalues of $P$ are written non-increasing as $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_n$, and the eigenvalues of $Q$ are written non-increasing as $\beta_1 \ge \beta_2 \ge \ldots \ge \beta_n$, then $\lambda_i \le \beta_i$ for each $i$. Then we have (see also \cite[Theorem 3.3]{mirageyer}):
\begin{proposition}\label{eigenefprop} If $P$ and $Q$ are irreducible and reversible with respect to $\pi$, and $P$ efficiency-dominates $Q$, then $P$ eigen-dominates $Q$. \end{proposition}
\newpartitle{Proof} By Theorem~\ref{linprop}, $\inn{f}{Pf} \le \inn{f}{Qf}$ for all $f:S\to{\bf R}$. Hence, the result follows from the ``min-max'' characterisation of eigenvalues (e.g.\ \cite[Theorem~4.2.6]{horn}) that $$ \lambda_i \ = \ \inf_{g_1,\ldots,g_{i-1}} \sup_{f:S\to{\bf R} \atop \inn{f}{g_j}=0 \, \forall \, j} {\inn{f}{Pf} \over \inn{f}{f}} \, . $$ (Intuitively, $g_1,\ldots,g_{i-1}$ represent the first $i-1$ eigenvectors, so that the new eigenvector $f$ will be orthogonal to them. But since the formula is stated in terms of \underbar{any} vectors $g_1,\ldots,g_{i-1}$, the same formula applies to both $P$ and $Q$, thus giving the result.) \qed
The \underbar{converse} of Proposition~\ref{eigenefprop} does \underbar{not} hold, contrary to a claim in \cite[Theorem~2]{mira}. For example, suppose the state space is $S=\{1,2,3\}$. Let $e>0$ be very small, and define $P$, $Q$, and $R$ as $$ P\ =\ \pmatrix{\,{1 \over 2}\, & {1 \over 2} & 0
\cr
\,{1 \over 2}\, & {1 \over 2}-e & e
\cr
\,0\, & e & 1\!-\!e\,},\ \ \ \ Q\ =\ \pmatrix{1\!-\!e & e & 0
\cr
e & {1 \over 2}-e & {1 \over 2}\,
\cr
0 & {1 \over 2} & {1 \over 2}\,},\ \ \ \ R\ =\ \pmatrix{1\!-\!e & e & 0
\cr
e & {1 \over 2} & {1 \over 2}-e\,
\cr
0 & {1 \over 2}-e & {1 \over 2}+e\,}
$$ \iffalse $Q(1,2)=Q(2,1)=e$, $Q(1,1)=1-e$, $Q(2,2)={1 \over 2}-e$, $Q(2,3)=Q(3,2)=Q(3,3)={1 \over 2}$, and meanwhile $P(3,2)=P(2,3)=e$, $P(3,3)=1-e$, $P(2,2)={1 \over 2}-e$, $P(2,1)=P(1,2)=P(1,1)={1 \over 2}$. \fi These all are reversible with respect to $\pi=\,$Uniform$(S)$, and are irreducible and aperiodic. One can see that $P$ eigen-dominates $Q$, since $P$ and $Q$ are equivalent upon swapping states~1 and~3, and so have the same eigenvalues. However, $P$ does \underbar{not} efficiency-dominate $Q$. For example, taking $e=0.01$, $P$ and $Q$ each have the same eigenvalues given (to six decimal places) by 1, 0.985076, $-0.005076$, but $Q\!-\!P$ has eigenvalues 0.848705, 0, $-0.848705$ which are \underbar{not} all non-negative.
Intuitively, in this example, $Q$ moves easily between states 2 and 3 but not to or from state 1, while $P$ moves easily between states 1 and 2 but not to or from state 3. Hence, if, for example, $f(1)=1/3$ and $f(2)=1/6$ and $f(3)=1/2$ so that $f(1)={1 \over 2}[f(2)+f(3)]$, then $v(f,Q) < v(f,P)$, since $Q$ moving slowly between $\{1\}$ and $\{2,3\}$ doesn't matter, but $P$ moving slowly between $\{1,2\}$ and $\{3\}$ \underbar{does} matter.
$R$ is a slight modification to $Q$ that has two smaller off-diagonal elements, and hence is Peskun-dominated (and efficiency-dominated) by $Q$. It's eigenvalues are 1, 0.985077, 0.014923, the later two of which are strictly larger than those of $P$. But the eigenvalues of $R-P$ are 0.850060, 0, $-0.830060$, not all positive, so $P$ does not efficiency-dominate $R$, even though it strictly eigen-dominates $R$.
However, the next result is in a sense a converse of Proposition~\ref{eigenefprop} for the special case where $Q$ is the independent sampling chain, $\Pi$, with all rows equal to $\pi$. A chain, $P$, is ``antithetic'' (as in, e.g., \cite{greenhan}), if all its eigenvalues (except $\lambda_1=1$) are negative or zero (with at least one negative). Such antithetic samplers are always more efficient than i.i.d.\ sampling:
\begin{theorem} If $P$ is irreducible and reversible with respect to $\pi$, and has eigenvalues $\lambda_1=1$ and $\lambda_2,\lambda_3,\ldots,\lambda_n \le 0$, then $P$ efficiency-dominates $\Pi$ (the operator corresponding to i.i.d.\ sampling from $\pi$). \end{theorem}
\newpartitle{Proof} Let $v_1,v_2,\ldots,v_n$ be an orthonormal basis of eigenvectors for $P$, with $v_1={\bf 1}$. Then $P\,{\bf 1}={\bf 1}$ and $\Pi\,{\bf 1}={\bf 1}$, so $v_1$ is also an eigenvector of $\Pi$ with eigenvalue $\lambda_1=1$. Since the rows of $\Pi$ are all equal, the other eigenvalues of $\Pi$ are all zero, and can be associated with the same orthonormal eigenvectors $v_2,\ldots,v_n$ as $P$. So $(\Pi-P)v_1 = 0$, and for $i \ge 2$, $Pv_i = \lambda_i v_i$ and $\Pi \, v_i = 0$ so that $(\Pi-P)v_i = -\lambda_i v_i$. Hence, the eigenvalues of $\Pi-P$ are $0,-\lambda_2,-\lambda_2,\ldots,-\lambda_n$, which are all non-negative since each $\lambda_i \le 0$. Hence, by Theorem~\ref{eigenthm}, $P$ efficiency-dominates $\Pi$. \qed
\newpartitle{Remark} Theorem~1 of \cite{frigessi} shows that if $\pi_{\min} = \min_x \pi(x)$, the maximum eigenvalue (other than $\lambda_1$) of a transition matrix reversible with respect to $\pi$ must be greater than or equal to $-\pi_{\min}\,/\,(1-\pi_{\min})$, which must be greater than or equal to $-1/(n\!-\!1)$, since $\pi_{\min} \le 1/n$. This limits the possible gain from antithetic sampling for the worst-case choice of $f$. On the other hand, we can still have, for example, $\lambda_1=1$, $\lambda_n=-1$, and all the other $\lambda_i=0$, which would represent a significant improvement in MCMC estimation for some functions.
Since practical interest focuses on whether or not some chain, $P$, efficiency-dominates another chain, $Q$, Proposition~\ref{eigenefprop} is perhaps most useful in its contrapositive form --- if $P$ and $Q$ are reversible, and $P$ does \underbar{not} eigen-dominate $Q$, then $P$ does \underbar{not} efficiency-dominate~$Q$. That is, if $Q$ has at least one eigenvalue greater than the corresponding eigenvalue of $P$, then $P$ does not efficiency-dominate $Q$. If both chains have an eigenvalue greater than the corresponding eigenvalue of the other chain, then neither efficiency-dominates the other.
But what if two different chains have exactly the same ordered set of eigenvalues --- that is, they both eigen-dominate the other? In that case, neither efficiency-dominates the other:
\begin{proposition}\label{eigensame} If $P$ and $Q$ are both irreducible transitions matrices, reversible with respect to $\pi$, and the eigenvalues (counting multiplicity) for both are identical,
and $P \ne Q$, then $P$ does not efficiency-dominate $Q$, and $Q$ does not efficiency-dominate $P$. \end{proposition}
\newpartitle{Proof} By Theorem~\ref{eigenthm}, if $P$ efficiency-dominates $Q$, then $Q\!-\!P$ has no negative eigenvalues. And it cannot have all zero eigenvalues, since then $Q\!-\!P=0$ (since $Q\!-\!P$ is self-adjoint), contradicting the premise that $P \ne Q$. So, $Q\!-\!P$ has at least one positive eigenvalue, and no negative eigenvalues, and hence the sum of eigenvalues of $Q\!-\!P$ is strictly positive. But the sum of the eigenvalues of a matrix is equal to its trace~\cite[p.~51]{horn}, so this is equivalent to $\mbox{trace}(Q\!-\!P)>0$. Since trace is linear, this implies that $\mbox{trace}(Q)\!-\!\mbox{trace}(P) > 0$, and hence $\mbox{trace}(P)<\mbox{trace}(Q)$. But $\mbox{trace}(P)=\mbox{trace}(Q)$, since the eigenvalues of $P$ and $Q$ are identical. So $P$ cannot efficiency-dominate $Q$, and by the same argument, $Q$ cannot efficiency-dominate $P$.\qed
The following lemma along with Propositions~\ref{eigenefprop} and~\ref{eigensame} will allow us to show that some reversible chains cannot be efficiency-dominated by \underbar{any} other reversible chain.
\begin{lemma}\label{eigensum} For any transition matrix $P$ on a finite state space $S$, for which $\pi$ is a stationary distribution, the sum of the diagonal elements of $P$ --- that is, $\mbox{trace}(P)$ --- must be at least $\max\,(0,\ (2\pi_{\max}-1)\,/\,\pi_{\max})$, where $\pi_{\max}=\max_x \pi(x)$. Furthermore, any $P$ attaining this minimum value will have at most one non-zero value on its diagonal, and any such non-zero diagonal value will be for a state $x^*$ for which $\pi(x^*)=\pi_{\max} > 1/2$. \end{lemma}
\newpartitle{Proof} The statement is trivial when $\pi_{\max} \le 1/2$, since the lower limit on $\mbox{trace}(P)$ is then zero, and any such $P$ has all zeroes on the diagonal. Otherwise, if $x^*$ is such that $\pi(x^*)=\pi_{\max} > 1/2$, then stationarity implies that \begin{eqnarray*}
\pi_{\max} \ = \ \pi(x^*) & = & \sum_{x\in S} \pi(x) P(x,x^*)
\ =\ \pi(x^*)P(x^*,x^*)\ +\!\!\sum_{x\in S,\,x\ne x^*}\!\!\!\pi(x)P(x,x^*)
\\[3pt] & \le & \pi(x^*)P(x^*,x^*)\ +\!\!\sum_{x\in S,\, x\ne x^*}\!\!\!\pi(x) \ =\ \pi(x^*)P(x^*,x^*)\ +\ (1-\pi(x^*)) \\[3pt] & =& \pi_{\max}\,P(x^*,x^*)\ + (1 - \pi_{\max}) \, . \end{eqnarray*} It follows that $P(x^*,x^*) \ge (2\pi_{\max}-1)\,/\,\pi_{\max}$, hence $\mbox{trace}(P) \ge \max\,(0,\ (2\pi_{\max}-1)\,/\,\pi_{\max})$. Furthermore, if $\mbox{trace}(P)=\max\,(0,\ (2\pi_{\max}-1)\,/\,\pi_{\max})$, then $\mbox{trace}(P)=P(x^*,x^*)$, and hence all other values on the diagonal of $P$ must be zero.
\qed
\newpartitle{Remark} For any $\pi$, the minimum value of $\mbox{trace}(P)$ in Lemma~\ref{eigensum} is attainable, and can indeed be attained by a $P$ that is reversible. Several methods for constructing such a $P$ are discussed in the companion paper~\cite{radfordnew}, including, for example, the ``shifted tower'' method of~\cite{suwa}, which produces a reversible $P$ when the shift is by $1/2$.
We can now state a criterion for a chain to not be efficiency-dominated by any other reversible chain:
\begin{theorem}\label{nodom} If $P$ is the transition matrix for an irreducible Markov chain on a finite state space that is reversible with respect to $\pi$, and the sum of the eigenvalues of $P$ (equivalently, the trace of $P$) equals $\max\,(0,\ (2\pi_{\max}-1)\,/\,\pi_{\max})$, where $\pi_{\max}=\max_x \pi(x)$, then no other reversible chain can efficiency-dominate $P$. \end{theorem}
\newpartitle{Proof} Suppose that $Q$ efficiency-dominates $P$, with $Q \ne P$. Then by Proposition~\ref{eigenefprop}, $Q$ must eigen-dominate $P$. But by Proposition~\ref{eigensame}, $Q$ cannot have exactly the same eigenvalues as $P$. Hence there must be at least one eigenvalue of $Q$ that is strictly less than the corresponding eigenvalue of $P$, while all other eigenvalues of $Q$ are at least as small as those of $P$. But this is not possible if the sum of the eigenvalues of $P$ is equal to $\max\,(0,\ (2\pi_{\max}-1)\,/\,\pi_{\max})$, which by Lemma~\ref{eigensum} is the minimum possible value. \qed
As an example of how this theorem can be applied, if the state space is $S=\{1,2,3\}$, with $\pi(1)=\pi(2)=1/5$ and $\pi(3)=3/5$, for which $\pi_{\max}=3/5$, then the transition matrix $$
P_1\ =\ \pmatrix{0 & 0 & 1
\cr 0 & 0 & 1
\cr 1/3 & 1/3 & 1/3} $$ is reversible with respect to $\pi$, and has eigenvalues of $1, 0, -2/3$, which sum to 1/3 (the trace). By Theorem~\ref{nodom}, $P_1$ cannot be efficiency-dominated by any other reversible transition matrix, since its sum of eigenvalues is equal to $(2\pi_{\max}-1)\,/\,\pi_{\max}$.
On the other hand, consider the following transition matrix, reversible with respect to the same $\pi$: $$
P_2\ =\ \pmatrix{0 & 1/4 & 3/4
\cr 1/4 & 0 & 3/4
\cr 1/4 & 1/4 & 1/2} $$ $P_2$ has eigenvalues of $1, -1/4, -1/4$, which sum to $1/2$, greater than $(2\pi_{\max}-1)\,/\,\pi_{\max}$, so Theorem~\ref{nodom} does not apply. However, $P_2$ is an instance of a transition matrix constructed according to a procedure of Frigessi, Hwang, and Younes \cite[Theorem~1]{frigessi}, which they prove has the property that the transition matrix produced has the smallest possible value for $\lambda_2$, and subject to having that value for $\lambda_2$, the smallest possible value for $\lambda_3$, etc. We can therefore again conclude from Propositions~\ref{eigenefprop} and~\ref{eigensame} that no other reversible chain can efficiency-dominate $P_2$.
It's easy to see that any reversible $P$ with at least two non-zero diagonal elements, say $P(x,x)$ and $P(y,y)$, can be efficiency-dominated by a chain, $Q$, that is the same as $P$ except that these diagonal elements are reduced, allowing $Q(x,y)$ and $Q(y,x)$ to be greater than $P(x,y)$ and $P(y,x)$, so that $Q$ Peskun-dominates $P$. Theorem~\ref{nodom} shows that \underbar{some} reversible $P$ in which only a single diagonal element is non-zero cannot be efficiency-dominated by any other reversible chain. This leaves open the question of whether \underbar{all} reversible $P$ with only one non-zero diagonal element cannot be efficiency-dominated by some other reversible chain. We know of no examples of a reversible $P$ with only one non-zero diagonal element that is dominated by another reversible chain, but we do not have a proof that this is impossible.
\section{Re-deriving Peskun's Theorem}\label{sec-peskun}
Recall that $P$ \underbar{Peskun-dominates} $Q$ if $P(x,y) \ge Q(x,y)$ for all $x\not=y$ --- i.e., that $Q\!-\!P$ has all non-positive entries off the diagonal (and hence also that $Q-P$ has all non-negative entries on the diagonal). It is known through several complicated proofs~\cite{peskun, tierney2, radfordnonrev} that if $P$ Peskun-dominates $Q$, then $P$ efficiency-dominates $Q$. We will see here that once Theorem~\ref{eigenthm} has been established, this fact can be shown easily.
\begin{proposition}\label{peskunprop} If $P$ and $Q$ are irreducible, and both are reversible with respect to some $\pi$, and $P$ Peskun-dominates $Q$, then $P$ efficiency-dominates $Q$. \end{proposition}
To prove Proposition~\ref{peskunprop}, we begin with a simple eigenvalue lemma.
\begin{lemma}\label{rowsumlemma} If $Z$ is an $n \times n$ matrix with $z_{ii} \ge 0$ and $z_{ij} \le 0$ for all $i\not=j$, and row-sums $\sum_j z_{ij} = 0$ for all $i$, then all eigenvalues of $Z$ must be non-negative. \end{lemma}
\newpartitle{Proof} Suppose $Zv=\lambda v$. Find the index $j$ which maximizes $|v_j|$, i.e.\ such that $|v_j| \ge |v_k|$ for all $k$. We can assume $v_j > 0$ (if not, replace $v$ by $-v$), so $v_j \ge |v_k|$ for all $k$. Then $$
\lambda \, v_j \ = \ (Zv)_j \ = \ \sum_i z_{ji} v_i
\ = \ z_{jj} v_j + \sum_{i \not= j} z_{ji} v_i
\ \ge \ z_{jj} v_j - \sum_{i \not= j} |z_{ji}| \, |v_i| $$ $$
\ \ge \ z_{jj} v_j - \sum_{i \not= j} |z_{ji}| \, v_j
\ = \ v_j \Big( z_{jj} + \sum_{i \not= j} z_{ji} \Big)
\ = \ v_j (0)
\ = \ 0 \, . $$ So, $\lambda \, v_j \ge 0$. Hence, since $v_j>0$, we must have $\lambda \ge 0$. \qed
\newpartitle{Proof of Proposition~\ref{peskunprop}} Let $Z=Q\!-\!P$. Since $P$ Peskun-dominates $Q$, $z_{ii} = Q(i,i) - P(i,i) \ge 0$ and $z_{ij} = Q(i,j) - P(i,j) \le 0$ for all $i\not=j$. Also $\sum_j z_{ij} = \sum_j P(i,j) - \sum_j Q(i,j) = 1 - 1 = 0$. Hence, by Lemma~\ref{rowsumlemma}, $Z=Q\!-\!P$ has all eigenvalues non-negative. Hence, by Theorem~\ref{eigenthm}, $P$ efficiency-dominates $Q$. \qed
\newpartitle{Remark} Proposition~\ref{peskunprop} can also be proven by transforming $Q$ into $P$ one step at a time, in the sequence $Q,Q',Q'',\ldots,P$, with each matrix in the sequence efficiency-dominating the previous matrix. At each step, say from $Q'$ to $Q''$, two of the off-diagonal transition probabilities that differ between $Q$ and $P$, say those involving states $x$ and $y$, will be increased from $Q(x,y)$ to $P(x,y)$ and from $Q(y,x)$ to $P(y,x)$, while $Q''(x,x)$ and $Q''(y,y)$ will decrease compared to $Q'(x,x)$ and $Q'(y,y)$. The difference $Q'-Q''$ will be zero except for a $2 \times 2$ submatrix involving states $x$ and $y$, which will have the form $\Big(\begin{array}{rr} a & \!-a \\[-2pt] \!\!-b & b \end{array}\Big)$ for some $a,b>0$, which has non-negative eigenvalues of 0 and $a+b$. Hence, by Theorem~\ref{eigenthm}, $Q''$ efficiency-dominates $Q'$. Since this will be true for all the steps from $Q$ to $P$, transitivity (see Theorem~\ref{partialord}(c)) implies that $P$ efficiency-dominates $Q$. \qed
Note that the converse to Proposition~\ref{peskunprop} is \underbar{false}. For example, let $S=\{1,2,3\}$, and $$ P = \pmatrix{ 0 & 1/2 & 1/2
\cr 1 & 0 & 0
\cr 1 & 0 & 0 } , \quad Q = \pmatrix{ 1/2 & 1/4 & 1/4
\cr 1/2 & 1/4 & 1/4
\cr 1/2 & 1/4 & 1/4 } , \quad Q\!-\!P = \pmatrix{ +1/2 & -1/4 & -1/4
\cr -1/2 & +1/4 & +1/4
\cr -1/2 & +1/4 & +1/4 } \, . $$ Here, $P$ does \underbar{not} Peskun-dominate $Q$, since, for example, $Q(2,3)=1/4 > 0 = P(2,3)$. However, the eigenvalues of $Q\!-\!P$ are $1,0,0$, all of which are non-negative, so $P$ \underbar{does} efficiency-dominate $Q$. Furthermore, Theorem~\ref{partialord}(b) implies that $P$ is strictly better than $Q$ --- there is some $f$ for which $v(f,P)<v(f,Q)$. (For example, the indicator function for the first state, which has asymptotic variance zero using $P$, and asymptotic variance $1/4$ using $Q$.) Peskun dominance therefore does not capture all instances of efficiency dominance that we are interested in, which motivates our investigation here.
One should note, however, that all our results concern only reversible chains. Non-reversible chains are often used, either in a deliberate attempt to improve performance (see \cite{radfordnonrev}), or somewhat accidentally, as a result of combining methods sequentially rather than by random selection. Investigation of non-reversible chains is another area for future work.
\section{Elementary Proof of Lemma~\ref{effequivlemma}} \label{sec-effequivproof}
We conclude by presenting the promised elementary proof of Lemma~\ref{effequivlemma} above, which states the surprising fact that, for any reversible irreducible transition matrices $P$ and $Q$, $\inn{f}{Pf} \le \inn{f}{Qf}$ for all $f$ \underbar{if and only if} $\inn{f}{P(I\!-\!P)^{-1}f} \le \inn{f}{Q(I\!-\!Q)^{-1}f}$ for all $f$.
As observed in \cite[pp.~16--17]{mirageyer}, this proposition follows from the more general result of Bendat and Sherman \cite[p.~60]{bendat}, using results of L\"owner \cite{lowner}, which states that that if $h(x) = {ax+b \over cx+d}$ where $ad-bc>0$, and $J$ and $K$ are any two self-adjoint operators with spectrum contained in $(-\infty,-d/c)$ or in $(-d/c,\infty)$, then if $\inn{f}{Jf} \le \inn{f}{Kf}$ for all $f$, then also $\inn{f}{h(J)f} \le \inn{f}{h(K)f}$ for all $f$. In particular, choosing $a=d=1$, $b=0$, and $c=-1$ gives that $h(J) = {J \over I-J}$, so if $\inn{f}{Jf} \le \inn{f}{Kf}$ for all $f$ then $\inn{f}{{J \over I-J} \, f} \le \inn{f}{{K \over I-K} \, f}$ for all $f$. Conversely, choosing $a=c=d=1$ and $b=0$ gives that $h({J \over I-J}) = J$, so if $\inn{f}{{J \over I-J} \, f} \le \inn{f}{{K \over I-K} \, f}$ for all $f$ then $\inn{f}{Jf} \le \inn{f}{Kf}$ for all $f$, finishing the proof.
However, the proof in \cite{bendat} is very technical, requiring analytic continuations of transition functions into the complex plane. Instead, we now present an elementary proof of Lemma~\ref{effequivlemma}. We begin with some lemmas about operators on a finite vector space ${\cal V}$, e.g.\ ${\cal V}=L^2_0(\pi)$.
\begin{lemma}\label{Zlemma} If $X,Y,Z$ are
operators on a finite vector space ${\cal V}$, with $Z$ self-adjoint, and $\inn{f}{Xf} \le \inn{f}{Yf}$ for all $f\in{\cal V}$, then $\inn{f}{ZXZf} \le \inn{f}{ZYZf}$ for all $f\in{\cal V}$. \end{lemma}
\newpartitle{Proof} Since $Z$ is self-adjoint, making the substitution $g=Zf$ gives $$ \inn{f}{ZXZf} \ = \ \inn{Zf}{XZf} \ = \ \inn{g}{Xg} \ \le \ \inn{g}{Yg} \ = \ \inn{Zf}{YZf} \ = \ \inn{f}{ZYZf} \, . \eqno\darkbox $$
Next, say a self-adjoint matrix $J$ is \underbar{strictly positive} if $\inn{f}{Jf} > 0$ for all $f\in{\cal V}$. Since $\inn{f}{Jf} = \sum_i (a_i)^2 \lambda_i$ (see~Section~\ref{sec-prelim}), this is equivalent to $J$ having all eigenvalues positive.
\begin{lemma}\label{inverselemmaoneway} If $J$ and $K$ are strictly positive self-adjoint
operators on a finite vector space ${\cal V}$,
and $\inn{f}{Jf} \le \inn{f}{Kf}$ for all $f\in{\cal V}$,
then $\inn{f}{J^{-1}f} \ge \inn{f}{K^{-1}f}$ for all $f\in{\cal V}$. \end{lemma}
\newpartitle{Proof}
Note (see Section~\ref{sec-prelim}) that $K^{-1/2}$ and $K^{-1/2}JK^{-1/2}$ are self-adjoint, and have eigenvalues that are positive, since $J$ and $K$ are strictly positive. Applying Lemma~\ref{Zlemma} with $X=J$, $Y=K$, and $Z=K^{-1/2}$ then gives that for all $f\in{\cal V}$, $$ \inn{f}{K^{-1/2}JK^{-1/2}f} \ \le \ \inn{f}{K^{-1/2}KK^{-1/2}f} \ = \ \inn{f}{If} \ = \ \inn{f}{f} \, . $$ It follows that all the eigenvalues of $K^{-1/2}JK^{-1/2}$ are in $(0,1]$ (since $\inn{v}{Av}\le \inn{v}{v}$ and $Av=\lambda v$ with $v\ne0$ imply $\lambda\le 1$). Hence, its inverse $\big( K^{-1/2}JK^{-1/2} \big)^{-1}$ has eigenvalues all $\ge 1$, so $ \inn{f}{\big( K^{-1/2}JK^{-1/2} \big)^{-1}f} \ \ge \ \inn{f}{If} \, , $ and therefore $$ \inn{f}{If} \ \le \ \inn{f}{\big( K^{-1/2}JK^{-1/2} \big)^{-1}f} \ = \ \inn{f}{K^{1/2}J^{-1}K^{1/2}f} \, . $$ Then, applying Lemma~\ref{Zlemma} again with $X=I$, $Y=K^{1/2}J^{-1}K^{1/2}$, and $Z=K^{-1/2}$ gives $$ \inn{f}{K^{-1/2}IK^{-1/2}f} \ \le \ \inn{f}{K^{-1/2}\, \big(K^{1/2}J^{-1}K^{1/2}\big)\, K^{-1/2}f} \, . $$ That is, $ \inn{f}{K^{-1}f} \ \le \ \inn{f}{J^{-1}f}\ \ \mbox{for all $f \in {\cal V}$,} $ giving the result. \qed
\newpartitle{Remark} Lemma~\ref{inverselemmaoneway} can be partially proven more directly. If $f=\sum_i a_i v_i$,
\underbar{Jensen's Inequality}
gives $
\Big( \sum_i (a_i)^2 \lambda_i \Big)^{-1} \le
\sum_i (a_i)^2 \, (\lambda_i)^{-1} $, so we always have $
{1 \over \inn{f}{Jf}} \le \inn{f}{J^{-1}f} \, . $ Hence, if $\inn{f}{Jf} \le \inn{f}{Kf}$, then
$\inn{f}{J^{-1}f} \ge {1 \over \inn{f}{Kf}}$. If $f$ is an \underbar{eigenvector} of $K$, then $ \inn{f}{K^{-1}f} \ = \ {1 \over \inn{f}{Kf}} $,
so this shows directly that
$\inn{f}{J^{-1}f} \ge \inn{f}{K^{-1}f}$.
However, it is unclear how to extend this argument to other $f$. \qed
Applying Lemma~\ref{inverselemmaoneway} twice gives a (stronger) two-way equivalence:
\begin{lemma}\label{inverselemma} If $J$ and $K$ are strictly positive self-adjoint
operators on a finite vector space ${\cal V}$, then $\inn{f}{Jf} \le \inn{f}{Kf}$ for all $f\in{\cal V}$ \underbar{if and only if} $\inn{f}{J^{-1}f} \ge \inn{f}{K^{-1}f}$ for all $f\in{\cal V}$. \end{lemma}
\newpartitle{Proof} The forward implication is Lemma~\ref{inverselemmaoneway}. And, the reverse implication follows from Lemma~\ref{inverselemmaoneway} by replacing $J$ with $K^{-1}$ and replacing $K$ with $J^{-1}$. \qed
\iffalse
Applying Lemma~\ref{inverselemma} again with $J$ replaced by $K^{-1}$ and $K$ replaced by $J^{-1}$ gives:
\begin{lemma}\label{equivlemma} If $J$ and $K$ are strictly positive self-adjoint
operators on a finite vector space ${\cal V}$, then $\inn{f}{Jf} \le \inn{f}{Kf}$ for all $f$ \underbar{if and only if} $\inn{f}{J^{-1}f} \ge \inn{f}{K^{-1}f}$ for all $f$. \end{lemma}
\fi
Using Lemma~\ref{inverselemma}, we easily obtain:
\newpartitle{Proof of Lemma~\ref{effequivlemma}} Recall that we can restrict to $f\in L^2_0(\pi)$, so $\pi(f)=0$, and $f$ is orthogonal to the eigenvector corresponding to eigenvalue~1. On that restricted subspace, the eigenvalues of $P$ and $Q$ are contained in $[-1,1)$. Hence, the eigenvalues of $I\!-\!P$ and $I\!-\!Q$ are contained in $(0,2]$, and in particular are all strictly positive. So, $I\!-\!P$ and $I\!-\!Q$ are strictly positive self-adjoint operators.
Now, $\inn{f}{Pf} \le \inn{f}{Qf}$ for all $f \in L^2_0(\pi)$ is equivalent to $$\inn{f}{(I\!-\!P)f} = \inn{f}{f}-\inn{f}{Pf} \ \ge\ \inn{f}{f}-\inn{f}{Qf} = \inn{f}{(I\!-\!Q)f}\ \ \mbox{for all $f \in L^2_0(\pi)$}.$$ Then, by Lemma~\ref{inverselemma} with $J = I\!-\!Q$ and $K = I\!-\!P$, this is equivalent to $$\inn{f}{(I\!-\!P)^{-1}f}\ \le\ \inn{f}{(I\!-\!Q)^{-1}f}\ \ \mbox{for all $f \in L^2_0(\pi)$}.$$ Since $(I\!-\!P)^{-1} = P(I\!-\!P)^{-1} + (I-P)(I-P)^{-1} = P(I\!-\!P)^{-1} + I$ and similarly $(I\!-\!Q)^{-1} = Q(I\!-\!Q)^{-1} + I$, this latter is equivalent to $$\inn{f}{P(I\!-\!P)^{-1}f}\ \le\ \inn{f}{Q(I\!-\!Q)^{-1}f}\ \ \mbox{for all $f \in L^2_0(\pi)$},$$ which completes the proof. \qed
\newpartitle{Acknowledgements} We are very grateful to Heydar Radjavi for the proof of Lemma~\ref{inverselemmaoneway} herein, and thank Rajendra Bhatia, Gareth Roberts, Daniel Rosenthal, and Peter Rosenthal for several very helpful discussions about eigenvalues.
\end{document} | arXiv |
There are 30 people in my math class. 12 of them have cool dads, 15 of them have cool moms, and 9 of them have cool dads and cool moms. How many people have moms and dads who are both uncool?
We can solve this with a Venn diagram. First we notice that there are 9 people with both cool dads and cool moms.
[asy]
label("Cool Dad", (2,75));
label("Cool Mom", (80,75));
draw(Circle((30,45), 22));
draw(Circle((58, 45), 22));
label(scale(0.8)*"$9$", (44, 45));
//label(scale(0.8)*"$33$",(28,45));
//label(scale(0.8)*"$23$",(63,45));
//label(scale(0.8)*"$17$", (70, 15));
[/asy]
Since 12 people have cool dads and 9 of those have cool moms, too, $12-9=3$ of the people have cool dads and uncool moms. Likewise, $15-9=6$ people have cool moms and uncool dads.
[asy]
label("Cool Dad", (2,75));
label("Cool Mom", (80,75));
draw(Circle((30,45), 22));
draw(Circle((58, 45), 22));
label(scale(0.8)*"$9$", (44, 45));
label(scale(0.8)*"$3$",(28,45));
label(scale(0.8)*"$6$",(63,45));
//label(scale(0.8)*"$17$", (70, 15));
[/asy]
This means that $3+9+6=18$ people have at least one cool parent. That leaves $30-18=\boxed{12}$ sad people with a pair of uncool parents. | Math Dataset |
Coordination contracts for a dual-channel supply chain under capital constraints
JIMO Home
Utility maximization with habit formation of interaction
May 2021, 17(3): 1471-1483. doi: 10.3934/jimo.2020030
Finite-horizon optimal control of discrete-time linear systems with completely unknown dynamics using Q-learning
Jingang Zhao , and Chi Zhang
School of Automation, Beijing Institute of Technology, Beijing, 100081, China
* Corresponding author: Jingang Zhao
Received March 2019 Revised September 2019 Published May 2021 Early access February 2020
Fund Project: The first author is supported by International Graduate Exchange Program of Beijing Institute of Technology; This paper is supported by the National Natural Science Foundation of China grant 61673065
Figure(8)
This paper investigates finite-horizon optimal control problem of completely unknown discrete-time linear systems. The completely unknown here refers to that the system dynamics are unknown. Compared with infinite-horizon optimal control, the Riccati equation (RE) of finite-horizon optimal control is time-dependent and must meet certain terminal boundary constraints, which brings the greater challenges. Meanwhile, the completely unknown system dynamics have also caused additional challenges. The main innovation of this paper is the developed cyclic fixed-finite-horizon-based Q-learning algorithm to approximate the optimal control input without requiring the system dynamics. The developed algorithm main consists of two phases: the data collection phase over a fixed-finite-horizon and the parameters update phase. A least-squares method is used to correlate the two phases to obtain the optimal parameters by cyclic. Finally, simulation results are given to verify the effectiveness of the proposed cyclic fixed-finite-horizon-based Q-learning algorithm.
Keywords: Finite-horizon, Optimal control, Discrete-time linear systems, Completely unknown dynamics, Q-learning.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Jingang Zhao, Chi Zhang. Finite-horizon optimal control of discrete-time linear systems with completely unknown dynamics using Q-learning. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1471-1483. doi: 10.3934/jimo.2020030
A. Al-Tamimi, F. L. Lewis and M. Abu-Khalaf, Model-free Q-learning design for discrete-time zero-sum games with application to H-infinity control, Automatica, 43 (2007), 473-481. doi: 10.1016/j.automatica.2006.09.019. Google Scholar
D. P. Bertsekas and J. N. Tsitsiklis, Neuro-Dynamic Programming, 1$^st$ edition, Athena Scientific, Belmont, 1996. Google Scholar
D. P. Bertsekas, Value and policy iterations in optimal control and adaptive dynamic programming, IEEE Transactions on Neural Networks and Learning Systems, 28 (2017), 500-509. doi: 10.1109/TNNLS.2015.2503980. Google Scholar
X. S. Chen, X. Li and F. H. Yi, Optimal stopping investment with non-smooth utility over an infinite time horizon, Journal of Industrial & Management Optimization, 15 (2019), 81-96. Google Scholar
Y. F. Chen and Y. G. Zhu, Indifinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems, Journal of Industrial & Management Optimization, 14 (2018), 913-930. Google Scholar
T. Cheng, F. L. Lewis and M. Abu-Khalaf, A neural network solution for fixed-final time optimal control of nonlinear systems, Automatica, 43 (2007), 482-490. doi: 10.1016/j.automatica.2006.09.021. Google Scholar
M. G. Gan, J. G. Zhao and C. Zhang, Extended adaptive optimal control of linear systems with unknown dynamics using adaptive dynamic programming, Asian J Control, (2019), 1–10. doi: 10.1002/asjc.2243. Google Scholar
W. N. Gao, Y. Jiang, Z. P. Jiang and Y. T. Chai, Output-feedback adaptive optimal control of interconnected systems based on robust adaptive dynamic programming, Automatica, 72 (2016), 37-45. doi: 10.1016/j.automatica.2016.05.008. Google Scholar
A. Heydari and S. N. Balakrishnan, Finite-horizon control-constrained nonlinear optimal control using single network adaptvie critics, IEEE Transactions on Neural Networks and Learning Systems, 24 (2013), 145-157. Google Scholar
Y. Jiang and Z. P. Jiang, Computational adaptive optimal control for continuous-time linear systems with completely unknown dynamics, Automatica, 48 (2012), 2699-2704. doi: 10.1016/j.automatica.2012.06.096. Google Scholar
Y. Jiang and Z. P. Jiang, Global adaptive dynamic programming for continuous-time nonlinear systems, IEE Trans on Automatic Control, 60 (2015), 2917-2929. doi: 10.1109/TAC.2015.2414811. Google Scholar
W. K. Jong, J. P. Byung, Y. Haeun, H. L. Jay and M. L. Jong, Deep reinforcement learning based finite-horizon optimal tracking control for nonlinear systems, in International Federation Automatic Control, (2018), 257–262. Google Scholar
R. Kamalapurkar, P. Walters and W. E. Dixon, Model-based reinforcement learning for approximate optimal control, in Reinforcement Learning for Optimal Feedback Control, Communications and Control Engineering. Springer, Cham, (2018), 99–148. Google Scholar
J. Y. Lee, B. P. Jin and Y. H. Chio, Integral Q-learning and explorized policy iteration for adaptive optimal control of continuous-time linear systems, Automatica, 48 (2012), 2850-2859. doi: 10.1016/j.automatica.2012.06.008. Google Scholar
F. L. Lewis and D. R. Liu, Reinforcement learning and approximate dynamic programming for feedback control, IEEE Circuits and Systems Magazine, 9 (2015), 32-50. Google Scholar
F. L. Levis, D. L. Vrabie and V. L. Syrmos, Optimal Control, 3$^th$ edition, John Wiley and Sons, Hoboken, 2015. Google Scholar
J. N. Li, T. Y. Chai, F. L. Lewis, Z. T. Ding and Y. Jiang, Off-policy interleaved Q-learning: optimal control for affine nonlinear discrete-time systems, IEEE Transactions on Neural Networks and Learning Systems, 30 (2019), 1308-1320. doi: 10.1109/TNNLS.2018.2861945. Google Scholar
J. N. Li, T. Y. Chai, F. L. Lewis, J. L. Fan, Z. T. Ding and J. L. Ding, Off-policy Q-learning: Set-point design for optimizing dual-rate rougher flotation operational processes, IEEE Transactions on Industrial Electronics, 65 (2018), 4092-4102. doi: 10.1109/TIE.2017.2760245. Google Scholar
X. X. Li, Z. H. Peng, L. Liang and W. Z. Zha, Policy iteration based Q-learning for linear nonzero-sum quadratic differential games, Science China(Information Sciences), 62 (2019), 195-213. Google Scholar
Q. Lin, Q. L. Wei and D. R. Liu, A novel optimal tracking control scheme for a class of discrete-time nonlinear systems using generalised policy iteration adaptive dynamic programming algorithm, International Journal of Systems Science, 48 (2017), 525-534. doi: 10.1080/00207721.2016.1188177. Google Scholar
H. L. Liu and Q. X. Zhu, New forms of Riccati equations and the further results of the optimal control for linear discrete-time systems, International Journal of Control, Automation, and Systems, 12 (2014), 1160-1166. doi: 10.1007/s12555-013-0202-x. Google Scholar
B. Luo, D. R. Liu, T. W. Huang and D. Wang, Data-based approximate policy iteration for nonlinear continuous-time optimal control design, Automatica, 50 (2014), 3281-3290. doi: 10.1016/j.automatica.2014.10.056. Google Scholar
B. Luo, D. R. Liu, T. W. Huang and D. Wang, Model-free optimal tracking control via critic-only Q-learning, IEEE Transactions on Neural Networks and Learning Systems, 27 (2016), 2134-2144. doi: 10.1109/TNNLS.2016.2585520. Google Scholar
B. Luo, D. R. Liu and H. N. Wu, Adaptive constrained optimal control design for data-based nonlinear discrete-time systems with critic-only structure, IEEE Transactions on Neural Networks and Learning Systems, 29 (2018), 2099-2111. doi: 10.1109/TNNLS.2017.2751018. Google Scholar
B. Luo, D. R. Liu, H. N. Wu, D. Wang and F. L. Lewis, Policy gradient adaptive dynamic programming for data-based optimal control, IEEE Transactions on Cybernetics, 47 (2017), 3341-3354. doi: 10.1109/TCYB.2016.2623859. Google Scholar
B. Luo, H. N. Wu and T. W. Huang, Optimal output regulation for model-free quanser helicopter with multistep Q-learning, IEEE Transactions on Industrial Electronics, 65 (2018), 4953-4961. doi: 10.1109/TIE.2017.2772162. Google Scholar
B. Luo, Y. Yang and D. R. Liu, Adaptive Q-Learning for data-based optimal output regulation with experience replay, IEEE Transactions on Cybernetics, 48 (2018), 3337-3348. Google Scholar
Y. F. Lv, N. Jing, Q. M. Yang, X. Wu and Y. Guo, Online adaptive optimal control for continuous-time nonlinear systems with completely unknown dynamics, International Journal of Control, 89 (2016), 99-112. doi: 10.1080/00207179.2015.1060362. Google Scholar
W. B. Powell, Approximate Dynamic Programming: Solving the Curses of Dimensionality, 2$^nd$ edition, Wiley Princeton, New Jersey, 2011. Google Scholar
A. Sahoo, H. Xu and S. Jagannathan, Approximate optimal control of affine nonlinear continuous-time systems using event-sampled neurodynamic programming, IEEE Transactions on Neural Networks and Learning Systems, 28 (2017), 639-652. Google Scholar
K. G. Vamvoudakis, Q-learning for continuous-time linear systems: A model-free infinite horizon optimal control approach, Systems and Control Letters, 100 (2017), 14-20. Google Scholar
K. G. Vamvoudakis and F. L. Lewis, Online actor-critc algorithm to solve the continuous-time infinite horizon optimal control problem, Automatica, 46 (2010), 878-888. Google Scholar
D. Wang, D. R. Liu and Q. L. Wei, Finite-horizon neuro-optimal tracking control for a class of discrete-time nonlinear systems using adaptive dynamic programming approach, Neurocomputing, 78 (2012), 14-22. Google Scholar
F. Y. Wang, N. Jin, D. R. Liu and Q. L. Wei, Adaptive dynamic programming for finite-horizon optimal control of discrete-time nonlinear systems with $\varepsilon -$ error bound, IEEE Transactions on Neural Networks, 22 (2011), 24-36. Google Scholar
Q. L. Wei and D. R. Liu, A novel policy iteration based deterministic Q-learning for discrete-time nonlinear systems, Science China(Information Sciences), 58 (2015), 1-15. Google Scholar
Q. L. Wei, D. R. Liu and H. Q. Lin, Value iteration adaptive dynamic programming for optimal control of discrete-time nonlinear systems, IEE Trans Cybern, 46 (2016), 840-853. Google Scholar
Y. Wu, Z. H. Yuan and Y. P. Wu, Optimal tracking control for networked control systems with random time delays and packet dropouts, Journal of Industrial & Management Optimization, 11 (2015), 1343-1354. Google Scholar
H. G. Zhang, J. He, Y. H. Luo and G. Y. Xiao, Data-driven optimal consensus control for discrete-time multi-agent systems with unknown dynamics using reinforcement learning method, IEEE Transactions on Industrial Electronics, 64 (2017), 4091-4100. Google Scholar
H. G. Zhang, Q. L. Wei and Y. H. Luo, A novel infinite-time optimal tracking control scheme for a class of discrete-time nonlinear systems via the greedy HDP iteration algorithm, IEEE Transactions on Systems Man and Cybernetics Part B, 38 (2008), 937-942. Google Scholar
Q. C. Zhang, D. B. Zhao and D. Wang, Event-based robust control for uncertain nonlinear systems using adaptive dynamic programming, IEEE Transactions on Neural Networks and Learning Systems, 29 (2018), 37-50. Google Scholar
J. G. Zhao, M. G. Gan and C. Zhang, Event-triggered ${{H}_{\infty }}$ optimal control for continuous-time nonlinear systems using neurodynamic programming, Neurocomputing, 360 (2019), 14-24. Google Scholar
Q. M. Zhao, Finite-horizon Optimal Control of Linear and a Class of Nonlinear Systems, Ph.D thesis, Missouri University of Science and Technology. 2013. Google Scholar
Q. M. Zhao, X. Hao and J. Sarangapani, Finite-horizon near optimal adaptive control of uncertain linear discrete-time systems, Optimal Control Applications and Methods, 36 (2016), 853-872. Google Scholar
X. N. Zhong, H. B. He, D. Wang and Z. Ni, Model-free adaptive optimal control for unknown nonlinear zero-sum differnetial game, IEEE Transactions on Cybernetics, 48 (2018), 1633-1646. Google Scholar
Q. X. Zhu and G. M. Xie, Finite-horizon optimal control of discrete-time switched linear systems, Mathematical Problems in Engineering, 2012 (2012), 1-12. Google Scholar
Figure 1. The flow chart of Algorithm 1
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Figure 2. Initial system state $x_0$ are randomly selected from a compact set $\Omega : = \{-1\le {{x}_{1}},{{x}_{2}}\le 1\}$
Figure 3. The convergence process of $\hat W$
Figure 4. The trajectories of system states
Figure 5. The optimal control input $u$
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\begin{document}
\begin{abstract} In this article, we show some new second main theorems for the mappings and moving hyperplanes of ${\mathbf{P}}^n({\mathbf{C}})$ with truncated counting functions. Our results are improvements of recent previous second main theorems for moving hyperplanes with the truncated (to level $n$) counting functions.
As their application, we prove a unicity theorem for meromorphic mappings sharing moving hyperplanes. \end{abstract}
\def\empty{\empty} \footnotetext{ 2010 Mathematics Subject Classification: Primary 32H30, 32A22; Secondary 30D35.\\ \hskip8pt Key words and phrases: Nevanlinna, second main theorem, meromorphic mapping, moving hyperplane.}
\maketitle
\section{Introduction}
\noindent The theory of the Nevanlinna's second main theorem for meromorphic mappings of ${\mathbf{C}}^m$ into the complex projective space ${\mathbf{P}}^n({\mathbf{C}})$ intersecting a finite set of fixed hyperplanes or moving hyperplanes in ${\mathbf{P}}^n({\mathbf{C}})$ was started about 70 years ago and has grown into a huge theory. For the case of fixed hyperplanes, maybe, the second main theorem given by Cartan-Nochka is the best possible. Unfortunately, so far there has been a few second main theorems with truncated counting functions for moving hyperplanes. Moreover, almost of them are not sharp.
We state here some recent results on the second main theorems for moving hyperplanes with truncated counting functions.
Let $\{a_i\}_{i=1}^q $ be meromorphic mappings of ${\mathbf{C}}^m$ into the dual space ${\mathbf{P}}^n({\mathbf{C}})^*$ in general position. For the case of nondegenerate meromorphic mappings, the second main theorem with truncated (to level $n$) counting functions states that.
\noindent \textbf{Theorem A} (see \cite[Theorem 2.3]{MR} and \cite[Theorem 3.1]{TQ05}). {\it Let $f :{\mathbf{C}}^m \to {\mathbf{P}}^n({\mathbf{C}})$ be a meromorphic mapping. Let $\{a_i\}_{i=1}^q \ (q\ge n+2)$ be meromorphic mappings of ${\mathbf{C}}^m$ into ${\mathbf{P}}^n({\mathbf{C}})^*$ in general position such that $f$ is linearly nondegenerate over $\mathcal {R}(\{a_i\}_{i=1}^{q}).$ Then
$$|| \ \dfrac {q}{n+2} T_f(r) \le \sum_{i=1}^q N_{(f,a_i)}^{[n]}(r) + o(T_f(r)) + O(\max_{1\le i \le q}T_{a_i}(r)).$$ }
We note that, Theorem A is still the best second main theorem with truncated counting functions for nondegenerate meromorphic mappings and moving hyperplanes available at present. In the case of degenerate meromorphic mappings, the second main theorem for moving hyperplanes with counting function truncated to level $n$ was first given by M. Ru-J. Wang \cite{RW} in 2004. After that in 2008, D. D. Thai-S. D. Quang \cite{TQ08} improved the result of M. Ru-J. Wang by proved the following second main theorem.
\noindent \textbf{Theorem B} (see \cite[Corollary 1]{TQ08}). {\it Let $f:{\mathbf{C}}^m\to{\mathbf{P}}^n({\mathbf{C}})$ be a meromorphic mapping. Let $\{a_i\}_{i=1}^q$ $(q\ge 2n+1)$ be $q$ meromorphic mappings of ${\mathbf{C}}^m$ into ${\mathbf{P}}^n({\mathbf{C}})^*$ in general position such that $(f,a_i)\not\equiv 0\ (1\le i\le {q}).$ Then
$$\bigl |\bigl |\quad\dfrac{q}{2n+1}\cdot T_f(r)\le \sum_{i=1}^{q}N^{[n]}_{(f,a_i)}(r)+O\bigl(\max_{1\le i\le {q}}T_{a_i}(r)\bigl)+O\bigl(\log^{+}T_f(r)\bigl).$$ }
These results play very essential roles in almost all researches on truncated multiplicity problems of meromorphic mappings with moving hyperplanes. Hovewer, in our opinion, the above mentioned results of these authors are still weak.
Our main purpose of the present paper is to show a stronger second main theorem of meromorphic mappings from ${\mathbf{C}}^m$ into ${\mathbf{P}}^n({\mathbf{C}})$ for moving targets. Namely, we will prove the following. \begin{theorem}\label{1.1} Let $f :{\mathbf{C}}^m \to {\mathbf{P}}^n({\mathbf{C}})$ be a meromorphic mapping. Let $\{a_i\}_{i=1}^q \ (q\ge 2n-k+2)$ be meromorphic mappings of ${\mathbf{C}}^m$ into ${\mathbf{P}}^n({\mathbf{C}})^*$ in general position such that $(f,a_i)\not\equiv 0\ (1\le i\le q),$ where $k+1=\mathrm{rank}_{\mathcal R\{a_i\}}(f)$. Then the following assertions hold: \begin{align*}
&\mathrm{(a)}\ || \ \dfrac {q}{2n-k+2}T_f(r) \le \sum_{i=1}^q N_{(f,a_i)}^{[k]}(r) + o(T_f(r)) + O(\max_{1\le i \le q}T_{a_i}(r)),\\
&\mathrm{(b)}\ || \ \dfrac {q-n+2k-1}{n+k+1}T_f(r) \le \sum_{i=1}^q N_{(f,a_i)}^{[k]}(r) + o(T_f(r)) + O(\max_{1\le i \le q}T_{a_i}(r)). \end{align*} \end{theorem} We may see that Theorem \ref{1.1}(a) is a generalization of Theorem A and also is an improvement of Theorem B. Theorem \ref{1.1}(b) is really stronger than Theorem B.
\noindent \textit{Remark.}
1) If $k\ge\dfrac{n+1}{2}$ then Theorem \ref{1.1}(a) is stronger than Theorem \ref{1.1}(b). Otherwise, if $k<\dfrac{n+1}{2}$ then Theorem \ref{1.1}(b) is stronger than Theorem \ref{1.1}(a).
2) If $k=0$ then $f$ is constant map, and hence $T_f(r)=0.$
3) Setting $t=\frac{2n-k+2}{3n+3}$ and $\lambda =\frac{n+k+1}{3n+3},$ we have $t+\lambda =1$. Thus, for all $1\le k\le n$ we have \begin{align*} \max\biggl \{\dfrac{q}{2n-k+2},\dfrac{q-n+2k-1}{n+k+1}\biggl\}&\ge \dfrac{q}{2n-k+2}\cdot t+\dfrac{q-n+2k-1}{n+k+1}\cdot\lambda\\ \\ &=\dfrac{2q-n+2k-1}{3n+3}\ge\dfrac{2q-n+1}{3n+3}. \end{align*}
4) If $k\ge 1$, we have the following estimates: \begin{itemize} \item $\min_{\frac{n+1}{2}\le k\le n, (k\in\mathbf{Z})}\left (\dfrac{q}{2n-k+2}\right )\ge\dfrac{q}{2n-\frac{n+1}{2}+2}=\dfrac{2q}{3(n+1)}$. \item $\min_{1\le k\le \frac{n+1}{2}, (k\in\mathbf{Z})}\left (\dfrac{q-n+2k-1}{n+k+1}\right )=\min_{1\le k\le \frac{n+1}{2}, (k\in\mathbf{Z})}\left (\dfrac{q-3n-3}{n+k+1}+2\right )$
\noindent \hspace{185pt}$\ge\begin{cases} \dfrac{2q}{3(n+1)}&\text{ if }q\ge 3n+3 \\ \dfrac{q-n+1}{n+2}&\text{ if }q< 3n+3 \end{cases}$ \end{itemize} Thus $$\min_{1\le k\le n}\biggl \{\max\bigl \{\dfrac{q}{2n-k+2},\dfrac{q-n+2k-1}{n+k+1}\bigl\}\biggl \} \ge\begin{cases} \dfrac{2q}{3(n+1)}&\text{ if }q\ge 3n+3 \\ \dfrac{q-n+1}{n+2}&\text{ if }q< 3n+3. \end{cases}$$ Therefore, from Theorem \ref{1.1} and Remark (1-4) we have the following corollary. \begin{corollary}\label{1.2} Let $f :{\mathbf{C}}^m \to {\mathbf{P}}^n({\mathbf{C}})$ be a meromorphic mapping. Let $\{a_i\}_{i=1}^q \ (q\ge 2n+1)$ be meromorphic mappings of ${\mathbf{C}}^m$ into ${\mathbf{P}}^n({\mathbf{C}})^*$ in general position such that $(f,a_i)\not\equiv 0\ (1\le i\le q).$
$\mathrm{(a)}$ Then we have
$$|| \dfrac{2q-n+1}{3(n+1)}T_f(r) \le \sum_{i=1}^q N_{(f,a_i)}^{[n]}(r) + o(T_f(r)) + O(\max_{1\le i \le q}T_{a_i}(r)).$$
$\mathrm{(b)}$ If $q\ge 3n+3$ then
$$|| \dfrac{2q}{3(n+1)}T_f(r) \le \sum_{i=1}^q N_{(f,a_i)}^{[n]}(r) + o(T_f(r)) + O(\max_{1\le i \le q}T_{a_i}(r)).$$
$\mathrm{(c)}$ If $q< 3n+3$ then
$$|| \dfrac{q-n+1}{n+2}T_f(r) \le \sum_{i=1}^q N_{(f,a_i)}^{[n]}(r) + o(T_f(r)) + O(\max_{1\le i \le q}T_{a_i}(r)).$$ \end{corollary}
As applications of these second main theorems, in the last section we will prove a unicity theorem for meromorphic mappings sharing moving hyperplanes regardless of multiplicities. To state our main result, we give the following definition.
Let $f:{\mathbf{C}}^m \to {\mathbf{P}}^n({\mathbf{C}})$ be a meromorphic mapping. Let $k$ be a positive integer or maybe $+\infty$. Let $\{a_i\}_{i=1}^{q}$ be ``slowly'' (with respect to $f$) moving hyperplanes in ${\mathbf{P}}^n({\mathbf{C}})$ in general position such that $$\dim\ \{z\in{\mathbf{C}}^m: (f,a_i)(z)\cdot (f,a_j)(z)= 0\} \leq m-2\quad (1\le i<j\le q).$$
Consider the set $\mathcal F(f,\{a_i\}_{i=1}^q,k)$ of all meromorphic maps $g: {\mathbf{C}}^m \to {\mathbf{P}}^n({\mathbf{C}})$ satisfying the following two conditions:
(a) $\min\{\nu_{(f,a_i)}(z),k\}=\min\{\nu_{(g,a_i)}(z),k\}\quad (1 \le i \le q),$ for all $z\in{\mathbf{C}}^m$,
(b) $f(z) = g(z)$ for all $z\in\bigcup_{i=1}^{q}\mathrm{Zero} (f,a_i)$.
We wil prove the following
\begin{theorem}\label{1.3} Let $f:{\mathbf{C}}^m \to {\mathbf{P}}^n({\mathbf{C}})$ be a meromorphic mapping. Let $\{a_i\}_{i=1}^{q}$ be slowly (with respect to $f$) moving hyperplanes in ${\mathbf{P}}^n({\mathbf{C}})$ in general position such that $$\dim\ \{z\in{\mathbf{C}}^m: (f,a_i)(z)\cdot (f,a_j)(z)= 0\} \leq m-2\quad (1\le i<j\le q).$$ Then the following assertions hold:
a) If $q>\frac{9n^2+9n+4}{4}$ then $\sharp\ \mathcal F(f,\{a_i\}_{i=1}^q,1)\le 2,$
b) If $q>3n^2+n+2$ then $\sharp\ \mathcal F(f,\{a_i\}_{i=1}^q,1)=1.$ \end{theorem}
{\bf Acknowledgements.} This work was done during a stay of the author at Vietnam Institute for Advanced Study in Mathematics. He would like to thank the institute for their support.
\section{Basic notions and auxiliary results from Nevanlinna theory}
\noindent {\bf (a)} Counting function of divisor.
For $z = (z_1,\dots,z_m) \in {\mathbf{C}}^m$, we set
$\Vert z \Vert = \Big(\sum\limits_{j=1}^m |z_j|^2\Big)^{1/2}$ and define \begin{align*} B(r) &= \{ z \in {\mathbf{C}}^m ; \Vert z \Vert < r\},\quad S(r) = \{ z \in {\mathbf{C}}^m ; \Vert z \Vert = r\},\\ d^c & = \dfrac{\sqrt{-1}}{4\pi}(\overline \partial - \partial),\quad \sigma = \big(dd^c \Vert z\Vert^2\big)^{m-1},\\ \eta &= d^c \text{log}\Vert z\Vert^2 \land \big(dd^c\text{log}\Vert z \Vert\big)^{m-1}. \end{align*}
Thoughout this paper, we denote by $\mathcal M$ the set of all meromorphic functions on ${\mathbf{C}}^m$. A divisor $E$ on ${\mathbf{C}}^m$ is given by a formal sum $E=\sum\mu_{\nu}X_{\nu}$, where $\{X_\nu\}$ is a locally family of distinct irreducible analytic hypersurfaces in ${\mathbf{C}}^m$ and $\mu_{\nu}\in \mathbf{Z}$. We define the support of the divisor $E$ by setting $\mathrm{Supp}\, (E)=\cup_{\nu\ne 0} X_\nu$. Sometimes, we identify the divisor $E$ with a function $E(z)$ from ${\mathbf{C}}^m$ into $\mathbf{Z}$ defined by $E(z):=\sum_{X_{\nu}\ni z}\mu_\nu$.
Let $k$ be a positive integer or $+\infty$. We define the truncated divisor $E^{[k]}$ by $$ E^{[k]}:= \sum_{\nu}\min\{\mu_\nu, k \}X_\nu , $$ and the {\it truncated counting function to level $k$} of $E$ by \begin{align*} N^{[k]}(r,E) := \int\limits_1^r \frac{n^{[k]}(t,E)}{t^{2m-1}}dt\quad (1 < r < +\infty), \end{align*} where \begin{align*} n^{[k]}(t,E): = \begin{cases} \int\limits_{\mathrm{Supp}\, (E) \cap B(t)} E^{[k]}\sigma &\text{ if } m \geq 2,\\
\sum_{|z| \le t} E^{[k]}(z)&\text{ if } m = 1. \end{cases} \end{align*} We omit the character $^{[k]}$ if $k=+\infty$.
For an analytic hypersurface $E$ of ${\mathbf{C}}^m$, we may consider it as a reduced divisor and denote by $N(r,E)$ its counting function.
Let $\varphi$ be a nonzero meromorphic function on ${\mathbf{C}}^m$. We denote by $\nu^0_{\varphi}$ (resp. $\nu^{\infty}_{\varphi}$) the divisor of zeros (resp. divisor of poles) of $\varphi$. The divisor of $\varphi$ is defined by $$\nu_{\varphi}=\nu^0_{\varphi}-\nu^{\infty}_{\varphi}.$$
We have the following Jensen's formula: \begin{align*} N(r,\nu^0_{\varphi}) - N(r,\nu^{\infty}_{\varphi}) =
\int\limits_{S(r)} \text{log}|\varphi| \eta
- \int\limits_{S(1)} \text{log}|\varphi| \eta . \end{align*} For convenience, we will write $N_{\varphi}(r)$ and $N^{[k]}_{\varphi}(r)$ for $N(r,\nu^0_{\varphi})$ and $N^{[k]}(r,\nu^0_{\varphi})$, respectively.
\noindent {\bf (b)} The first main theorem.
Let $f$ be a meromorphic mapping of ${\mathbf{C}}^m$ into ${\mathbf{P}}^n({\mathbf{C}})$. For arbitrary fixed homogeneous coordinates $(w_0: \cdots : w_n)$ of ${\mathbf{P}}^n({\mathbf{C}} )$, we take a reduced representation $f = (f_0 : \cdots : f_n)$, which means that each $f_i$ is holomorphic function on ${\mathbf{C}}^m$ and $f(z) = (f_0(z) : \cdots : f_n(z))$ outside the analytic set $I(f):=\{ z ; f_0(z) = \cdots = f_n(z) = 0\}$ of codimension at least $2$.
Denote by $\Omega$ the Fubini Study form of ${\mathbf{P}}^n({\mathbf{C}} )$. The characteristic function of $f$ (with respect to $\Omega$) is defined by \begin{align*} T_f(r) := \int_1^r\dfrac{dt}{t^{2m-1}}\int_{B(t)}f^*\Omega\wedge\sigma ,\quad\quad 1 < r < +\infty. \end{align*} By Jensen's formula we have \begin{align*}
T_f(r)=\int_{S(r)}\log ||f||\eta +O(1), \end{align*}
where $\Vert f \Vert = \max \{ |f_0|,\dots,|f_n|\}$.
Let $a$ be a meromorphic mapping of ${\mathbf{C}}^m$ into ${\mathbf{P}}^n({\mathbf{C}})^*$ with reduced representation $a = (a_0 : \dots : a_n)$. We define
$$m_{f,a}(r)=\int\limits_{S(r)} \text{log}\dfrac {||f||\cdot ||a||}{|(f,a)|}\eta -
\int\limits_{S(1)}\text{log}\dfrac {||f||\cdot ||a||}{|(f,a)|}\eta,$$
where $\Vert a \Vert = \big(|a_0|^2 + \dots + |a_n|^2\big)^{1/2}$ and $(f,a)=\sum_{i=0}^nf_i\cdot a_i.$
Let $f$ and $a$ be as above. If $(f,a)\not \equiv 0$, then the first main theorem for moving hyperplaness in value distribution theory states $$T_f(r)+T_a(r)=m_{f,a}(r)+N_{(f,a)}(r)+O(1)\ (r>1).$$
For a meromorphic function $\varphi$ on ${\mathbf{C}}^m$, the proximity function $m(r,\varphi)$ is defined by
$$ m(r,\varphi) = \int\limits_{S(r)} \log^+ |\varphi| \eta , $$ where $\log^+ x = \max \big\{ \log x, 0\big\}$ for $x \geqslant 0$. The Nevanlinna's characteristic function is defined by $$T(r, \varphi ) = N(r, \nu^{\infty}_\varphi) + m(r,\varphi ).$$ We regard $\varphi$ as a meromorphic mapping of ${\mathbf{C}}^m$ into ${\mathbf{P}}^1({\mathbf{C}} )^*$, there is a fact that $$ T_\varphi (r)=T(r,\varphi )+O (1). $$ {\bf (c)} Lemma on logarithmic derivative.
As usual, by the notation $``|| \ P"$ we mean the assertion $P$ holds for all $r \in [0,\infty)$ excluding a Borel subset $E$ of the interval $[0,\infty)$ with $\int_E dr<\infty$. Denote by $\mathbf{Z}_+$ the set of all nonnegative integers. The lemma on logarithmic derivative in Nevanlinna theorey is stated as follows. \begin{lem}[{see \cite[Lemma 3.11]{Shi}}]\label{2.1} Let $f$ be a nonzero meromorphic function on ${\mathbf{C}}^m.$ Then
$$\biggl|\biggl|\quad m\biggl(r,\dfrac{\mathcal{D}^\alpha (f)}{f}\biggl)=O(\log^+T_f(r))\ (\alpha\in \mathbf{Z}^m_+).$$ \end{lem}
\noindent {\bf (d)} Family of moving hyperplanes.
We assume that thoughout this paper, the homogeneous coordinates of ${\mathbf{P}}^n({\mathbf{C}})$ is chosen so that for each given meromorphic mapping $a=(a_0:\cdots :a_n)$ of ${\mathbf{C}}^m$ into ${\mathbf{P}}^n({\mathbf{C}})^*$ then $a_{0}\not\equiv 0$. We set $$ \tilde a_i=\dfrac{a_i}{a_0}\text{ and }\tilde a=(\tilde a_0:\tilde a_1:\cdots:\tilde a_n).$$ Let $f:{\mathbf{C}}^m\rightarrow{\mathbf{P}}^n({\mathbf{C}})$ be a meromorphic mapping with the reduced representation $f=(f_0:\cdots :f_n).$ We put $(f,a):=\sum_{i=0}^{n}f_ia_{i}$ and $(f,\tilde a):=\sum_{i=0}^{n}f_i\tilde a_{i}.$
Let $\{a_i\}_{i=1}^q$ be $q$ meromorphic mappings of ${\mathbf{C}}^m$ into ${\mathbf{P}}^n({\mathbf{C}})^*$ with reduced representations $a_i=(a_{i0}:\cdots :a_{in})\ (1\le i\le q).$ We denote by $\mathcal R(\{a_i\})$ (for brevity we will write $\mathcal R$ if there is no confusion) the smallest subfield of $\mathcal M$ which contains ${\mathbf{C}}$ and all ${a_{i_j}}/{a_{i_k}}$ with $a_{i_k}\not\equiv 0.$
\begin{definition} The family $\{a_i\}_{i=1}^q$ is said to be in general position if $\dim (\{a_{i_0},\ldots ,a_{i_n}\})_{\mathcal {M}}=n+1$ for any $1\le i_0\le\cdots\le i_n\le q$, where $(\{a_{i_0},\ldots ,a_{i_n}\})_{\mathcal {M}}$ is the linear span of $\{a_{i_0},\ldots ,a_{i_N}\}$ over the field $\mathcal{ M}.$ \end{definition}
\begin{definition} A subset $\mathcal {L}$ of $\mathcal {M}$ (or $\mathcal {M}^{n+1}$) is said to be minimal over the field $\mathcal R$ if it is linearly dependent over $\mathcal {R}$ and each proper subset of $\mathcal L$ is linearly independent over $\mathcal {R}.$ \end{definition}
Repeating the argument in (\cite[Proposition 4.5]{Fu}), we have the following: \begin{proposition}[{see \cite[Proposition 4.5]{Fu}}]\label{2.2} Let $\Phi_0,\ldots,\Phi_k$ be meromorphic functions on ${\mathbf{C}}^m$ such that $\{\Phi_0,\ldots,\Phi_k\}$ are linearly independent over ${\mathbf{C}}.$
Then there exists an admissible set $\{\alpha_i=(\alpha_{i1},\ldots,\alpha_{im})\}_{i=0}^k \subset \mathbf{Z}^m_+$ with $|\alpha_i|=\sum_{j=1}^{n}|\alpha_{ij}|\le k \ (0\le i \le k)$ such that the following are satisfied:
(i)\ $\{{\mathcal D}^{\alpha_i}\Phi_0,\ldots,{\mathcal D}^{\alpha_i}\Phi_k\}_{i=0}^{k}$ is linearly independent over $\mathcal M,$\ i.e, \ $\det{({\mathcal D}^{\alpha_i}\Phi_j)}\not\equiv 0.$
(ii) $\det \bigl({\mathcal D}^{\alpha_i}(h\Phi_j)\bigl)=h^{k+1}\det \bigl({\mathcal D}^{\alpha_i}\Phi_j\bigl)$ for any nonzero meromorphic function $h$ on ${\mathbf{C}}^m.$ \end{proposition}
\section{Proof of Theorem \ref{1.1}}
In order to prove Theorem \ref{1.1} we need the following. \begin{lem}\label{3.1} Let $f:{\mathbf{C}}^m\rightarrow{\mathbf{P}}^n({\mathbf{C}})$ be a meromorphic mapping. Let $\{a_i\}_{i=1}^q$ $(q\ge n+1)$ be $q$ meromorphic mappings of ${\mathbf{C}}^m$ into ${\mathbf{P}}^n({\mathbf{C}})^*$ in general position. Assume that there exists a partition $\{1,\ldots,q\}=I_1\cup I_2\cdots\cup I_l$ satisfying:
$\mathrm{(i)}$ \ $\{(f,\tilde a_i)\}_{i\in I_1}$ is minimal over $\mathcal R$, and $\{(f,\tilde a_i)\}_{i\in I_t}$ is linearly independent over $\mathcal {R}\ (2\le t \le l), $
$\mathrm{(ii)}$ \ For any $2\le t\le l,i\in I_t,$ there exist meromorphic functions $c_i\in \mathcal {R}\setminus\{0\}$ such that $$\sum_{i\in I_t}c_i(f,\tilde a_i)\in \biggl(\bigcup_{j=1}^{t-1}\bigcup_{i\in I_j}(f,\tilde a_i) \biggl)_{\mathcal {R}}.$$ Then we have $$ T_f(r)\le\sum_{i=1}^qN^{[k]}_{(f,a_i)}+ o(T_f(r)) + O(\max_{1\le i \le q}T_{a_i}(r)),$$ where $k+1=\mathrm{rank}_{\mathcal R}(f)$. \end{lem} \textbf{Proof.}\ Let $f=(f_0:\cdots :f_n)$ be a reduced representation of $f$. By changing the homogeneous coordinate system of ${\mathbf{P}}^n({\mathbf{C}})$ if necessary, we may assume that $f_0\not\equiv 0.$ Without loss of generality, we may assume that $I_1=\{1,\ldots.,k_1\}$ and $$I_t=\{k_{t-1}+1,\ldots, k_t\}\ (2\le t \le l),\text{ where }1=k_0<\cdots< k_l=q.$$
Since $\{(f,\tilde a_i)\}_{i\in I_1}$ is minimal over $\mathcal R$, there exist $c_{1i}\in\mathcal {R}\setminus \{0\}$ such that $$\sum_{i=1}^{k_1}c_{1i}\cdot (f, \tilde a_i)=0.$$ Define $c_{1i}=0$ for all $i>k_1.$ Then $$\sum_{i=1}^{k_l} c_{1i}\cdot (f,\tilde a_i)=0.$$
Because ${\{c_{1i}(f,\tilde a_i)\}}_{i=k_0+1}^{k_1}$ is linearly independent over $\mathcal R,$ Lemma \ref{2.2} yields that there exists an admissible set $\{\alpha_{1(k_0+1)},\ldots,\alpha_{1k_1}\}\subset \mathbf{Z}^m_+$ \ $(|\alpha_{1i}|\le k_1-k_0-1\le \mathrm{rank}_{\mathcal R}f-1=k)$ such that the matrix $$\ A_1=\left (\mathcal {D}^{\alpha_{1i}}(c_{1j}(f,\tilde a_j));k_0+1\le i,j\le k_1 \right)$$ has nonzero determinant.
Now consider $t\ge 2.$ By constructing the set $I_t$, there exist meromorphic mappings $c_{ti}\not\equiv 0\ (k_{t-1}+1\le i\le k_t)$ such that $$\sum_{i=k_{t-1}+1}^{k_t}c_{ti}\cdot (f,\tilde a_i)\in \biggl(\bigcup_{j=1}^{t-1}\bigcup_{i\in I_t}{(f,\tilde a_i)}\biggl)_{\mathcal {R}}.$$ Therefore, there exist meromorphic mappings $c_{ti}\in \mathcal {R}\ (1\le i\le k_{t-1})$ such that $$\sum_{i=1}^{k_t}c_{ti}\cdot (f,\tilde a_i)=0.$$ Define $c_{ti}=0$ for all $i>k_t.$ Then $$\sum_{i=1}^{k_l}c_{ti}\cdot (f,\tilde a_i)=0.$$
Since $\{c_{ti}(f,\tilde a_i)\}_{i=k_{t-1}+1}^{k_t}$ is $\mathcal {R}$-linearly independent, by again Lemma \ref{2.2} there exists an admissible set $\{\alpha_{t(k_{t-1}+1)},\ldots,\alpha_{tk_t}\}\subset \mathbf{Z}^m_+$ \ $(|\alpha_{ti}|\le k_t-k_{t-1}-1\le \mathrm{rank}_{\mathcal R}f-1=k)$ such that the matrix $$\ A_t=\left (\mathcal {D}^{\alpha_{ti}}(c_{1j}(f,\tilde a_j));k_{t-1}+1\le i,j\le k_t \right)$$ has nonzero determinant.
Consider the following $(k_l-1)\times k_l$ matrix \begin{align*} T&=\left (\mathcal {D}^{\alpha_{ti}}(c_{1j}(f,\tilde a_j));k_0+1\le i\le k_t,1\le j\le k_t \right)\\ \\
&=\left [ \begin {array}{cccc} \mathcal {D}^{\alpha_{12}}(c_{11}(f,\tilde a_1)) &\cdots & \mathcal {D}^{\alpha_{12}}(c_{1k_l}(f,\tilde a_{k_l})) \\ \mathcal {D}^{\alpha_{13}}(c_{11}(f,\tilde a_1)) &\cdots & \mathcal {D}^{\alpha_{13}}(c_{1k_l}(f,\tilde a_{k_l})) \\ \vdots &\vdots &\vdots\\ \mathcal {D}^{\alpha_{1k_1}}(c_{11}(f,\tilde a_1))&\cdots &\mathcal {D}^{\alpha_{1k_1}}(c_{1k_l}(f,\tilde a_{k_l}))\\ \mathcal {D}^{\alpha_{2k_1+1}}(c_{21}(f,\tilde a_1)) &\cdots & \mathcal {D}^{\alpha_{2k_1+1}}(c_{2k_l}(f,\tilde a_{k_l})) \\ \mathcal {D}^{\alpha_{2k_1+2}}(c_{21}(f,\tilde a_1)) &\cdots & \mathcal {D}^{\alpha_{2k_1+2}}(c_{2k_l}(f,\tilde a_{k_l})) \\ \vdots &\vdots &\vdots\\ \mathcal {D}^{\alpha_{2k_2}}(c_{21}(f,\tilde a_1))&\cdots &\mathcal {D}^{\alpha_{2k_2}}(c_{2k_t}(f,\tilde a_{k_l}))\\ \vdots &\vdots &\vdots\\ \mathcal {D}^{\alpha_{lk_{l-1}+1}}(c_{l1}(f,\tilde a_1)) &\cdots & \mathcal {D}^{\alpha_{lk_{l-1}+1}}(c_{lk_l}(f,\tilde a_{k_l})) \\ \mathcal {D}^{\alpha_{lk_{l-1}+2}}(c_{l1}(f,\tilde a_1)) &\cdots & \mathcal {D}^{\alpha_{lk_{l-1}+2}}(c_{lk_l}(f,\tilde a_{k_l})) \\ \vdots &\vdots &\vdots\\ \mathcal {D}^{\alpha_{lk_l}}(c_{lk}(f,\tilde a_1))&\cdots &\mathcal {D}^{\alpha_{lk_l}}(c_{lk_l}(f,\tilde a_{k_l}))\\ \end {array} \right]. \end{align*} \vskip0.3cm Denote by $D_i$ the subsquare matrix obtained by deleting the $(i+1)$-th column of the minor matrix $T$. Since the sum of each row of $T$ is zero, we have $$\det D_i={(-1)}^{i-1}\det D_1={(-1)}^{i-1}\prod_{j=1}^{l}\det A_j.$$
Since $\{a_i\}_{i=1}^q$ is in general position, we have $$\det (\tilde a_{ij}, \ 1\le i\le n+1,0\le j\le n )\not\equiv 0.$$ By solving the linear equation system $(f,\tilde a_i)=\tilde a_{i0}\cdot f_0+\ldots +\tilde a_{in}\cdot f_n \ (1\le i\le n+1),$ we obtain \begin{align}\label{+}f_v=\sum_{i=1}^{n+1}A_{vi}(f,\tilde a_{i})\ (A_{vi}\in\mathcal R)\text{ for each }0\le v \le n. \end{align} Put
$\Psi(z)=\sum_{i=1}^{n+1}\sum_{v=0}^n |A_{vi}(z)|\ (z\in {\mathbf{C}}^m).$ Then
$$\ \ ||f(z)||\le \Psi (z)\cdot \max_{1\le i\le n+1}\bigl (|(f,\tilde a_i)(z)|\bigl )\le \Psi (z)\cdot \max_{1\le i\le q}\bigl (|(f,\tilde a_i)(z)|\bigl )\ (z\in {\mathbf{C}}^m),$$ and \begin{align*}
\int\limits_{S(r)} \log^+\Psi (z) \eta &\le \sum_{i=1}^{n+1}\sum_{v=0}^n \int\limits_{S(r)}\log^+|A_{vi}(z)|\eta+O(1)\\ &\le \sum_{i=1}^{n+1}\sum_{v=0}^n T(r,A_{vi}) +O(1)\\ &= O(\max_{1\le i\le q}T_{a_i}(r))+O(1). \end{align*}
Fix $z_0 \in {\mathbf{C}}^m\setminus\bigcup_{j=1}^q\biggl (\mathrm{Supp}\, (\nu^0_{(f,\tilde a_j)})\cup \mathrm{Supp}\, (\nu^\infty_{(f,\tilde a_j)})\biggl ).$ Take $i\ (1\le i \le q)$ such that
$$|(f,\tilde a_i)(z_0)|=\max_{1\le j\le q}(|f,\tilde a_j)(z_0)|.$$
Then \begin{align*}
\dfrac{|\det D_1(z_0)|\cdot ||f(z_0)||}{\prod_{j=1}^{q}|(f,\tilde a_i)(z_0)|}
&=\dfrac{|\det D_i(z_0)|}{\prod_{\underset{j\ne i}{j=0}}^{q}|(f,\tilde a_j)(z_0)|}\cdot \biggl(\dfrac
{||f(z_0)||}{|(f,\tilde a_i)(z_0)|}\biggl)\\
&\le \Psi (z_0)\cdot \dfrac{|\det D_i(z_0)|}{\prod_{\underset{j\ne i}{j=1}}^{q}|(f,\tilde a_j)(z_0)|}. \end{align*} This implies that \begin{align*}
\log\dfrac{|\det D_1(z_0)|.||f(z_0)||}{\prod_{j=1}^{q}|(f,\tilde a_j)(z_0)|}& \le\log^+\biggl ( \Psi (z_0)\cdot \biggl(\dfrac{|\det D_i(z_0)|}{\prod_{j=1,j\ne i}^{q}|(f,\tilde a_j)(z_0)|}\biggl)\biggl )\\
& \le\log^+\biggl(\dfrac{|\det D_i(z_0)|}{\prod_{j=1,j\ne i}^{q}|(f,\tilde a_j)(z_0)|}\biggl)+\log^+\Psi (z_0). \end{align*}
Thus, for each $z\in {\mathbf{C}}^m\setminus\bigcup_{j=1}^q\biggl (\mathrm{Supp}\, (\nu^0_{(f,\tilde a_j)})\cup \mathrm{Supp}\, (\nu^\infty_{(f,\tilde a_j)})\biggl ),$ we have \begin{align*}
\log\dfrac{|\det D_1(z)|.||f(z)||}{\prod_{i=1}^{q}|(f,\tilde a_i)(z)|} \le\sum_{i=1}^{q}\log^+\biggl(\dfrac{|\det D_i(z)|}{\prod_{j=1,j\ne i}^{q}|(f,\tilde a_j)(z)|}\biggl)+\log^+ \Psi (z) \end{align*} Hence \begin{align}\label{*}
\log ||f(z)||+\log \dfrac{|\det D_1(z)|}{\prod_{i=1}^{q}|(f,\tilde a_i)(z)|}\le \sum_{i=1}^{q}\log^+\biggl(\dfrac{|\det D_i(z)|}{\prod_{j=1,j\ne i}^{q}|(f,\tilde a_j)(z)|}\biggl)+\log^+ \Psi (z). \end{align} Note that \begin{align*} \dfrac{\det D_i}{\prod_{j=1,j\ne i}^{q}(f,\tilde a_j)} &=\dfrac{\det D_i/f_0^{q-1}}{\prod_{j=1,j\ne i}^{q}\biggl ((f,\tilde a_j)/f_0\biggl )}\\ &=\left [ \begin {array}{cccc} \dfrac {\mathcal {D}^{\alpha_{12}}\biggl(\dfrac {c_{11}(f,\tilde a_1)}{f_0}\biggl)}{\dfrac{(f,\tilde a_1)}{f_0}} &\cdots & \dfrac {\mathcal {D}^{\alpha_{12}}\biggl(\dfrac {c_{1k_l}(f,\tilde a_{k_l})}{f_0}\biggl)}{\dfrac{(f,\tilde a_{k_l})}{f_0}} \\ \vdots &\vdots &\vdots \\ \dfrac {\mathcal {D}^{\alpha_{lk_l}}\biggl(\dfrac {c_{l1}(f,\tilde a_1)}{f_0}\biggl)}{\dfrac{(f,\tilde a_1)}{f_0}} &\cdots & \dfrac {\mathcal {D}^{\alpha_{lk_l}}\biggl(\dfrac {c_{lk_l}(f,\tilde a_{k_l})}{f_0}\biggl)}{\dfrac {(f,\tilde a_{k_l})}{f_0}} \end {array} \right] \end{align*}
\quad (The determinant is counted after deleting the $i$-th column in the above matrix).
Each element of the above matrix has a form $$\dfrac {\mathcal {D}^{\alpha}\biggl(\dfrac {c(f,\tilde a_j)}{f_0}\biggl)}{\dfrac{(f,\tilde a_j)}{f_0}}= \dfrac {\mathcal {D}^{\alpha}\biggl(\dfrac {c(f,\tilde a_j)}{f_0}\biggl)}{\dfrac{c(f,\tilde a_j)}{f_0}}\cdot c \ (c \in \mathcal {R}).$$ By lemma on logarithmic derivative lemma, we have \begin{align*}
\biggl|\biggl| \quad\quad m \biggl(r,\dfrac {\mathcal {D}^{\alpha}\biggl(\dfrac {c(f,\tilde a_j)}{f_0}\biggl)}{\dfrac{(f,\tilde a_j)}{f_0}}\biggl)&\le m \biggl(r,\dfrac {\mathcal {D}^{\alpha}\biggl(\dfrac {c(f,\tilde a_j)}{f_0}\biggl)}{\dfrac{c(f,\tilde a_j)}{f_0}}\biggl)+m(r,c)\\ &= O\biggl(\log^+T\biggl(r,\dfrac {c(f,\tilde a_j)}{f_0}\biggl)\biggl)+O(\max_{1\le i\le q}T(r,a_i))\\ &= O(\log^+T_f(r))+O(\max_{1\le i \le q}T(r,a_i)). \end{align*} This yields that
$$\biggl|\biggl| \quad m\left (r,\dfrac{\det D_i}{\prod_{j=1,j\ne i}^{q}(f,\tilde a_j)}\right )=
O(\log^+T_f(r))+O(\max_{1\le j \le q}T_{a_j}(r))\ (1 \le i \le q).$$ Hence
$$\biggl|\biggl| \quad\quad \sum_{i=1}^{q} m\left (r,\dfrac{\det D_i}{\prod_{j=1,j\ne i}^{q}(f,\tilde a_j)}\right )= O(\log^+T_f(r))+O(\max_{1\le j \le q}T_{a_j}(r)).$$
Integrating both sides of the inequality (\ref{*}), we have \begin{align*}
\biggl|\biggl| \ \int_{S(r)}\log ||f|| \eta &+ \int_{S(r)}\log \biggl(\dfrac{|\det{D}_0|}{\prod_{i=1}^{q} |(f,\tilde a_i)|} \biggl)\eta\\
&\le \sum_{i=1}^{q} \int_{S(r)}\log^+ \biggl(\dfrac{|\det D_i|}{\prod_{j=1,j\ne i}^{q}
|(f,\tilde a_j)|}\biggl)\eta +\int_{S(r)}\log^+ \Psi(z)\eta\\ &= \sum_{i=1}^{q} m\biggl(r,\dfrac{\det D_i}{\prod_{j=1,j\ne i}^{q}(f,\tilde a_j)}\biggl) +O(\max_{1\le i \le q}T_{a_i}(r))\\ &= O(\log^+T_f(r))+O(\max_{0\le i \le q-1}T_{a_i}(r)). \end{align*} Hence
$$||\ \ T_f(r)+ \int\limits_{S(r)} \text{log}\dfrac{|\det D_1|}{\prod_{i=1}^{q}|(f,\tilde a_i)|} \eta =O(\log^+T_f(r))+O(\max_{1\le i\le q}T_{a_i}(r)),\ \text {i.e, }$$ \begin{align}\nonumber
||\ T_f(r) &= \int\limits_{S(r)} \text{log}\dfrac{\prod_{i=1}^{q}|(f,\tilde a_i)|}{|\det D_1|} \eta+ O(\log^+T_f(r))+O(\max_{1\le i\le q}T_{a_i}(r))\\ \nonumber
&= \int\limits_{S(r)} \text{log}\prod_{i=1}^{q}|(f,\tilde a_i)|\eta- \int\limits_{S(r)} \text{log}|\det D_1| \eta +O(\log^+T_f(r))+O(\max_{1\le i\le q}T_{a_i}(r))\\ \label{3.3} &\le N_{\prod_{i=1}^{q}(f, \tilde a_i)}(r)-N(r,\nu_{\det D_1})+ O(\log^+T_f(r))+O(\max_{1\le i\le q}T_{a_i}(r)). \end{align} \begin{claim}
$||\ N_{\prod_{i=1}^{q}(f, \tilde a_i)}(r)-N(r,\nu_{\det D_1})\le \sum_{i=1}^qN^{[k]}_{(f,a_i)}(r)+O(\max_{1\le i\le q}T_{a_i}(r)).$ \end{claim} Indeed, fix $z\in{\mathbf{C}}^m\setminus I(f)$, where $I(f)=\{f_0=\cdots f_n=0\}$. We call $i_0$ the index satisfying $$\nu^0_{(f,\tilde a_{i_0})}(z)=\min_{1\le i\le n+1}\nu^0_{(f,\tilde a_i)}(z).$$ For each $i\ne i_0, i\in I_s$, we have \begin{align*} \nu^0_{\mathcal {D}^{\alpha_{sk_{s-1}+j}}(c_{si}(f,\tilde a_{i}))}(z) &\ge\min_{\beta\in \mathbf{Z}_+^m \text { with } \alpha_{sk_{s-1}+j}-\beta \in\mathbf{Z}_+^m} \{\nu^0_{\mathcal{D}^{\beta}c_{si}\mathcal D^{\alpha_{st_{s-1}+j}-\beta }(f,\tilde a_{i})}(z)\}\\
&\ge\min_{\beta\in \mathbf{Z}_+^n \text { with } \alpha_{sk_{s-1}+j}-\beta \in\mathbf{Z}_+^n}\bigl{\{}\max\{0,\nu^0_{(f,\tilde a_{i})}(z)-|\alpha_{sk_{s-1}+j}-\beta|\}\\ &\hspace{90pt}-(\beta+1)\nu^{\infty}_{c_{si}}(z)\bigl{\}}\\ &\ge\max\{0,\nu_{(f\tilde a_i)}^{0}(z)-k\}-(k+1)\nu_{c_{si}}^{\infty}(z) \end{align*} On the other hand, we also have \begin{align*}
\nu^\infty_{\mathcal {D}^{\alpha_{sk_{s-1}+j}}(c_{si}(f,\tilde a_{i}))}(z)\le (|\alpha_{sk_{s-1}+j}|+1)\nu^\infty_{c_{si}(f,\tilde a_{i})}(z)\le (k+1)(\nu^\infty_{c_{si}}(z)+\nu^0_{a_{i0}}(z)). \end{align*} Thus $$ \nu_{\mathcal {D}^{\alpha_{sk_{s-1}+j}}(c_{si}(f,\tilde a_{i}))}(z)\ge \max\{0,\nu_{(f\tilde a_i)}^{0}(z)-k\}-(k+1)\bigl (2\nu_{c_{si}}^{\infty}(z)+\nu^0_{a_{i0}}(z)\bigl )$$ Since each element of the matrix $D_{i_0}$ has a form $\mathcal {D}^{\alpha_{sk_{s-1}+j}}(c_{si}(f,\tilde a_{i}))\ (i\ne i_0)$, one estimates \begin{align}\label{3.4} \nu_{D_1}(z)=\nu_{D_{i_0}}(z) \ge\sum_{i\ne i_0}\left (\max\{0,\nu_{(f\tilde a_i)}^{0}(z)-k\}-(k+1)\bigl (2\nu_{c_{si}}^{\infty}(z)+\nu^0_{a_{i0}}(z)\bigl )\right ). \end{align} We see that there exists $v_0\in\{0,\ldots,n\}$ with $f_{v_0}(z)\ne 0$. Then by (\ref{+}), there exists $i_1\in\{1,\ldots,n+1\}$ such that $A_{v_0i_1}(z)\cdot (f,\tilde a_{i_1})(z)\ne 0$. Thus \begin{align}\label{3.5} \nu^0_{(f,\tilde a_{i_0})}(z)\le \nu^0_{(f,\tilde a_{i_1})}(z)\le\nu^\infty_{A_{v_0i_1}}(z)\le\sum_{A_{vi}\not\equiv 0}\nu^\infty_{A_{vi}}(z). \end{align} Combining the inequalities (\ref{3.4}) and (\ref{3.5}), we have \begin{align*} \nu^{0}_{\prod_{i=1}^{q}(f,\tilde a_i)}(z)&-\nu_{\det D_1}(z)\\ &\le\sum_{i\ne i_0}\left (\min\{\nu_{(f,\tilde a_i)}^{0}(z),k\}+(k+1)\bigl (2\nu_{c_{si}}^{\infty}(z)+\nu^0_{a_{i0}}(z)\bigl )\right )+\sum_{A_{vi}\not\equiv 0}\nu^\infty_{A_{vi}}(z)\\ &\le\sum_{i=1}^q\left (\min\{\nu_{(f,\tilde a_i)}^{0}(z),k\}+(k+1)\bigl (2\nu_{c_{si}}^{\infty}(z)+\nu^0_{a_{i0}}(z)\bigl )\right )+\sum_{A_{vi}\not\equiv 0}\nu^\infty_{A_{vi}}(z), \end{align*} where the index $s$ of $c_{si}$ is taken so that $i\in I_s$. Integrating both sides of this inequality, we obtain \begin{align}\nonumber
||\ \ N_{\prod_{i=1}^{q}(f, \tilde a_i)}(r)&-N(r,\nu_{\det D_1})\\ \nonumber &\le \sum_{i=1}^q\left (N^{[k]}_{(f,\tilde a_i)}(r)+(k+1)\biggl (2N_{\frac{1}{c_{si}}}(r)+N_{a_{i0}}(r)\biggl )\right )+\sum_{A_{vi}\not\equiv 0}N_{{1}/{A_{vi}}}(r)\\ \label{3.6} &= \sum_{i=1}^qN^{[k]}_{(f,a_i)}(r)+O(\max_{1\le i\le q}T_{a_i}(r)). \end{align} The claim is proved.
From the inequalities (\ref{3.3}) and the claim, we get
$$ ||\ \ T_f(r)\le \sum_{i=1}^qN^{[k]}_{(f,a_i)}(r)+O(\log^+T_f(r))+O(\max_{1\le i\le q}T_{a_i}(r)). $$ The lemma is proved.
$\square$
\vskip0.2cm \noindent {\bf Proof of Theorem \ref{1.1}.}
(a). We denote by $\mathcal I$ the set of all permutations of $q-$tuple $(1,\ldots,q)$. For each element $I=(i_1,\ldots,i_q)\in\mathcal I$, we set $$N_I=\{r\in{\mathbf{R}}^+;N^{[k]}_{(f,a_{i_1})}(r)\le\cdots\le N^{[k]}_{(f,a_{i_q})}(r)\}.$$
We now consider an element $I=(i_1,\ldots,i_q)$ of $\mathcal I$. We will construct subsets $I_t$ of the set $A_1=\{1,\ldots,{2n-k+2}\}$ as follows.
We choose a subset $I_1$ of $A$ which is the minimal subset of $A$ satisfying that $\{(f,\tilde a_{i_j})\}_{j\in I_1}$ is minimal over $\mathcal R$. If $\sharp I_1\ge n+1$ then we stop the process.
Otherwise, set $A_2=A_1\setminus I_1$. We consider the following two cases: \begin{itemize} \item Case 1. Suppose that $\sharp A_2\ge n+1$. Since $\{\tilde a_{i_j}\}_{j\in A_2}$ is in general position, we have $$ \left ((f,\tilde a_{i_j}); j\in A_2\right )_{\mathcal R}=\left (f_0,\ldots,f_n\right )_{\mathcal R}\supset \left ((f,\tilde a_{i_j}); j\in I_1\right )_{\mathcal R}\not\equiv 0.$$ \item Case 2. Suppose that $\sharp A_2< n+1$. Then we have the following: \begin{align*} &\dim_{\mathcal R}\left ((f,\tilde a_{i_j}); j\in I_1\right )_{\mathcal R}\ge k+1-(n+1-\sharp I_1)=k-n+\sharp I_1,\\ & \dim_{\mathcal R}\left ((f,\tilde a_{i_j}); j\in A_2\right )_{\mathcal R}\ge k+1-(n+1-\sharp A_2)=k-n+\sharp A_2. \end{align*} We note that $\sharp I_1+\sharp A_2=2n-k+2$. Hence the above inequalities imply that \begin{align*} \dim_{\mathcal R}&\biggl (\bigl ((f,\tilde a_{i_j}); j\in I_1\bigl )_{\mathcal R}\cap\bigl ((f,\tilde a_{i_j}); j\in A_2\bigl )_{\mathcal R}\biggl )\\ &\ge\dim_{\mathcal R}\left ((f,\tilde a_{i_j}); j\in I_1\right )_{\mathcal R}+\dim_{\mathcal R}\left ((f,\tilde a_{i_j}); j\in A_2\right )_{\mathcal R}-(k+1)\\ &=k-n+\sharp I_1+k-n+\sharp A_2-(k+1)=1. \end{align*} \end{itemize} Therefore, from the above two case, we see that $$ \bigl ((f,\tilde a_{i_j}); j\in I_1\bigl )_{\mathcal R}\cap\bigl ((f,\tilde a_{i_j}); j\in A_2\bigl )_{\mathcal R}\ne \{0\}. $$ Therefore, we may chose a subset $I_2\subset A_2$ which is the minimal subset of $A_2$ satisfying that there exist nonzero meromorphic functions $c_i\in\mathcal R\ (i\in I_2)$, $$\sum_{i\in I_2}c_i(f,\tilde a_i)\in \biggl(\bigcup_{i\in I_1}(f,\tilde a_i) \biggl)_{\mathcal {R}}.$$ By the minimality of the set $I_2$, the family $\{(f,\tilde a_{i_j})\}_{j\in I_2}$ is linearly independent over $\mathcal R$, and hence $\sharp I_2\le k+1$ and $$\sharp (I_2\cup I_2)\le\min\{2n-k+2, n+k+1\}.$$ If $\sharp (I_2\cup I_2)\ge n+1$ then we stop the process.
Otherwise, by repeating the above argument, we have a subset $I_3$ of $A_3=A_1\setminus (I_1\cup I_2)$, which satisfies the following: \begin{itemize} \item there exist nonzero meromorphic functions $c_i\in\mathcal R\ (i\in I_3)$ so that $$\sum_{i\in I_3}c_i(f,\tilde a_i)\in \biggl(\bigcup_{i\in I_1\cup I_2}(f,\tilde a_i) \biggl)_{\mathcal {R}},$$ \item $\{(f,\tilde a_{i_j})\}_{j\in I_3}$ is linearly independent over $\mathcal R$, \item $\sharp I_3\le k+1$ and $\sharp (I_1\cup\cdots\cup I_3)\le \min\{2n-k+2, n+k+1\}$. \end{itemize}
Continuing this process, we get the subsets $I_1,\ldots,I_l$, which satisfy: \begin{itemize} \item $\{(f,\tilde a_{i_j})\}_{j\in I_1}$ is minimal over $\mathcal R$, $\{(f,\tilde a_{i_j})\}_{j\in I_t}$ is linearly independent over $\mathcal {R}\ (2\le t \le l), $ \item for any $2\le t\le l, j\in I_t,$ there exist meromorphic functions $c_j\in \mathcal {R}\setminus\{0\}$ such that $$\sum_{j\in I_t}c_j(f,\tilde a_{i_j})\in \biggl(\bigcup_{s=1}^{t-1}\bigcup_{j\in I_s}(f,\tilde a_{i_j}) \biggl)_{\mathcal {R}},$$ \item $n+1\le\sharp (I_1\cup\cdots\cup I_l)\le \min\{2n-k+2, n+k+1\}$. \end{itemize}
Then the family of subsets $I_1,\ldots,I_t$ satisfies the assumptions of the Lemma \ref{3.1}. Therefore, we have \begin{align*}
||\ T_f(r)\le\sum_{j\in J}N^{[k]}_{(f,a_{i_j})}+o(T_f(r))+O(\max_{1\le i \le q}T_{a_i}(r)), \end{align*} where $J=I_1\cup\cdots\cup I_l$. Then for all $r\in N_I$ (may be outside a finite Borel measure subset of ${\mathbf{R}}^+$) we have \begin{align}\nonumber
||\ T_f(r)&\le\dfrac{\sharp J}{q-(2n-k+2)+\sharp J}\biggl (\sum_{j\in J}N^{[k]}_{(f,a_{i_j})}(r)+\sum_{j=2n-k+3}^qN^{[k]}_{(f,a_{i_j})}(r)\biggl )\\ \label{3.7} &+o(T_f(r)) + O(\max_{1\le i \le q}T_{a_i}(r)). \end{align} Since $\sharp J\le 2n-k+2$, the above inequality implies that \begin{align}\label{3.8}
||\ T_f(r)\le\dfrac{2n-k+2}{q}\sum_{i=1}^qN^{[k]}_{(f,a_{i})}(r)+o(T_f(r))+O(\max_{1\le i \le q}T_{a_i}(r)),\quad r\in N_I. \end{align}
We see that $\bigcup_{I\in\mathcal I}N_I={\mathbf{R}}^+$ and the inequality (\ref{3.8}) holds for every $r\in N_I, I\in\mathcal I$. This yields that $$ T_f(r)\le\dfrac{2n-k+2}{q}\sum_{i=1}^qN^{[k]}_{(f,a_{i})}(r)+o(T_f(r))+O(\max_{1\le i \le q}T_{a_i}(r)) $$ for all $r$ outside a finite Borel measure subset of ${\mathbf{R}}^+$. Thus
$$ ||\ \dfrac{q}{2n-k+2}T_f(r)\le\sum_{i=1}^qN^{[k]}_{(f,a_{i})}(r)+o(T_f(r))+O(\max_{1\le i \le q}T_{a_i}(r)). $$ The assertion (a) is proved.
(b) We repeat the same argument as in the proof of the assertion (a). If $n+k+1>2n-k+1$ then the assertion (b) is a consequence of the assertion (a). Then we now only consider the case where $n+k+1\le 2n-k+1$.
From (\ref{3.7}) with a note that $\sharp J\le n+k+2$, we have \begin{align*}
||\ T_f(r)&\le\dfrac{n+k+1}{q-(2n-k+2)+n+k+1)}\sum_{i=1}^qN^{[k]}_{(f,a_{i})}(r)+o(T_f(r))+O(\max_{1\le i \le q}T_{a_i}(r))\\ &=\dfrac{n+k+1}{q-n+2k-1}\sum_{i=1}^qN^{[k]}_{(f,a_{i})}(r)+o(T_f(r))+O(\max_{1\le i \le q}T_{a_i}(r))\ r\in N_I. \end{align*} Repeating again the argument in the proof of assertion (a), we see that the above inequality holds for all $r\in{\mathbf{R}}^+$ outside a finite Borel measure set. Then the assertion (b) is proved.
$\square$
\section{Proof of Theorem \ref{1.3}}
In order to prove Theorem \ref{1.3}, we need the following.
\noindent {\bf 4.1.}\ Let $f:{\mathbf{C}}^m \to {\mathbf{P}}^n({\mathbf{C}})$ be a meromorphic mapping with a reduced representation $f=(f_0:\ldots :f_n)$. Let $\{a_i\}_{i=1}^{q}$ be ``slowly'' (with respect to $f$) moving hyperplanes of ${\mathbf{P}}^n({\mathbf{C}})$ in general position such that $$\dim \{ z \in {\mathbf{C}}^m : (f,a_i)(z)=(f,a_j)(z)=0 \} \le m-2 \quad (1 \le i<j \le q).$$
For $M+1$ elements $f^0,\ldots ,f^M \in \mathcal {F}(f,\{a_j\}_{j=1}^q,1)$, we put $$T(r)=\sum_{k=0}^M T(r,f^k).$$
Assume that $a_i$ has a reduced representation $a_i=(a_{i0}:\cdots :a_{in}).$ By changing the homogeneous coordinate system of ${\mathbf{P}}^n({\mathbf{C}}),$ we may assume that $a_{i0}\not \equiv 0\ (1\le i \le q).$
We set $\ F^{jk}_{i}:=\dfrac{(f^k,a_j)}{(f^k,a_i)}\quad (1 \le i,j \le q, \ 0\le k\le M).$
\begin{lemma}\label{4.1} Suppose that $q\ge 2n+1$. Then
$$|| \ T_g(r)=O(T_f(r))\text { for each } g \in \mathcal {F}(f,\{a_i\}_{i=1}^q,1).$$ \end{lemma} \noindent {\bf Proof.} By Corollary \ref{1.2}(a), we have \begin{align*} \parallel\ \dfrac {2q-n+1}{3(n+1)}T_g(r)&\le \sum_{i=1}^qN_{(g,a_i)}^{[n]}(r)+o(T_g(r)+T_f(r))\\ &\le n\sum_{i=1}^qN_{(g,a_i)}^{[1]}(r)+o(T_g(r)+T_f(r))\\ &= \sum_{i=1}^qnN_{(f,a_i)}^{[1]}(r)+o(T_g(r)+T_f(r))\\ &\le qn T_f(r)+o(T_g(r)+T_f(r)). \end{align*}
Hence \quad $|| \quad T_g(r)=O(T_f(r)).$
$\square$
\begin{definition}[{see \cite[p. 138]{Fu3}}]
Let $F_0,\ldots ,F_M$ be nonzero meromorphic functions on ${\mathbf{C}}^m$, where $M\ge 1$. Take a set $\alpha:=(\alpha^0,\ldots ,\alpha^{M-1})$ whose components $\alpha^k$ are composed of $m$ nonnegative integers, and set $|\alpha|=|\alpha^0|+\ldots +|\alpha^{M-1}|.$ We define Cartan's auxiliary function by
$$\Phi^\alpha \equiv \Phi^\alpha(F_0,\ldots ,F_M):=F_0F_1\cdots F_M\left | \begin {array}{cccc} 1&1&\cdots &1\\ \mathcal {D}^{\alpha^0}(\frac {1}{F_0})&\mathcal {D}^{\alpha^0}(\frac {1}{F_1}) &\cdots &\mathcal {D}^{\alpha^0}(\frac {1}{F_M})\\ \vdots &\vdots &\vdots &\vdots \\ \mathcal {D}^{\alpha^{M-1}}(\frac {1}{F_0}) &\mathcal {D}^{\alpha^{M-1}}(\frac {1}{F_1}) &\cdots &\mathcal {D}^{\alpha^{M-1}}(\frac {1}{F_M})\\ \end {array}
\right| $$ \end{definition}
\begin{lemma}[{see \cite[Proposition 3.4]{Fu3}}]\label{4.4}
If $\Phi^\alpha(F,G,H)=0$ and $\Phi^\alpha(\frac {1}{F},\frac {1}{G},\frac {1}{H})=0$ for all $\alpha$ with $|\alpha|\le1$, then one of the following assertions holds :
(i) \ $F=G, G=H$ or $H=F$
(ii) \ $\frac {F}{G},\frac {G}{H}$ and $\frac {H}{F}$ are all constant. \end{lemma}
\begin{lemma}[{see \cite[Lemma 4.7]{TQ05}}]\label{4.5}
Suppose that there exists $\Phi^\alpha=\Phi^\alpha(F_{i_0}^{j_00},\ldots , F_{i_0}^{j_0M})\not\equiv 0$ with $1\le i_0,j_0\le q,\ |\alpha|\le \dfrac {M(M-1)}{2}, \ d\ge |\alpha|.$ Assume that $\alpha$ is a minimal element such that $\Phi^\alpha(F_{i_0}^{j_00},\ldots , F_{i_0}^{j_0M})\not\equiv 0$. Then, for each $0 \le k \le M$, the following holds:
$$\parallel N_{(f^k,a_{j_0})}^{[d-|\alpha|]}(r)+M \sum_{j\ne{j_0,i_0}}N_{(f^k,a_j)}^{[1]}(r)\le N_{\Phi^\alpha}(r) \le T(r)-M\cdot N^{[1]}_{(f^k,a_{i_0})}(r)+o(T(r)).$$ And hence
$$|| \quad N_{(f^k,a_{j_0})}^{[d-|\alpha|]}(r)+M \sum_{j\ne{j_0}}N_{(f^k,a_j)}^{[1]}(r)\le T(r)+o(T(r)).$$ \end{lemma}
\vskip0.2cm \noindent {\bf 4.2. Proof of Theorem \ref{1.3}}
\noindent a) Assume that $q>\frac{9n^2+9n+2}{2}$. Suppose that there exist three distinct elements $f^0,f^1,f^2 \in \mathcal {F}(f,\{a_j\}_{j=1}^{q},1).$
Suppose that there exist two indices $i,j\in\{1,\ldots ,q\}$ and $\alpha=(\alpha_0,\alpha_1)\in (\mathbf{Z}_+^n)^2 $ with $|\alpha|\le 1$ such that $\Phi^\alpha (F_{j}^{i0},F_{j}^{i1},F_{j}^{i2}) \not \equiv 0$. By Lemma \ref{4.5}, we have $$2\sum_{t\ne i}N_{(f^0,a_t)}^{[1]}(r)\le T(r)+o(T_f(r)).$$ Hence, by Corollary \ref{1.2}(b) we have \begin{align*} \parallel T(r)&\ge \dfrac{2}{3}\sum_{k=1}^3 \sum_{t\ne i}N_{(f^k,a_t)}^{[1]}(r)+o(T_f(r))\ge \dfrac{2}{3n}\sum_{k=1}^3 \sum_{t\ne i}N_{(f^k,a_t)}^{[n]}(r)+o(T_f(r))\\ &\ge\dfrac{4(q-1)}{9n(n+1)}T(r)+o(T_f(r)). \end{align*} Letting $r\longrightarrow +\infty$, we get $1\ge\frac{4(q-1)}{9n(n+1)}$, i.e., $q\le\frac{9n^2+9n+4}{4}$. This is a contradiction.
Then for two indices $i,j$ $(1\le i<j\le q)$, we have $$\Phi^\alpha (F_{j}^{i0},F_{j}^{i1},F_{j}^{i2}) \equiv 0\text{ and }\Phi^\alpha (F_{i}^{j0},F_{i}^{j1},F_{i}^{j2}) \equiv 0$$
for all $\alpha=(\alpha_0,\alpha_1)\ \text { with }|\alpha|\le 1.$ By Lemma \ref{4.4}, there exists a constant $\lambda$ such that $$F_{j}^{i0}=\lambda F_{j}^{i1},F_{j}^{i1}=\lambda F_{j}^{i2}, \text { or } F_{j}^{i2}=\lambda F_{j}^{i0}.$$ For instance, we assume that $F_{j}^{i0}=\lambda F_{j}^{i1}$. We will show that $ \lambda=1.$
Indeed, assume that $\lambda \ne 1$. Since $F_{j}^{i0}=F_{j}^{i1}$ on the set $\bigcup_{k\ne j}\{z : (f,a_k)(z)=0\},$ we have that $F_{j}^{i0}=F_{j}^{i1}=0$ on the set $\bigcup_{k\ne j}\{z : (f,a_k)(z)=0\}.$ Hence $\bigcup_{k\ne j}\{z : (f,a_k)(z)=0\} \subset \{z : (f,a_i)(z)=0\}.$ It follows that $\{z : (f,a_k)(z)=0\}=\emptyset \ (k\ne i,j).$ We obtain that $$\parallel\dfrac{2(q-2)}{3(n+1)}T_f(r) \le \sum_{k\ne i,k\ne j}N_{(f,a_k)}^{[n]}(r)+o(T_f(r))=o(T_f(r)).$$ This is a contradiction. Thus $ \lambda =1\ (1\le i<j\le q).$
Define $$I_1= \{i\in \{1,\ldots ,q-1\}: F_{q}^{i0}=F_{q}^{i1}\},$$ $$I_2= \{i\in \{1,\ldots ,q-1\}: F_{q}^{i1}=F_{q}^{i2}\},$$ $$I_3= \{i\in \{1,\ldots ,q-1\}: F_{q}^{i2}=F_{q}^{i0}\}.$$ Since $\sharp (I_1\cup I_2\cup I_3)=\sharp \{1,\ldots ,q-1\}=q-1\ge 3n-2$, there exists $1\le k\le 3$ such that $\sharp \ I_k \ge n$. Without loss of generality, we may assume that $\sharp \ I_1 \ge n$. This implies that $f^0=f^1$. This is a contradiction.
Thus, we have $\sharp \ \mathcal {F}(f,\{a_i\}_{i=1}^{q},1)\le 2.$
\vskip0.2cm \noindent b) Assume that $q>3n^2+n+2$.
Take $g\in\mathcal{F}(f,\{a_i\}_{i=1}^{q},1).$ Suppose that $f\ne g.$ By changing indices if necessary, we may assume that $$\underbrace{\dfrac{(f,a_1)}{(g,a_1)}\equiv \dfrac{(f,a_2)}{(g,a_2)}\equiv \cdots\equiv \dfrac{(f,a_{k_1})} {(g,a_{k_1})}}_{\text { group } 1}\not\equiv \underbrace{\dfrac{(f,a_{k_1+1})}{(g,a_{k_1+1})}\equiv \cdots\equiv \dfrac{(f,a_{k_2})}{(g,a_{k_2})}}_{\text { group } 2}$$ $$\not\equiv \underbrace{\dfrac{(f,a_{k_2+1})}{(g,a_{k_2+1})}\equiv \cdots\equiv \dfrac{(f,a_{k_3})}{(g,a_{k_3})}}_{\text { group } 3}\not\equiv \cdots\not\equiv \underbrace{\dfrac{(f,a_{k_{s-1}+1})}{(g,a_{k_{s-1}+1})}\equiv\cdots \equiv \dfrac{(f,a_{k_s})}{(g,a_{k_s})}}_{\text { group } s},$$ where $k_s=q.$
For each $1\le i \le q,$ we set \begin{equation*} \sigma (i)= \begin{cases} i+n& \text{ if $i+n\leq q$},\\ i+n-q&\text{ if $i+n> q$} \end{cases} \end{equation*} and $$P_i=(f,a_i)(g,a_{\sigma (i)})-(g,a_i)(f,a_{\sigma (i)}).$$ By supposition that $f\ne g$, the number of elements of each group is at most $n$. Hence $\dfrac{(f,a_i)}{(g,a_i)}$ and $\dfrac{(f,a_{\sigma (i)})}{(g,a_{\sigma (i)})}$ belong to distinct groups. This means that $P_i\not\equiv 0\ (1\le i\le q)$.
Fix an index $i$ with $1\le i \le q.$ It is easy to see that \begin{align*} \nu_{P_i}(z) \ge \min\{\nu_{(f,a_i)},\nu_{(g,a_i)}\}+\min\{\nu_{(f,a_{\sigma (i)})},\nu_{(g,a_{\sigma (i)})}\} +\sum_{\underset{v\ne i,\sigma (i)}{v=1}}^{q}\nu_{(f,a_v)}^{[1]}(z) \end{align*} outside a finite union of analytic sets of dimension $\le m-2.$ Since $\min\{a,b\}+n\ge\min\{a,n\}+\min\{b,n\}$ for all positive integers $a$ and $b$, the above inequality implies that \begin{align*} N_{P_i}(r)\geq \sum_{v=i,\sigma (i)}\left ( N^{[n]}_{(f,a_v)}(r)+N^{[n]}_{(g,a_v)}(r)-nN^{[1]}_{(f,a_v)}(r) \right ) +\sum_{\underset{v\ne i,\sigma (i)}{v=1}}^{q}N^{[1]}_{(f,a_v)}(r). \end{align*} On the other hand, by the Jensen formula, we have \begin{align*}
N_{P_i}(r)=&\int_{S(r)}\log |P_i|\eta + O(1)\\
\le &\int_{S(r)}\log (|(f,a_i)|^2+|(f,a_{\sigma (i)}|^2)^{\frac{1}{2}}\eta
+ \int_{S(r)}\log (|(g,a_i)|^2+|(g,a_{\sigma (i)}|^2)^{\frac{1}{2}}\eta +O(1)\\ \le &T_f(r)+T_g(r) +o(T_f(r)). \end{align*} This implies that \begin{align*} T_f(r)+T_g(r)\ge &\sum_{v=i,\sigma (i)}\left ( N^{[n]}_{(f,a_v)}(r)+N^{[n]}_{(g,a_v)}(r)-nN^{[1]}_{(f,a_v)}(r) \right )\\ + &\sum_{\underset{v\ne i,\sigma (i)}{v=1}}^{q}N^{[1]}_{(f,a_v)}(r) +o(T_f(r)). \end{align*} Summing-up both sides of the above inequality over $i=1,\ldots ,q$ and by Corollary \ref{1.2}(b), we have \begin{align*} q(T_f(r)+T_g(r))\ge &2\sum_{v=i}^q\left ( N^{[n]}_{(f,a_v)}(r)+N^{[n]}_{(g,a_v)}(r) \right )\\ &+ (q-2n-2)\sum_{v=1}^{q}N^{[1]}_{(f,a_v)}(r)+o(T_f(r))\\ \ge & (2+\frac{q-2n-2}{2n})\sum_{v=i}^q\left ( N^{[n]}_{(f,a_v)}(r)+N^{[n]}_{(g,a_v)}(r) \right )+o(T_f(r))\\ \ge &(2+\frac{q-2n+2}{2n})\dfrac{2q}{3(n+1)}(T_f(r)+T_g(r))+o(T_f(r)). \end{align*} Letting $r \to \infty$, we get $q\ge (2+\frac{q-2n-2}{2n})\dfrac{2q}{3(n+1)}\Leftrightarrow q\le 3n^2+n+2.$ This is a contradiction.
Then $f=g$. This implies that $\sharp \mathcal{F}(f,\{a_i\}_{i=1}^{q},1)=1$. The theorem is proved.
$\square$
\end{document} | arXiv |
\begin{document}
\title{Bayesian Additive Adaptive Basis Tensor Product Models for Modeling
High Dimensional Surfaces: An application to high-throughput toxicity testing.}
\author {Matthew W. Wheeler Risk Analysis Branch\\ National Institute for Occupational Safety and Health, Cincinnati, OH } \maketitle
\label{firstpage}
\begin{abstract} Many modern data sets are sampled with error from complex high-dimensional surfaces. Methods such as tensor product splines or Gaussian processes are effective/well suited for characterizing a surface in two or three dimensions but may suffer from difficulties when representing higher dimensional surfaces. Motivated by high throughput toxicity testing where observed dose-response curves are cross sections of a surface defined by a chemical's structural properties, a model is developed to characterize this surface to predict untested chemicals' dose-responses. This manuscript proposes a novel approach that models the multidimensional surface as a sum of learned basis functions formed as the tensor product of lower dimensional functions, which are themselves representable by a basis expansion learned from the data. The model is described, a Gibbs sampling algorithm proposed, and is investigated in a simulation study as well as data taken from the US EPA's ToxCast high throughput toxicity testing platform. \end{abstract}
\textbf{Keywords:} Dose-response Analysis;EPA ToxCast; Functional Data Analysis; Machine Learning; Nonparametric Bayesian Analysis.
\maketitle
\section{Introduction} \label{s:intro}
Chemical toxicity testing is vital in determining the public health hazards posed by chemical exposures. However, the number of chemicals far outweighs the resources available to adequately test all chemicals, which leaves knowledge gaps when protecting public health. For example, there are over $80,000$ chemicals in industrial use with less than $600$ of these chemicals subject to long term \textit{in vivo} studies conducted by the National Toxicology Program, and most of these studies occur only after significant public exposures.
As an alternative to long term studies, which are expensive and take years to complete, there has been an increased focus on the use of high throughput bioassays to determine the toxicity of a given chemical. In an effort to understand the utility of these approaches for risk assessment, many agencies have developed rich databases to study this problem. To such an end, the US EPA ToxCast chemical prioritization project \citep{judson2010} was created to collect dose-response information on thousands of chemicals for hundreds of \textit{in vitro} bioassays, and it has been used to develop screening methods that prioritize chemicals for study based upon \textit{in vitro} toxicity. Though these methods have shown utility in predicting toxicity for many chemicals that pose a risk to the public health, there are many situations where the \textit{in vitro} bioassay information may not be available (e.g, a chemical may be so new that it has not been studied). In these cases, it would be ideal if toxicity could be estimated \textit{in silico} by chemical structural activity relationship (SAR) information. Here the goal is to develop a model based on SAR information that predicts the entire dose-response for a given assay. This manuscript is motivated by this problem.
\subsection{Quantitative Structure Activity Relationships}
There is a large literature estimating chemical toxicity from SAR information. These approaches, termed Quantitative Structure Activity Relationships (QSAR) (for a recent review of the models and the statistical issues encountered see \citet{emmert2012}), estimate a chemical's toxicity from the chemical's structural properties. Multiple linear regression has played a role in QSAR modeling since its inception\citep[p. 191]{roy2015}, but models where the predictor enters into the relationship as a linear function often fail to describe the non-linear nature of the relationship. To address the non-linearity of the predictor-response relationship, approaches such as neural networks \citep{devillers1996}, regression trees \citep{deconinck2005}, support vector machines \citep{czerminski2001,norinder2003}, and Gaussian processes \citep{burden2001} have been applied to the problem with varying levels of success. These approaches have been tailored to scalar responses, and, save one instance, have not been used to model the dose-response relationship, which may result in a loss of information.
The only QSAR approach that has addressed the problem of estimating a dose-response curve is the work by \citet{low2015}. This approach defined a Bayesian regression tree over functions where the leaves of the tree represent a different dose-response surface. It was used to identify chemical properties related to the observed dose-response, and when this approach was applied to prediction the approach sometimes performed poorly in a leave one out analysis. Further, it is computationally intensive and was estimated to take more than a week to analyze the data in motivating problem.
\subsection{Relevant Literature} Assume that one obtains a $P$ dimensional vector $s \in \mathcal{S}$, and one wants to predict a $Q$ dimensional response over $d \in \mathcal{D}$ from $s$. Given $s$ (e.g., SAR characteristics in the motivating problem) and $d$ (e.g., doses in the motivating problem) one is interested in
estimating an unknown $P+Q$ dimensional surface $h:(\mathcal{S} \times \mathcal{D}) \rightarrow \mathbb{R}$ where response curve $i$ is a cross section of $h(s,d)$ at $s_i.$ Figure (\ref{fig:cross-sections}) describes this in the case of the motivating example. Here two 1-dimensional cross sections (black lines) of a 2-dimensional surface are observed and one is interested the entire 2-dimensional surface. \begin{figure}
\caption{Example of the problem for a 2-dimensional surface. Here two
dose-response curves (black lines), which are cross sections of a larger surface,
are observed, and one is interested in this surface. }
\label{fig:cross-sections}
\end{figure}
One may use a Gaussian process (GP, \citep{rasmussen2006} ) to characterize the entire $P+Q$ dimensional surface, but there are computational problems that make the use of a GP impractical. In the motivating example, there are over $4000$ unique $(s,d)$ pairs. GP regression requires inversion of the covariance matrix, which is inherently an $\mathcal{O}(n^3)$ operation; inverting a $4000 \times 4000$ matrix in each iteration of a Gibbs sampler is challenging, and, though the covariance matrix may be approximated leading to reduced computational burden (\citet{quinonero2005},\citet{banerjee2013}), it is the author's experience that, in the higher dimensional case of the data example, such approximations became accurate when the dimension of the approximation matrix approaches that of the matrix it is approximating. This leads to minimal computational savings. Further, if the approximation approach can be used resulting in computational benefits, the the proposed method can be used in in conjunction with such approximations.
As an alternative to GPs, one can use tensor product splines (\citet{deBoor2000}, Chapter 17); however, even if one defines a functional basis having only two basis functions for each dimension, the resulting tensor spline basis would have dimension $2^{P+Q},$ which is often computationally intractable. The proposed model sidesteps these issues by defining a tensor product of learned multi-dimsensional surfaces defined on $\mathcal{S}$ and $\mathcal{D}.$
One could consider the problem from a functional data perspective \citep{ramsay2006,morris2015} by assuming a functional response over $\mathcal{S}\times\mathcal{D}.$ Approaches such as the functional linear array model \citep{brockhaus2015} and functional additive mixed models \citep{scheipl2015} are closest to the proposed approach as they allow the surface to be modeled as a basis defined on the entire space. For high dimensional spaces where interactions are appropriate, these approaches suffer the drawbacks because they do not allow for interactions in all dimensions. Additionally, these approaches assume that for any two observations $i,i'$ when $s_i = s_{i'}$, loadings are drawn from a distribution independent of $s_{i}.$ This differs from the proposed approach assumes, which correlates loadings through $\mathcal{S}.$
Clustering the functional responses is also a possibility. Here, one would model the surface using $\mathcal{D}$ and cluster using $\mathcal{S}.$ There are many functional clustering approaches (see for example \citet{hall2001}, \citet{ferraty2006},\citet{zhang2010},\citet{sprechmann2010} \citet{delaigle2012}, \citet{delaigle2013}, and references therein), but these methods predict $s \in \mathcal{S}$ based upon observing the functional response $f(d)$, which is the opposite of the problem at hand. In this problem, one observes information on the group (i.e., $s$) and one wishes to estimate the new response (i.e., $h(s,d)$). However different, such approaches can be seen as motivating the proposed method. Instead of clustering the loadings, the proposed approach assumes these loading for each basis is a value on a continuous function over $\mathcal{S}.$ This is similar to clustering as similarity is induced for any two responses having values in $\mathcal{S}$ that are sufficiently close.
As the number of basis functions used in the model is choice that may impact the model's ability to represent an arbitrary surface, a sufficiently large number of basis functions are included. Parsimony in this basis set is ensured by adapting to the number components in the model using the multiplicative gamma prior \citep{bhattacharya2011}. This is a global shrinkage prior is used to remove components from the sum by stochastically decreasing the prior variance of consecutive loadings in the sum to zero. The sum adapts to the actual number of basis functions needed to model the data and the choice in the number of basis functions is less important as long as the number is sufficiently large.
In what follows, section 2 defines the model. Section $3$ gives the data model for normal responses and outlines a sampling algorithm. Section 4 shows through a simulation study the method outperforms many traditional machine learning approaches, and section $5$ is a data example applying the method to data from the US EPA's ToxCast database.
\section{Model} \subsection{Basic Approach} \label{s:model} Consider modeling the surface $h: (\mathcal{S} \times \mathcal{D}) \rightarrow \mathbb{R}$ where $\mathcal{S} \subset \mathbb{R}^P,$ $ P \geq 1,$ and $\mathcal{D} \subset \mathbb{R}^Q,$ $Q \geq 1$. Given $s \in \mathcal{S}$ and $d \in \mathcal{D},$ tensor product spline approaches \citep[chapter 17]{deBoor2000} approximate $h$ as a product of spline functions defined over $\mathcal{S}$ and $\mathcal{D},$ i.e., \begin{align}
g \otimes f &= g(s)f(d) \hspace{1cm} \forall s \in \mathcal{S}, d \in \mathcal{D}, \label{m:TP1} \end{align} for $g: \mathcal{S} \rightarrow \mathbb{R}^P$ and $f: \mathcal{D} \rightarrow \mathbb{R}^Q.$ The tensor product spline defines $g$ and $f$ to be in the span of a spline basis. Assuming $g$ and $f$ are functions defined as \begin{align*} g(s) &= \sum_{j=1}^J \lambda_{j} \phi_{j}(s) \end{align*} and \begin{align*} f(d) &= \sum_{l=1}^{L} \gamma_{l} \nu_{l}(d), \end{align*} where $\{\phi_{j}(s)\}_{j=1}^{J}$ and $\{\nu_j(d)\}_{l=1}^{L}$ are spline bases. The tensor product spline is \begin{align*} g \otimes f = \sum_{j=1}^J \sum_{l=1}^{L} \rho_{jl} \phi_{j}(s)\nu_{l}(d), \end{align*} where $\rho_{jl} = \lambda_{j} \gamma_{l}.$ As this approach typically uses 1- or 2-dimensional spline bases, when the dimension of $(\mathcal{S} \times \mathcal{D})$ is large the tensor product becomes impractical as the number of functions in the tensor product increases exponentially.
Where tensor product spline models define the basis through a spline basis \textit{a priori}, many functional data approaches (e.g, \citet{montagna2012}) model functions from a basis learned from the data. Often the function space is not defined over $(\mathcal{S}\times \mathcal{D})$ directly, but is constructed on a smaller dimensional subspace, defined to be $\mathcal{D}$ in the present discussion. For cross section $i,$ this approach models $h(s_i,d),$ to be in the span of a finite basis $\{f_1(d),\ldots,f_K(d)\}.$ That is, \begin{align} h(s_i,d) = \sum_{k=1}^{K} z_{ik} f_k(d), \label{m:eq1} \end{align} where $(z_{i1},\ldots,z_{iK})'$ is a vector of basis coefficients. This effectively ignores $\mathcal{S}$ which may not be reasonable in many applications.
To model $h(s,d)$ over $( \mathcal{S} \times \mathcal{D}),$ the functional data and tensor product approaches can be combined. The idea is to define a basis over $(\mathcal{S}\times \mathcal{D})$ where each basis function is the tensor product of two surfaces defined on $\mathcal{S}$ and $\mathcal{D}$ respectively. Extending (\ref{m:eq1}), define $\{g_1(s),\ldots,g_K(s)\}$ to be surfaces on $\mathcal{S},$ and replace each $z_{ik}$ with $\zeta_k g_k(s_i).$ Now, loadings are continuous function indexed by $s$, and a new basis $\{g_1\otimes f_1,\ldots, g_K \otimes f_K \}$ with \begin{align}
h(s,d) &= \sum^{K}_{k=1} \zeta_k g_k(s) f_k(d) \label{S2:model}. \end{align}
As pointed out by a referee, an alternative way to look at (\ref{S2:model}) is that of an ensemble learner \citep{sollich1996}, which includes techniques such as bagging \cite{breiman1996} and random forests \citep{breiman2001}. Ensemble learners describe the estimate as a weighted sum of learners (see \citet{murphy2012} and references therein). In this way, model (\ref{S2:model}) can be looked as a weighted sum over tensor tensor product learners, and related to an ensemble based approach.
To define a tensor product, the functions
$\{g_k(s)\}_{k=1}^{K}$ and $\{f_k(d)\}_{k=1}^{K}$ must be in of a linear function space \citep[pg 293]{deboor2001}. For flexibility, let \begin{align*} g_k &\sim \mathcal{GP}(0,\sigma^{g}_k(\cdot,\cdot)) \\ f_k &\sim \mathcal{GP}(0,\sigma^{f}_k(\cdot,\cdot)) \end{align*} where $\sigma^{g}_k(\cdot,\cdot)$ and $\sigma^{f}_k(\cdot,\cdot)$ are positive definite kernel functions. This places $g_k$ and $f_k,$ $k = 1,\ldots, K,$ in a reproducing kernel Hilbert space defined by $\sigma^{g}_k(\cdot,\cdot)$ or $\sigma^{f}_k(\cdot,\cdot),$ and embeds $h(s,d)$ in a linear space defined by the tensor product of these functions. By estimating the hyperparameters, this approach learns the basis of each function in (\ref{S2:model}), which may be preferable to a tensor spline approach that defines the basis \textit{a priori}.
One may mistake this definition to be a GP defined by the tensor product of covariance kernels (e.g., see \citet{bonilla2007}). Such an approach forms a new covariance kernel over $(\mathcal{S} \times \mathcal{D})$ as a product of individual covariance kernels defined on $\mathcal{S}$ and $\mathcal{D}$. For the proposed model, the product is the modeled function and not the individual covariance kernels.
\subsection{Selection of K}
The number of elements in the basis is determined by $K.$ The larger $K$, the richer the class of functions the model can entertain. In many cases, one would not expect a large number of functions to
contribute to the surface and would prefer as few components as possible. One could place a prior on $K,$ but it is difficult to find efficient sampling algorithms in this case. As an alternative, the multiplicative gamma process \citep{bhattacharya2011} is used to define a prior over the $\zeta_1, \ldots \zeta_k$ that allows the sum to adapt to the necessary number of components. Here, \begin{align*}
\zeta_k \sim N\bigg(0, \bigg[\phi \prod_{j=1}^{k} \delta_j\bigg]^{-1} \bigg) \end{align*} with $\phi \sim \text{Ga}(1,1) $ and $\delta_j \sim \text{Ga}(a_1,1),$ $1 \leq j \leq K.$ This is an adaptive shrinkage prior over the functions. If $a_1 > 1,$ the variances are stochastically decreasing favoring more shrinkage as $k$ increases. The choice of this prior implies the surface defined by $\zeta_k g_k(s)f_k(d)$ is increasingly close to zero as $k$ increases. As many of the basis functions contribute negligibly to modeling the surface, this induces effective basis selection.
\subsection{Relationships to other Models} Though GPs are used in the specification of the model, one may use alternative approaches such as polynomial spline models or process-convolution approaches \citep{higdon2002}. Depending on the choice (\ref{S2:model}) can degenerate into other methods. For example, if $\sigma_1^g,\ldots ,$ and $\sigma_K^g$ are defined as white noise processes and $f_1(s),\cdots,$ and $f_k(s)$ defined to be in the linear span of the same finite basis, the model is identical to the approach of \citet{montagna2012}, which connects the model to functional data approaches. In this way, the additive adaptive tensor product model can be looked at as a functional model with loadings correlated by a continuous stochastic process over $\mathcal{S},$ instead of a white noise process.
If $P+Q=2$ and the functions $g_1$ and $f_1$ are defined using a spline basis, this approach trivially degenerates to the tensor product spline model. Similarly, let each function in $\{f_1(d),\ldots,f_k(d)\}$ be defined using a common basis, with, \begin{align*}
f_k(d) = \sum_{\ell=1}^{L} \beta_{\ell} \phi_\ell(d), \end{align*} where $\{\phi_\ell(d)\}_{\ell=1}^{L}$ is a basis used for all $f_k(d)$. In this case, model (\ref{S2:model}) can be re-written as \begin{align*}
g(s,d) = \sum_{k=1}^{K} \sum_{\ell=1}^{L} \zeta_k \beta_{\ell} g_k(s) \phi_\ell(d). \end{align*} Letting $\beta^{\ast}_{lk} = \zeta_k \beta_{\ell}$, one arrives at a tensor product model with learned basis $\{g_1(s),\ldots,g_{K}(s)\}$ and specified basis $\{\phi_\ell(d)\}_{\ell=1}^{L}.$
\subsection{Computational Benefits} When $\mathcal{D}$ is observed on a fixed number of points and $s$ is the same for each cross section, the proposed approach can deliver substantial reductions in the computational resources needed when compared to GP regression. Let $r$ be the number of unique replicates on $\mathcal{D},$ and let $n$ be the total number of observed cross sections. For a GP, the dimension of the corresponding covariance matrix is $rn.$ Inverting this matrix is an $\mathcal{O}( [rn]^3)$ operation. For the proposed approach, there are $K$ inversions of a matrix of dimension $r$ and $K$ inversions of a matrix of dimension $n.$ This results in a computational complexity of $\mathcal{O}( K[r^3+n^3]),$ which can be significantly less than a GP based method. In the data example this results in approximately $1/20th$ the resources needed as compared to the GP approach, here $K=15$, $n=669$, and $r=7$. Savings increase as the experiment becomes more balanced. For example, if $n=r$ and there are the same number of observations as in the data example, then a GP approach would require $10,000$ times more computing time than the proposed method.
\section{Data Model and Sampling Algorithm} \subsection{Data Model} A Markov chain Monte Carlo (MCMC) sampling approach is outlined for normal errors. Assume that for cross section $i,$ $i = 1,\ldots, n,$ one observes $C_i$ measurements at $\{(s_i,d_{ic})\}_{c=1}^{C_i}.$ For error prone observation $y_{i}(s_i,d_{ic}),$ let \begin{align*}
y_{i}(s_i,d_{ic}) = h(s_i,d_{ic}) + \epsilon_{ic}, \end{align*} where $\epsilon_{ic} \sim N(0,\tau^{-1}).$ Define $N= \sum_{i=1}^{n} C_i.$ Model (\ref{S2:model}) assumes the surface is centered at zero; here, let the model be centered at $f_0(d).$
In defining $h_k(s)$ and $f_k(d)$, the covariance kernel, along with its hyper-parameters, determines the smoothness of the function. The squared exponential kernel is used to model smooth response surfaces. Let \begin{align}
\sigma^{g}_k(s,s') = \varsigma_k \exp\bigg( -\theta_k \|s-s'\|^2\bigg) \label{S3:covmodel1} \end{align} and \begin{align}
\sigma^{f}_k(d,d') = \exp\bigg( -\omega_k \|d-d'\|^2 \bigg), \label{S3:covmodel2} \end{align}
where $\| \cdot \|$ is the Euclidean norm, $\varsigma_k$ is the prior variance, $\theta_k$ and $\omega_k$ are scale parameters, and $1 \leq k \leq K$. In this specification, the parameter $\zeta_k$ is not necessary; it is equivently defined through the variance of the GP where $\varsigma_k = (\phi \prod_{j=1}^{k} \delta_j)^{-1}.$ To allow for a variance other than one for the process $f_0$, let
$\sigma^{f}_0(d,d') = \nu \exp( -\omega_0 \|d-d'\|^2 ).$
Depending on the application any positive definite kernel may be used. For example, one may wish to use a Matern kernel, which is frequently used in spatial statistics. As $\mathcal{S}$ is described in the application as the distance between spatial locations in an abstract chemical space, such an approach may be appropriate. Inpreliminary tests, a Matern kernel provides results that are qualitatively identical to the squared exponential kernel, and it is not considered further.
Given these choices the data model is \begin{align}
h(s,d) &= f_0(d) + \sum^{K}_{k=1} g_k(s) f_k(d). \label{S3:model} \end{align} Appropriate priors are placed over the hyperparameters of $\{\sigma_k^f\}_{k=0}^{K}$ and $\{\sigma_k^g\}_{k=1}^{K}.$ For the length parameter of the squared exponential kernels in (\ref{S3:covmodel1}) and (\ref{S3:covmodel2}), uniform distributions, i.e., $\text{Unif}(a,b),$ $0 < a < b,$ are placed over the scale parameters $\theta_k$ and $\omega_k.$ This places equal prior probability over a range of plausible values allowing the smoothness of the constituent basis to be learned. The value $a$ is chosen so that correlations between any two points in the space of interest are close to $1,$ and $b$ is chosen so that correlation between any two points in the resultant correlation is approximately to $0.$
\subsection{Sampling Algorithm}
Define the vector of observations $Y = \{y_1(s_1,d_{11}),\ldots,y_1(s_1,d_{1C_1}),$ $\ldots, y_1(s_n,d_{n1}),$$\ldots,y_1(s_1,d_{nC_n})\}'$ to be the $(N\times1)$ vector of measurements across the $n$ observed curves. Likewise define \begin{align*}
\mathbf{G} &= \left[ \begin{array}{ccccc}
1&g_1(s_1) & g_2(s_1) & \cdots & g_K(s_1) \\
\vdots&\vdots & \vdots & & \vdots \\
1&g_1(s_1) & g_2(s_1) & \cdots & g_K(s_1) \\
\vdots&\vdots & \vdots & & \vdots \\
1&g_1(s_n) & g_2(s_n) & \cdots & g_K(s_n) \\
\vdots&\vdots & \vdots & & \vdots \\
1&g_1(s_n) & g_2(s_n) & \cdots & g_K(s_n) \\
\end{array} \right], \end{align*} to be an $(N \times K+1)$ matrix, where each row corresponds to the loadings for observation $y_i(s_i,d_{ic}).$ Let \begin{align*}
\mathbf{F} = \left[ \begin{array}{cccc}
f_0(d_{11})& f_1(d_{11}) & \cdots & f_K(d_{11}) \\
\vdots &\vdots & & \vdots \\
f_0(d_{1C_1})& f_1(d_{1C_1}) & \cdots & f_K(d_{1C_1})\\
\vdots &\vdots & & \vdots \\
f_0(d_{n1})& f_1(d_{n1}) & \cdots & f_K(d_{n1})\\
\vdots &\vdots & & \vdots \\
f_0(d_{nC_n})& f_1(d_{nC_n}) & \cdots & f_K(d_{nC_n})\\
\end{array} \right] \end{align*} be an $(N \times K+1)$ matrix, where each row represents the basis functions evaluated at $d_{ic}.$ Using these definitions, (\ref{S3:model}) is expressible as \begin{align}
Y = \bigg(\mathbf{G}\circ\mathbf{F}\bigg) J' + \mathbf{\epsilon}, \label{S3:sampf} \end{align} where $\circ$ is the Schur product, $J = (1,1,\ldots,1)$ is a $K+1$ row vector, and $\mathbf{\epsilon} = (\epsilon_{11},\ldots,\epsilon_{iC_1},$ $\ldots,\epsilon_{n1},\ldots,\epsilon_{nC_n})'$ is the $(N \times 1)$ vector of error terms.
Let $D = \{d^{\ast}_r\}_{r=1}^{R}$ be the set of $R$ uniquely observed inputs across all observations, and define $\mathcal{I}^f$ to be an $(N \times R)$ matrix where the rows corresponds to each element in $Y.$ For each row in $\mathcal{I}^f$, all entries are set to zero except at column $r.$ This entry is set to one, and it corresponds to the observation $y_i(s_i,d_{ic})$ such that $d^{\ast}_r = d_{ic}.$ Likewise, define the matrix $\mathcal{I}^{g}$ to be an $(N \times n)$ matrix. For each row $r$, each entry is set to zero except at column $i,$ which is set to one. This entry corresponds to $Y_r = y_i(s_i,d_{ic}).$
Sample from $\{g_k\}_{k=1}^K,$ $\{f_k\}_{k=0}^K,$ $\phi$, and $\{\delta_k\}_{k=1}^{K}$ in a series of Gibbs steps as follows:
\begin{enumerate}
\item For each $k$, $0 \leq k \leq K$ ,
letting $Y^{\ast} = Y - (\mathbf{G}_{-k}\circ\mathbf{F}_{-k} ) J'_{-k},$
where $G_{-k},$ $F_{-k},$ and $J_{-k}$ are $G,$ $F$ and $J$
without column $k,$ sample $f_k \sim N(M,V)$ at
$\{d^{\ast}_r\}_{r=1}^{R}.$ Here
\begin{align*}
V &= \Sigma_k(\tau\mathcal{G}'\mathcal{G}\Sigma_k + I)^{-1},\\
M &= V (\tau \mathcal{G} Y^{\ast}),
\end{align*}
were $\Sigma_k$ is the $(R \times R)$
covariance function constructed from $\sigma^f_k(\cdot,\cdot),$
$\mathcal{G}$ is an $(N \times R)$ matrix
defined as $G_k \circ \mathcal{I}^f,$ and $I$ is an
$R \times R$ identity matrix.
\item For each $k$, $1 \leq k \leq K$ and letting $Y^{\ast} = Y - (\mathbf{G}_{-k}\circ\mathbf{F}_{-k} ) J'_{-k}$,
sample $g_k \sim N(M,V)$ at $\{s_i\}_{i=1}^{n}.$
Here
\begin{align*}
V &= \Sigma_k(\tau\mathcal{F}'\mathcal{F}\Sigma_k + I)^{-1}, \\
M &= V (\tau \mathcal{F}' Y^{\ast}),
\end{align*}
where $\Sigma_k$ is the $(n \times n)$ covariance matrix formed
from $\sigma^g_k(\cdot,\cdot)$, $\mathcal{F}$ is
an $(N \times n)$ matrix defined to be $(FJ')\circ \mathcal{I}^g,$ and $I$ is the
$n \times n$ identity matrix.
\item Sample $\phi \sim \mbox{Ga}(c,d)$, where
\begin{align*}
c &= K\frac{n}{2}+ a_1,\\
d &= \bigg[\sum_{i=1}^K \bigg(\prod_{j=1}^i \delta_j\bigg) G_i^{'} \Sigma_i G_i \bigg]+ 1,
\end{align*}
$G_i$ is a column vector from column $k$ of $G,$ and $\Sigma_i$ is the
matrix formed from $\exp(-\theta_i||s-s'||^2)$
\item For each $k$ sample $\delta_k \sim \mbox{Ga} (c,d),$ where
\begin{align*}
c &= (K-k+1)\frac{n}{2}+ a_1\\
d &= \bigg[\sum_{i=k}^K \phi \bigg(\prod_{j=1,j \neq k}^i \delta_j\bigg) G_i^{'} \Sigma_i G_i\bigg] + 1
\end{align*}
$G_i$ is a column vector from column $i$ of $G,$ $\Sigma_i$ is the
matrix formed from $\exp(-\theta_i||s-s'||^2),$ and $\bigg(\prod_{j=1,j \neq 1}^1 \delta_j\bigg) = 1$.
\end{enumerate}
The other parameters in the model are sampled using Gibbs or Metropolis steps. The algorithm is written in the \texttt{R} programming language \citep{r2015} using the \texttt{Rcpp} \texttt{C++} extensions \citep{eddelbuettel2013} and the \texttt{RcppArmadillo} \citep{eddelbuettel2014} linear algebra extensions and is available in the supplement.
When the number of unique observations is small, it is possible to develop a block Gibbs sampler for all of the $\{f_k\}_{k=0}^K$ and $\{g_k\}_{k=1}^{K}.$ In large problems, this requires the inversion of a large matrix offsetting the benefits of the increased computational efficiency.
\subsection{Predictive Inference}
The posterior predictive distribution of $n^{\ast}$ unobserved cross sections of $h(s,d)$ at $\grave{S} = \{\grave{s}_1,\ldots,\grave{s}_{n^{\ast}}\}$ is estimated through MCMC. Let the vector $g_k = (g_k(s_1),\ldots,g_k(s_n))'$ be observed at $S =(s_1,\ldots, s_n)'$ and one is interested in $\grave{g}_k = (g_k(\grave{s}_1),\ldots,g_k(\grave{s}_{n^{\ast}}))$ evaluated at $\grave{S}$; $g_k$ and $\grave{g}_k$ are jointly distributed as
\begin{align*}
\left[ \begin{array}{c} g_k \\ \grave{g}_k \end{array} \right]
\sim N\left(0,\left[ \begin{array}{cc} \Sigma^{g}_k(S,S) & \Sigma^{g}_k(S,\grave{S}) \\
\Sigma^{g}_k(\grave{S},S) & \Sigma^{g}_k(\grave{S},\grave{S})
\end{array} \right] \right). \end{align*} Here $\Sigma^{g}_k(S,S),$ $\Sigma^{g}_k(\grave{S},\grave{S}),$ $\Sigma^{g}_k(\grave{S},S)$ and $\Sigma^{g}_k(S,\grave{S})$ represent covariance matrices given $S,$ $\grave{S}$ and $\sigma^{g}_k(\cdot,\cdot).$ Using properties of the multivariate normal distribution, conditionally on $g_k$
\begin{align*} \grave{g}_k \mid g_k \sim &N\bigg[\Sigma^{g}_k(\grave{S},S) \Sigma^{g}_k(S,S)^{-1} g_k, \Sigma^{g}_k(\grave{S},\grave{S}) \\&- \Sigma^{g}_k(\grave{S},S) \Sigma^{g}_k(S,S)^{-1} \Sigma^{g}_k(S,\grave{S})\bigg]. \end{align*}
For each iteration, this expression is used to draw $\{\grave{g}_1,\ldots,\grave{g}_K\}.$ Given this draw, as well as $\{f_0,\ldots,f_K \}$ which represents $f_0(d),f_1(d)$ etc., evaluated at $D = \{d^{\ast}_r\}_{r=1}^R$, one can estimate the posterior predicted distribution of $h(\grave{S},D).$ If new $\grave{D}$ are of interest, the same technique can be applied to estimating $\{\grave{f}_0,\ldots,\grave{f}_K.$ These values can be used with $\{\grave{g}_k\}_{k=1}^{K}$ to provide estimates for $h(\grave{S},\grave{D}).$
\section{Simulation} This approach was tested on synthetic data. Here the dimension of $\mathcal{S}$ was chosen to be 2 or 3, and for a given dimension, $50$ synthetic data sets were created, For each data set, a total of $1000$ cross sections of $h(s,d)$ were observed at seven dose groups. Each data set contained a total of $7000$ observations.
To create a data set, the chemical information vector $s_i$, for chemical $i = 1, \ldots, 1000$, was sampled uniformly over the unit square/cube. At each $s_i,$ $h(s_i,d)$ was sampled at $d= 0, 0.375, 0.75, 1.5,3.0,4.5,$ and $ 6,$ from \begin{align*}
h(s_i,d) = \frac{\nu(s_i) d^m}{\kappa(s_i)^m+d^m}, \end{align*} where $\nu(s_i)$ is the magnitude of the response and $\kappa(s_i)$ is the dose $d$ where the response is at $50\%$ of the maximum. To vary the response over over $\mathcal{S},$ a different zero centered Gaussian process, $z(s)$, was sampled at $\{s_i\}_{i=1}^{1000},$ and for each simulation; $\nu(s_i)$ and $\kappa(s_i)$ were functions of $z(s)$ with $\nu(s_i) = 11 \max( z(s_i),0)$ and $\kappa(s_i) = \max(4.5 - \nu(s_i),0).$ This resulted in the maximum response being between $0$ and approximately $50$ and the dose resulting in $50\%$ response being placed closer to zero for steeper dose-responses. Sample data sets used in the simulation are available in the supplement.
In specifying the model, priors were placed over parameters that reflect assumptions based upon the smoothness of the curve. For $\{\sigma_k^g(\cdot,\cdot)\}_{k=1}^{K},$ each $\theta_k$ was drawn from a discrete uniform distribution over the set $\{0.05,0.1,0.15,\ldots,4.05\}.$ Additionally, for $\{\sigma_k^f(\cdot,\cdot)\}_{k=0}^K,$ each $\omega_k \sim \text{Unif}(0.1,1.5).$ For $\{\sigma_k^g(\cdot,\cdot)\}_{k=1}^K$, $\delta_k \sim \text{Ga}(2,1),$ $1 \leq k \leq K$, which were the choices used in \citep{bhattacharya2011}. The choice of the parameters for the prior over the $\delta_k$ were examined, and the results were nearly identical with $\delta_k \sim \text{Ga}(5,1).$ The prior specification for the model was completed by letting $\tau \sim \text{Ga}(1,1).$
A total of $12,000$ MCMC samples were taken with the first $2,000$ disregarded as burn in. Trace plots from multiple chains were monitored for convergence. Convergence was fast usually occurring after $500$ iterations with approximately $4$ functions in the learned basis defining the surface (i.e. having posterior variance at or above 1).
To analyze the choice of $K$, the performance of the model was evaluated for $K=1, 2, 3$ and $15.$ The estimates produced from these models were compared against bagged multivariate adaptive regression splines (MARS) \citep{friedman1991} and bagged neural networks \citep{zhou2002} using the `caret' package in R \citep{caret2016}. Additionally, treed Gaussian Processes \citep{gramacy2008} ,using the R `tgp' package \citep{gramacy2007}, were used in the comparison. All packages were run using their default settings; $100$ bagged samples were used for both the MARS and neural network models. For the neural network model, $100$ hidden layers were used. Posterior predictions from the treed Gaussian process were obtained from the maximum \textit{a posteriori} (MAP) estimate. This was done as initial tests revealed estimates sampled from the posterior distribution were no better than the MAP estimate, but sampling from the full posterior dramatically increased computation time making the full simulation impossible.
For each data set, $N= 75,125,$ and $175$ curves were randomly sampled and used to train the respective model; the remaining curves were used as a hold out sample for posterior predictive inference with the true curve being compared against the predictions. All methods were compared using the mean squared predicted error (MSPE).
\begin{table}
\caption{Mean squared prediction error in the simulation of the adaptive tensor product approach for four
values of K as well as treed Gaussian processes, bagged neural networks, and bagged
multivariate regression splines (MARS).}\label{sim:tab1}
\begin{tabular}{ccccccccc}
& & \multicolumn{4}{c}{Adaptive TP}& & & \\
& & K=1 & K=2 & K=3 & K=15 & Neural Net & MARS & Treed GP \\
\hline \hline \multirow{ 3}{*}{2-dimensions} & N=75 & 76.2 & 69.1 & 69.5 & 69.8 & 108.3 & $215.2^{1}$ &$839.7^{1}$ \\
& N=125 & 56.9 & 48.5 & 48.8 & 48.7 & 92.4 & 205.8 &$158.0^{1}$ \\
& N=175 & 48.5 & 37.7 & 38.4 & 38.3 & 85.1 & 198.8 & 61.1 \\
\hline \hline \multirow{ 3}{*}{3-dimensions} & N=75 & 164.9 & 162.0 & 155.4 & 155.4 & 185.2 &$246.6^{1}$ &$1521.5^{1}$ \\
&N=125 & 128.6 & 125.0 & 121.0 & 121.0 & 160.4 & 223.4 &$421.3^{1}$ \\
&N=175 & 106.3& 102.6 & 99.7 & 100.1 & 150.0 & 217.7 &$163.5^{1}$ \\
\multicolumn{7}{l}{\footnotesize$^{1}$ Trimmed mean used with $5\%$ of the upper and lower tails removed.} \\
\end{tabular}
\end{table}
Table (\ref{sim:tab1}) describes the average mean squared predicted error across all simulations. For the adaptive tensor product model there is an improvement in prediction when increasing $K.$ For the 2-dimensional case the improvment occurs from $1$ to $2,$ but the results of $K=3$ and $15$ are almost identical to $K=2.$ For the 3-dimensional case, improvements are seen up to $K=3$ with identical results for $K=15$. These results support the assertion that one can make $K$ large and let the model adapt to the number of tensor products in the sum.
This table also gives the MSPE for the other methods. Compared to the other approaches, the adaptive tensor product approach is superior. For $N=175$, the treed Gaussian process, which is often the closest competitor, produces mean square prediction errors that are about $1.5$ times greater than the adaptive TP approach. For smaller values of $N,$ the treed GP as well as the bagged MARS approach failed to produce realistic predictions. In these cases, the trimmed mean with $5\%$ of the tails removed was used to provide a better estimate of center.
\begin{figure}
\caption{Comparison of the predictive performance between the adaptive tensor product and the treed Gaussian process. In the figure, the corresponding model's root mean squared predicted error is given as a contour plot. The heat map represents the maximum dose response given the coordinate pair; lighter colors represent greater dose-response activity. }
\label{fig:sim}
\end{figure}
Figure (\ref{fig:sim}), shows where the gains in prediction take place. As a surrogate for the intensity of the true dose-response function, the intensity of the dose-response at $d=6$ across $\mathcal{S}$ is shown in the 2-dimensional case. Dark gray regions are areas of little or no dose-response activity; lighter regions have the steeper dose-responses. Contour plots of the model's root MSPE are overlaid on the heat maps. The top plot shows the performance of the adaptive TP and the bottom plot shows the performance of the treed GP, which was the closest competitor for this data set. The figure shows the adaptive TP is generally better at predicting the dose-response curve across $\mathcal{S}$ and that larger gains are made in regions of high dose response activity. This is most evident in the lower left corner of both plots, where regions of high dose response activity are located. At the peaks, the root MSPE is almost double that for the treed Gaussian process.
\section{Data Example} The approach is applied to data released from Phase II of the ToxCast high throughput platform. The AttaGene PXR assay was chosen as it has the highest number of active dose-response curves across the chemicals tested. This assay targets the Preganene X receptor, which detects the presence of foreign substances and up regulates proteins involved with detoxification in the body; an increased response for this assay might relate to the relative toxicity of the given chemical.
Chemical descriptors were calculated using Mold$^2$ \citep{hong2008} where chemical structure was described from simplified molecular-input line-entry system (SMILES) information \citep{weininger1988}. Mold$^2$ computes $777$ unique chemical descriptors. For the set of descriptors, a principal component analysis was performed across all chemicals. This is a standard technique in the QSAR liturature \citep[pg 44]{emmert2012}. Here, the first $38$ principal components, representing approximately $95\%$ of the descriptor variability, were used as a surrogate for the chemical descriptor $s_i$.
The database was restricted to $969$ chemicals having SMILES information available. In the assay, each chemical was tested across a range of doses between $0\hspace{2pt} \mu M$ and $250\hspace{2pt} \mu M$ with no tests done at exactly a zero dose. Eight doses were used per chemical, with each chemical being tested at
different doses within the above range. Most chemicals had one observation per dose; however, some of the chemicals tested had multiple observations per dose. In total, the data set consisted of $9111$ data points from the $969$ distinct dose-response curves.
A random sample of $669$ chemicals was taken and the model wad trained to this data; in evaluating posterior predictive performance, the remaining $300$ chemicals were used as a hold out sample. In this analysis, $d$ was the log dose, where this value was rounded to two significant digits, and the same prior specification was in the simulation was used to train the model. To compare the prediction results, boosted MARS and neural networks were used; treed Gaussian processes were attempted, but the R package `tgp' crashed after 8 hours during burn-in. The method of \citet{low2015} was also attempted, but, the code was designed such that each chemical is tested at the same doses with the same number of replications per dose point. As the ToxCast data are not in this format, the method could not be applied to the data.
\begin{figure}
\caption{Four posterior predicted dose-response curves (black line)
with corresponding $90\%$ predicted credible intervals (dotted lines)
for four chemicals in the hold out samples having repeated measurements per
dose. Grey dash-dotted line represents the predicted response from the bagged
neural network.}
\label{fig:4chem-pred-mult}
\end{figure}
Figures (\ref{fig:4chem-pred-mult}) and (\ref{fig:4chem-pred-sing}) show the posterior predicted curves (solid black line) with equal tail $90\%$ posterior predicted quantiles (dashed line) for eight chemicals in the hold out sample. Figure (\ref{fig:4chem-pred-mult}) describes the predictions for chemicals having multiple measurements per dose group, and figure (\ref{fig:4chem-pred-sing}) gives predictions having only a single observation per dose group. As compared to the observed data, these figures show the model provides accurate dose-response predictions across a variety of shapes and chemical profiles. Additionally the grey dashed-dotted line represents the prediction using the bagged neural network. These estimates are frequently less smooth and further off from the observed data than the adaptive tensor product splines. As additional confirmation the model is predicting dose-response curves based upon the chemical information, one can look a the chemicals from a biological mode of action perspective. For example in (\ref{fig:4chem-pred-mult}), it is interesting to note that the dose-response predictions for both Bisphenol A and $17\alpha$ Hydroxyprogesterone are similar, because both act similarly and are known to bind the estrogen receptor.
\begin{figure}
\caption{Four posterior predicted dose-response curves (black line)
with corresponding $90\%$ predicted credible intervals (dotted lines)
for four chemicals in the hold out samples having repeated measurements per
dose. Grey dash-dotted line represents the predicted response from the bagged
neural network.}
\label{fig:4chem-pred-sing}
\end{figure}
In comparison to the other models, the adaptive tensor product approach also had the lowest predicted mean squared error and the predicted mean absolute error for the data in the hold out sample. Here the model had a predicted mean squared error of $342.1$ and mean absolute error of $11.7$, as compared to values of $354.7$ and $12.4$ for neural networks as well as $383.6$ and $13.4$ for MARS. These results are well in line with the simulation.
One can also compare the ability of the posterior predictive distribution to predict the observations in the hold out sample. To do this, lower and upper tail cut-points defined by $p$ were estimated from the posterior predictive distribution. The number of observations that were below the lower cut-point or above the upper cut-point were counted. Under the assumption the posterior predictive distribution adequately describes the data, this count is $\mbox{Binomial}(2p,n)$ where $n$ equal to the number of observations for that chemical. To test this assumption, the $90\%$ critical value was computed and compared to the count. This was done for $p = 5, 10$ and $15\%;$ here, $88, 89,$ and $90\%$ of the posterior predictive distributions were at or below the $90\%$ critical value. This supports the assertion that the model is accurately predicting the dose-response relationship given the chemical descriptor information $\mathcal{S}$.
The method was also compared to standard QSAR approaches, which model a single data point. Similar to the standard QSAR methodology, a model was fit to each data-set $i$ and the response associated with a given dose was computed to be observation $y_i$. This value was then used in the model \begin{align*} y_i = x(s_i) + \epsilon_i, \end{align*} where $x(s_i)$ is a Gaussian process with squared exponential covariance kernel. The model was fit on the same $669$ observations in the training set and predictions were made of the response at that dose. Both this QSAR approach as well as the adaptive tensor product approach were compared to the hold out samples using the correlation coefficient, a standard practice in the QSAR literature. For the dose of $20 \mu M$ the standard QSAR approach had a correlation of $0.42$ as compared to the co-mixture approaches' correlation of $0.48.$ For the dose of $100 \mu M$ a similar $0.06$ increase was seen, and, when observations that have a chemical within the training set defined as `close' (i.e., a relative distance between two chemicals less than 2.2), this improvement in the correlation coefficient is almost $0.1$ (i.e, $0.5$ compared to $0.6$). This indicates that $10\%$ to $20\%$ improvements in the correlation coefficient using the proposed approach.
\section{Conclusions} The proposed approach allows one to model higher dimensional surfaces as a sum of a learned basis, where the effective number of components in the basis adapts to the complexity of the surface. In the simulation and the motivating problem, this method is shown to be superior to competing approaches, and, given the design of the experiment, it is shown to require significantly less computational resources than GP approaches. Though this approach is demonstrated for high throughput data, it is anticipated it can be used for any multi-dimensional surface.
In terms of the application, this model shows that dose-response curves can be estimated from chemical SAR information, which is a step forward in QSAR modeling. Though such an advance is useful investigating toxic effects, it can also be used in therapeutic effects. It is conceivable, that such an approach can be used \textit{in silico} to find chemicals that have certain therapeutic effects in certain pathways without eliciting toxic effects in other pathways. Such an approach may be of significant use in drug development as well as chemical toxicity research.
Future research may focus on extending this model to multi-output functional responses. For example, multiple biossays dose-responses may be observed, and, as they target similar pathways, are correlated. In such cases, it may be reasonable to assume their responses are both correlated to each other and related to the secondary input, which is the chemical used in the bioassay. Such an approach may allow for lower level \textit{in vitro} bioassays, like the ToxCast endpoint studied here, to be used to model higher level \textit{in vivo} responses.
\appendix
\label{lastpage}
\end{document} | arXiv |
\begin{document}
\title{ Individual Fairness in Feature-Based Pricing for Monopoly Markets }
\begin{abstract} We study fairness in the context of feature-based price discrimination in monopoly markets. We propose a new notion of individual fairness, namely, $\alpha$-\textsf{fairness }, which guarantees that individuals with similar features face similar prices. First, we study discrete valuation space and give an analytical solution for optimal fair feature-based pricing. We show that the cost of fair pricing is defined as the ratio of expected revenue in an optimal feature-based pricing to the expected revenue in an optimal fair feature-based pricing (\textsc{CoF}) can be arbitrarily large in general. When the revenue function is continuous and concave with respect to the prices, we show that one can achieve \textsc{CoF}\ strictly less than $2$, irrespective of the model parameters. Finally, we provide an algorithm to compute fair feature-based pricing strategy that achieves this \textsc{CoF}. \end{abstract}
\section{Introduction} \label{sec:intro2}
The Internet has transformed the way markets function. Today's Internet-based ecosystems such as entertainment and e-commerce marketplaces are more consumer-centric and information-driven than ever before. Data and AI systems are primarily used to power advertising, consumer retention, and personalized experience. These AI systems are deployed to aggregate individual choices and preferences to make personalized experiences possible. It is a common practice to use aggregated information about consumers to offer different prices to different consumers or segments of the market; this practice is commonly termed \emph{price discrimination}~\cite{varian92}.
Price discrimination has come under ethical scrutiny on multiple instances in the recent past. For example, it was found that Orbitz, an online travel agency, charges Mac users more than Windows users~\cite{Orbitz}. Uber's strategy to charge personalized prices came under heavy consumer backlash~\cite{uberOne,uberTwo}, and thanks to the fine-grained data analysis of consumer behavior, several such instances were reported in e-commerce and retail industry~\cite{Hinz11}. More recently, \cite{akshat2021} showed that neighborhoods with high non-white populations, higher poverty, younger residents, and high education levels faced higher cab trip fares in Chicago. Not surprisingly, the regulatory bodies and research community has taken notice. Economists have raised concerns on fairness issues of personalized pricing~\cite{stefan}. Price discrimination based on nationality or residence is made illegal in the ~\cite{EU-PD}. In the USA, a white house report provides guidelines for enforcing existing anti-discrimination, privacy, and consumer protection laws while practicing discriminatory pricing~\cite{usa}. Given the overwhelming evidence and rising concerns, there is an urgent need to study price discrimination and fairness formally.
Sellers or firms use price discrimination for multiple reasons, including increasing revenue, covering transportation and storage costs, increasing market reach, rewarding loyal consumers, promoting a social cause, and so on \cite{cassady46}. In general, price discrimination does not always raise ethical and fairness issues and hence requires a careful inspection to categorize situations where this practice may lead to treatment disparity and invite regulatory intervention \cite{kaplan}. In this work, we focus on designing the pricing strategies for a seller (monopolist) who wants to maximize the revenue via price discrimination while ensuring fairness amongst the consumers.
A revenue-maximizing seller with complete knowledge of consumer valuations without fairness consideration would charge each consumer her valuation for the product. This pricing strategy, otherwise called \emph{first-degree} price discrimination, may result in wild fluctuations in prices and is considered unfair in general \cite{Moriarty21}. Also, in practice, sellers do not have full access to individual consumer valuations but may have a distribution over valuations through \emph{features}. In such \emph{feature-based pricing} (FP), the seller segregates the market into segments through the consumer features. The seller's problem then reduces to finding optimal pricing for each segment~\cite{bergemann2015limits,algorithmic-pricedisc}. Such FP is referred to as \emph{third-degree price discrimination} in the literature. In this paper, our goal is to ensure fairness issues in feature-based personalized pricing.
\paragraph{Our Contributions} We introduce the notion of \emph{$\alpha$-\textsf{fairness }} in price discrimination which ensures that similar individuals face similar prices. We emphasize that if individuals with similar features are charged differently by segregating them into different segments, the interpersonal price comparison based on their features renders fairness issues. With this, we introduce a model for optimal \emph{fair feature-based pricing} (FFP) as the problem of maximizing revenue while ensuring $\alpha$-\textsf{fairness }$\!\!$. We begin with two segments in the market and discrete valuations and propose an optimal FFP scheme (Section \ref{ssec:optdiscreteffp}). To quantify the loss in the revenue due to fairness, we then introduce \emph{cost of fairness} (\textsc{CoF}) -- the ratio of expected revenue in an optimal FFP to the expected revenue in an optimal FP. We prove that a constant lower bound on \textsc{CoF}\ is impossible to achieve in general.
Next, in Section \ref{sec:ffpcontinuous}, under the assumption that the revenue function is concave in offered prices \cite{bergemann2021thirddegree}\footnote{this assumption is standard in economics as a large number of probability distributions follow this},
we show that one can achieve a constant upper bound on \textsc{CoF}. Here, first, we show that the seller can compute optimal FFP using a convex program if it has access to distributional information (knows all consumers' valuation distribution functions). We then identify a class of FFP strategies, namely \textsc{LinP-FFP}\ that satisfy $\alpha$-\textsf{fairness }$\!\!$. With the help of these pricing strategies, we then show that the \textsc{CoF}\ is strictly less than $2$ irrespective of model parameters. Finally, we propose OPT-\textsc{LinP-FFP}, an $O(K\log(K))$ time algorithm where $K$ is the number of segments that does not need access to complete distributional information and computes $\alpha$-fair pricing that achieves the aforementioned \textsc{CoF}\ (Algorithm~\ref{alg:ouralgo} and Theorem~\ref{thm:opt_cofbound}).
\if 0
In summary, our contributions are as follows.
We introduce the notion $\alpha$-\textsf{fairness }\ in discriminatory pricing. With this, \begin{itemize}
\item We propose a model for optimal fair feature-based pricing.
\item For markets with two segments and discrete valuations, we propose an analytical solution to optimal fair feature-based pricing.
\item We prove that, in general, \textsc{CoF}\ could be arbitrarily bad (Theorem~\ref{}).
\item For a well-studied economic model of concave revenue, we show that one can achieve \textsc{CoF}\ strictly less than 2 (Theorem~\ref{}).
\item Finally, \end{itemize} \fi
\section{Related Work}
The impact of discriminatory pricing on consumer and seller surplus was first considered by \cite{bergemann2015limits} when the consumer characteristics are known to the seller. The authors proposed a method to provide the optimal market segmentation. The generalized problem was then considered by \cite{algorithmic-pricedisc} which extended the work of \cite{bergemann2015limits} to the case where only partial information about the consumer's valuation was known to the seller.
When the valuations of the consumers are not known, \cite{value-personalized,value-personalpricing} propose feature-based pricing and provides bounds on the value generated using idealized personalized pricing and Feature-based pricing over Uniform pricing. The value of feature-based pricing depends on the correlation of valuations and consumer features. \cite{random-network} consider the first-degree price discrimination over the social network where the centrality measures in social networks determine the features of the consumers. They provide bounds on the value of network-based personalized pricing in large random social networks with varying edge densities. Our work follows a similar approach because we derive personalized pricing from the features. However, naive feature-based pricing can be very unfair to the consumers, as we show in Proposition \ref{prop:discretecof}. Our focus is to design feature-based pricing that is fair at the same time.
Recently, many questions have been raised on the ethical side of price discrimination methods. \cite{Moriarty21} strongly criticizes online personalized pricing and suggests that personalized prices compete unfairly for social surplus created by transactions. \cite{ethical-legal} points out the need to design personalized pricing with ethical considerations, which can provide win-win outcomes for both organizations and consumers. \cite{fairness-tackled} discusses that discriminatory pricing leads to the perception of unfairness amongst the consumers, which undermines the stability of retail platforms. They discuss that when consumers are involved in forming the prices, this leads to improved fairness perception, thus leading to better retentivity. \cite{design-against-discrimination} discusses that web-based platforms typically use many private features of user profiles to connect buyers and sellers. When users interact on such platforms, it leads to discrimination regarding race, gender, and possibly other protected characteristics. All these studies lead to understanding the optimal price discriminatory strategies under the fairness constraint, which is the focus of our work.
Finally, \cite{personalfairness} presents a list of metrics like \emph{price disparity}, \emph{equal access}, \emph{allocative efficiency fairness} to measure and analyze fairness in feature-based pricing and study its interplay with welfare. The metrics discussed are mainly the group fairness notions which are entirely different from $\alpha$-\textsf{fairness } discussed in this paper. We emphasize that though the above papers discuss the ethical issues in price discrimination, none of them provides a systematic approach to design the pricing strategy that maximizes the revenue and ensures the fairness guarantee.
\section{Preliminaries} \label{sec:prelim} We consider a market with a monopolist seller seeking to price a single product available in infinite supply. The market is divided into finite number of segments $\mathcal{X} = \{x_1, x_2, \dots , x_K\}$, where $x_i$ represents the $i^{\text{th}}$ segment. The seller, given access to $\mathcal{X}$, can choose to price discriminate across segments to extract maximum revenue.
\begin{table*}[ht!]
\centering
\renewcommand{1}{1.2}
\begin{tabular}{|c|c|}
\hline \textbf{Notation} & \textbf{Description} \\
\hline
FP & Feature-based Pricing \\ \hline
FFP & Fair Feature-based Pricing \\
\hline
$\mathcal{F}_k$, $f_k()$ & Valuations CDF, PDF for $k^{\text{th}}$ consumer segment respectively\\ \hline
$\mathcal{X}$ & Set of all consumer features/types \\ \hline
$\mathcal{V}$ & Support set of consumers' valuations \\ \hline
$x_k$ & Consumer feature of the $k^{\text{th}}$ segment \\ \hline
$\beta_k$ & The fraction of consumers in the $k^{\text{th}}$ segment \\ \hline
$\mathbf{p} = (p_1, p_2, \ldots p_K)$ & Feature-based price vector \\ \hline
$\pi_k(p_k)$ & Revenue generated per consumer in the $k^{\text{th}}$ segment \\ \hline
$\Pi(p)$ & Revenue generated by $p$ across all consumer segments \\ \hline
$\mathbf{\widehat{p}} = (\widehat{p}_1, \widehat{p}_2, \ldots \widehat{p}_K)$ & Price function in optimal price discrimination \\ \hline
$d_{ij}:=d(x_i, x_j)$ & A real-valued metric on the consumer feature space $\mathcal{X}$ \\ \hline
$\alpha$ & Fairness parameter \\ \hline
$\mathbf{p^{\star}} = (p^{\star}_1, p^{\star}_2, \ldots p^{\star}_K)$ & Optimal fair feature-based price function \\ \hline
$\widetilde{\mathbf{p}} = (\widetilde{p}_1,\widetilde{p}_2,\ldots,\widetilde{p}_K)$ & Price vector for OPT-\textsc{LinP-FFP} \\ \hline
\textsc{CoF} & Cost of Fairness \\ \hline
$L_m$ & Linear approximation of concave revenue curve with $m$ as parameter \\ \hline
\end{tabular}
\renewcommand{1}{1}
\caption{Notation Table}
\label{tab:notations} \end{table*}
Consumers' valuations for the single product are non-negative random variables drawn from the set $\mathcal{V}$ (same across all segments). Let $\mathcal{F}_i(\cdot)$ be the cumulative distribution function for the valuation of the consumers in $i^{\text{th}}$ segment, and $f_i(\cdot)$ be corresponding probability density function (probability mass function when $\mathcal{V}$ is discrete). In this paper, we consider the following two cases separately, (a) $\mathcal{V}$ is discrete and finite, and (b) $\mathcal{V}$ is continuous. Next, we present feature-based pricing model.
\subsection{Feature-based Pricing Model} \label{ssec:featureprice} In feature-based pricing (FP), one can consider, without loss of generality, that the consumer feature is a representative of the market segment to which she belongs. Note that multiple consumers may have the same feature vector, and all the consumers having identical features belong to the same market segment. For simplicity, we will write $p_i:=$ price offered to the consumer in the $i^{\text{th}}$ segment. A consumer makes the purchase only if her valuation is equal to or more than the offered price. The expected revenue per consumer generated from the $i^{\text{th}}$ segment with a price $p_i\in \mathbb{R}_{+}$ is given by \begin{equation}
\pi_i(p_i) = p_i \cdot (1 - \mathcal{F}_i(p_i)) \\ \end{equation} Whenever it is clear from the context we refer to expected revenue per consumer from a segment to be expected revenue from that segment. Let $\beta_i$ be the fraction of consumers in the $i^{\text{th}}$ segment, then the expected revenue per consumer generated across all segments is given as $\Pi(\mathbf{p}) = \sum_{x_i \in \mathcal{X}} \beta_i\pi_i(p_i) $. We assume that $\beta_i$s are known to the seller. We call the sellers problem of revenue maximization as OPT$_{FP}(\mathcal{V,\mathcal{X},F,\beta})$ where $\mathcal{F}=(\mathcal{F}_1,\ldots,\mathcal{F}_K)$ and $\beta=(\beta_1,\ldots,\beta_K)$.
In the absence of fairness constraints, OPT$_{FP}(\cdot)$ reduces to charging each segment separately and optimal FP strategy $\widehat{\mathbf{p}}$ consisting $\widehat{p_i}$ for segment $i$ is given by
$
\widehat{p}_i \in \underset{p_i \in \mathbb{R}_{+} }{\operatorname{argmax}} \ \pi_i(p_i).$
\paragraph{Fairness in Feature-based Pricing} \label{ssec:ind-fair}
Let $d:\mathcal{X} \times \mathcal{X} \rightarrow \mathbb{R}_{+}$ be a distance function over $\mathcal{X}$. We assume that such a function exists and is well defined in $\mathcal{X}$, i.e., $(\mathcal{X},d)$ is a metric space. The distance function quantifies the dissimilarity between feature vectors of individuals belonging to market segments. For simplicity we write $d(x_i, x_j) := d_{ij}$. Individual fairness in FP strategy is defined as: \begin{definition}[$\alpha$-\textsf{fairness }$\!\!$] \label{def:fairness} A price function $\mathbf{p}:\mathcal{X} \rightarrow \mathbb{R}_+^{K}$ is $\alpha$-fair with respect to $d$ iff for all $x_i,x_j \in \mathcal{X}$, we have \begin{equation}\label{eq:alphafair}
| p_i - p_j| \leq \alpha \cdot d_{ij}. \end{equation} \end{definition} We call a pricing strategy Fair Feature-based Pricing ($\alpha$-FFP) that satisfies \cref{eq:alphafair} with a given value of $\alpha$. It is easy to see from the definition that any $\alpha$-FFP is also $\alpha{'}$-FFP for any $\alpha{'} \geq \alpha$. We will drop the quantifier $\alpha$ and call it FFP when it is clear from the context.
\paragraph{Cost of Fairness (\textsc{CoF}) } Next, we define \textsc{CoF}\ as the deviation from optimality due to fairness constraints given in \cref{eq:alphafair}. It is defined as the ratio of expected revenue generated by optimal feature-based pricing and fair feature-based pricing.
\begin{definition}[\textsc{Cost of Fairness }(\textsc{CoF})] Cost of fairness for an FFP strategy $\mathbf{p}$ is defined as \begin{equation}
\textsc{CoF} = \frac{\Pi(\mathbf{\widehat{p}})}{\Pi(\mathbf{\mathbf{p}})}. \end{equation} \end{definition}
In the following sections, we analyze FP and FFP strategies and their \textsc{CoF}\ when $\mathcal{V}$ is discrete (\cref{sec:discrete}) and continuous (\cref{sec:cont}).
\section{FFP for Discrete Valuations}\label{sec:discrete} We want to ensure $\alpha$-\textsf{fairness }\ in the pricing strategy given the optimal FP. $\alpha$-\textsf{fairness }\ is achieved by maximizing revenue while satisfying the fairness constraints. In this section, we derive optimal FP (\cref{ssec:optdiscretefp}), propose how to achieve $\alpha$-\textsf{fairness }\ (\cref{ssec:optdiscreteffp}), and provide an upper bound on \textsc{CoF}\ (\cref{ssec:cofdiscrete}) for discrete valuation setting.
We consider the simplest setting described as follows: Let the consumer segments be given by $\mathcal{X} = \{ x_1, x_2\}$ and their valuations are drawn from a discrete set $\mathcal{V} = \{ v_1, v_2\}$, we assume $v_1 < v_2$ without loss of generality. Let $\beta_1 = \beta$ and $\beta_2 = 1-\beta$. Further, let $f_1(v_1) = q_1$ ($f_2(v_1) = q_2$) denote the probability that a consumer has valuation $v_1$ in segment 1 (segment 2). The expected revenue generated by $\mathbf{p}$ is given by: \begin{align} \label{eq:opt-price} \Pi(\mathbf{p})=&\beta p_1 [ q_1 \mathbbm{1}(v_1 \geq p_1) + (1-q_1) \mathbbm{1}(v_2 \geq p_1) ] \nonumber \\ & + (1-\beta) p_2 [ q_2 \mathbbm{1}(v_1 \geq p_2) + (1-q_2) \mathbbm{1}(v_2 \geq p_2) ] \end{align}
\subsection{Optimal Feature-based Pricing} \label{ssec:optdiscretefp}
As discussed earlier, $\Pi(\mathbf{p})$ can be maximized by maximizing $\pi_i(p_i)$ for each market segment independently if there are no fairness constraints. This problem is an integer program with price for each consumer type being a discrete variable. The revenue generated depends on $\beta_i$ and $f_i(\cdot)$ ($\beta$, $q_1$, $q_2$ in the current simplest case). The optimal FP is then given as \begin{equation} \label{eq:idealprice1}
\text{For }\, i \in \{1,2\}:\; \widehat{p}_i = \begin{cases}
v_1 & \text{if } q_i\geq 1-\frac{v_1}{v_2} \\
v_2 & \text{otherwise}
\end{cases} \end{equation} \begin{proof} For a market segment $i$, $\pi_i(v_1) = v_1$ and $\pi_i(v_2) = v_2(1-q_i)$. So, $\widehat{p}_i = v_1$ if \begin{equation*} \pi_i(v_1) \geq \pi_i(v_2) \implies v_1 \geq v_2(1-q_i) \implies q_i \geq 1 - \frac{v_1}{v_2} \end{equation*} otherwise, $\widehat{p}_i = v_2$.
\end{proof} Next, we analyze the fairness aspects of the above pricing strategy.
\subsection{Optimal Fair Feature-based Pricing} \label{ssec:optdiscreteffp}
Let $(\mathcal{X},d)$ be a metric space. We model the Optimal fair feature-based pricing (FFP) problem as integer program which maximizes $\Pi(\mathbf{p})$ with $\alpha$-\textsf{fairness } constraints described in Eq.\eqref{eq:alphafair}. We denote this problem as OPT$_{FFP}(\mathcal{V}, \mathcal{X}, d, \mathcal{F}, \mathbf{\beta}, \alpha)$ and the corresponding optimal FFP strategy is denoted as $\mathbf{p}^{\star}$. First we make an interesting and very useful claim for binary valuations.
\begin{lemma} When $\mathcal{V}=\{v_1,v_2\}$, and if $\widehat{\mathbf{p}}$ is not $\alpha$-fair, OPT $_{\mbox{FFP}} (\mathcal{V},\mathcal{X},d,\mathcal{F}, \mathbf{\beta},\alpha)$ reduces to OPT$_{\mbox{FP}} (\widetilde{\mathcal{V}},\mathcal{X},\mathcal{F}, \mathbf{\beta}) $ where $\widetilde{\mathcal{V}} \text{ is either } \{v_1\}, \text{ or } \{v_2\}, \text{ or } \{v_1, v_1+\alpha d_{12}\} $.
\label{lem:reduction} \end{lemma} \iffalse \begin{proof} If the price function is not trivially fair (when $v_2 - v_1 > \alpha d_{12}$ ), then it is easy to see that prices are drawn from one of the sets $\{ v_1\}$, $\{ v_2\}$, or $\{ v_1, v_1 + \alpha d_{12}\}$. The support set $\{ v_2, v_2 - \alpha d_{12}\}$ also ensures $\alpha$-\textsf{fairness } but optimal FFP does not take values from this set as no one with valuation $v_1$ would buy if set is $\{ v_2 - \alpha d_{12}, v_2 \}$. \end{proof} \fi \begin{proof} Let $(p_1,p_2)$ be the tuple of offered prices. Note that if $v_2 - v_1 \leq \alpha d_{12}$ or $\widehat{p}_1 = \widehat{p}_2$, then the optimal $\mathbf{p}^{\star} = \mathbf{\widehat{p}}$ with support $\{v_1, v_2\}$ and $\mathbf{\widehat{p}}$ will be trivially fair. We consider a more interesting case when $v_2 - v_1 > \alpha d_{12}$ and $\widehat{p}_1 \ne \widehat{p}_2$. In this case, the only candidate support sets for optimal fair pricing strategy are: $\{ v_1\}$, $\{ v_2\}$, $\{ v_1, v_1 + \alpha d_{12}\}$, $\{v_2-\alpha d_{12}, v_2\}$. The optimal FFP does not take values from the set $\{v_2-\alpha d_{12}, v_2\}$ as the consumers with valuation $v_1$ would not make any purchase. Hence, the expected revenue with support $\{v_2-\alpha d_{12}, v_2\}$ will be less than or equal to the expected revenue with support $\{v_2\}$. \end{proof}
We now relax the constraint of binary valuation and analyze the optimal fair pricing scheme for $n$ valuations. The consumer segments are $\mathcal{X} = \{x_1,x_2\}$ with $\beta_1 = \beta$ and $\beta_2 = 1-\beta$, the valuations are drawn from the set $\mathcal{V} = \{v_1, v_2, \dots , v_n\}$, and $f_1(v_i) = q_{i,1}$ and $f_2(v_i) = q_{i,2}$. This is a simple extension of the pricing problem, OPT$_{FP} (\mathcal{V},\mathcal{X},\mathcal{F}, \mathbf{\beta})$ modelled as an integer program where the prices are drawn from the set $\mathcal{V}$. If $\mathbf{\widehat{p}}$ is not $\alpha$-fair then, the corresponding OPT$_{FFP} (\mathcal{V},\mathcal{X},d,\mathcal{F}, \mathbf{\beta}, \alpha)$ can be solved by reducing it to OPT$_{FP} (\mathcal{\widetilde{V}},\mathcal{X},\mathcal{F}, \mathbf{\beta})$ with $\widetilde{\mathcal{V}}$ given by:
\begin{equation*}
\widetilde{\mathcal{V}} = \begin{cases}
\{v_i\}, v_i \in \mathcal{V} & \text{if } p_1^{\star} = p_2^{\star} \\
\{v_j, v_j+\alpha d_{12}, v_j-\alpha d_{12}\}, v_j \in \mathcal{V} & \text{if } p_1^{\star} \neq p_2^{\star}
\end{cases} \end{equation*} Given the set $\mathcal{\widehat{V}}$, the pricing problem OPT$_{FP} (\mathcal{\widetilde{V}},\mathcal{X},\mathcal{F}, \mathbf{\beta})$ can be solved in constant time. It is easy to see that computing $\mathcal{\widehat{V}}$ takes $\mathcal{O}(n^{2})$ time for $n$ valuations and $2$ consumer types. Therefore, the fair pricing problem OPT$_{FFP} (\mathcal{V},\mathcal{X},d,\mathcal{F}, \mathbf{\beta}, \alpha)$ can be solved in $\mathcal{O}(n^{2})$ time.
\subsection{\textsc{CoF}\ Analysis} \label{ssec:cofdiscrete} For $n=2$, based on the values of $q_1, q_2$ we have the following cases: \begin{multicols}{2} \begin{enumerate}[leftmargin=*]
\item $p_1^{\star} = p_2^{\star} = v_1$
\item $p_1^{\star} = p_2^{\star} = v_2$
\item $p_1^{\star} = v_1 + \alpha d_{12}$, $p_2^{\star} = v_1$
\item $p_1^{\star} = v_1$, $p_2^{\star} = v_1 + \alpha d_{12}$ \end{enumerate} \end{multicols} \noindent In cases 1 and 2, optimal fair pricing is equivalent to uniform pricing and therefore are `trivially' fair with \textsc{CoF}\ = 1, i.e., $\Pi(\widehat{\mathbf{p}}) = \Pi(\mathbf{p}^{\star})$. For case 3, $\Pi(\widehat{\mathbf{p}})$ and $\Pi(\mathbf{p}^{\star})$ are given as: \begin{align*}
& \Pi(\widehat{\mathbf{p}}) = \beta(v_2)(1-q_1) + (1-\beta)v_1 \\
& \Pi(\mathbf{p}^{\star}) = \beta(v_1 + \alpha d_{12})(1-q_1) + (1-\beta)v_1 \end{align*} Then the cost of fairness for case 3 is given as:
\begin{align}
\textsc{CoF}\ & = \frac{\Pi(\widehat{\mathbf{p}})}{\Pi(\mathbf{p}^{\star})} = \frac{\beta(v_2)(1-q_1) + (1-\beta)v_1}{\beta(v_1 + \alpha d_{12})(1-q_1) + (1-\beta)v_1} \nonumber \\
& = \frac{\beta(v_2-v_1) + v_1 - \beta v_2 q_1}{\beta \alpha d_{12}(1-q_1) - \beta v_1 q_1 + v_1 } \nonumber \\
& = \frac{\beta \left( 1 - \frac{v_1}{v_2}\right) + \frac{v_1}{v_2} - \beta q_1}{\beta \left (\frac{\alpha d_{12}}{v_2}\right)(1-q_1) - \beta \left( \frac{v_1}{v_2}\right)q_1 + \frac{v_1}{v_2}}
\label{eq:cof} \end{align}
Replacing $\beta$ with $(1-\beta)$ and $q_1$ with $q_2$ in the above expression, we get a similar approximation of \textsc{CoF}\ for case 4.
\begin{restatable}{proposition}{} Cost of fairness with discrete valuations can go arbitrarily bad. \label{prop:discretecof} \end{restatable} \begin{proof}
From \cref{eq:cof} when $\frac{v_1}{v_2} \rightarrow 0$, we have $ \textsc{CoF} = \frac{v_2}{\alpha d_{12}}$. The \textsc{CoF}\ (in Case 3 and/or Case 4) is arbitrarily bad if $d_{12}>0$ when there is a large difference between $v_1$ and $v_2$. Note that $d_{12}=0$ is uninteresting as the seller is unable to distinguish between two segments. \end{proof}
Note that $v_2$ being arbitrarily large need not be a commonly occurring setting. Hence, we work with bounded support valuations in the backdrop of the above negative results. In the next section, we make assumptions based on standard economic literature about the revenue functions $\pi_i(\cdot)$, i.e., concave revenue functions and common support~\cite{bergemann2021thirddegree}. As argued in Section 3 of \cite{dhangwatnotai2015}, valuation distributions satisfying Monotone Hazard Rate (MHR) satisfy the assumptions as mentioned above regarding revenue functions. It is also observed that the revenue functions are concave for another commonly analyzed family of distributions in literature called the regular distributions in which the virtual valuation is non-decreasing (Section 4.3 of \cite{bergemann2021thirddegree}). MHR is a common assumption in Econ-CS~\cite{hartline2009}. Therefore, in the following section, we analyze the cost of fairness for such valuation distributions and the associated concave revenue functions.
\section{FFP for Continuous Valuations} \label{sec:cont} In this section, we consider feature-based pricing with continuous valuations. We impose a standard restriction on the revenue functions $\pi_i(\cdot)$ such that they are concave on the common support $\mathcal{V} = [\underline{v}, \Bar{v}]$ \cite{bergemann2021thirddegree}. The consumer segments are identified by the associated feature vectors $x_i \in \mathcal{X}$. $\underline{v}$ is the marginal cost defined as a minimum feasible valuation for which a seller is willing to sell the product. The marginal cost may include the cost of production, transportation, etc. On the other hand, $\Bar{v}$ is the maximum consumer valuation. Without loss of generality, we consider that maximum consumer valuation is greater than marginal cost; i.e., trade occurs.
We begin with a tight upper bound on the \textsc{CoF}\ under conditions as mentioned above (\cref{sec:ffpcontinuous}) followed by two pricing schemes based on the available information about the revenue functions (\cref{sec:ourffp}), and finally, we present an algorithm that achieves the \textsc{CoF}\ bound in \cref{sec:algorithm}.
\subsection{Optimal FFP for Continuous Valuations} \label{sec:ffpcontinuous} The problem of determining optimal FFP can be modeled as a convex program with $\alpha$-\textsf{fairness }\ as linear constraints. The convex program below describes OPT$_{FFP}(\mathcal{V}, \mathcal{X},d, \mathcal{F},\mathbf{\beta},\alpha)$ model with complete knowledge of revenue functions $\pi_i(\cdot)$. \begin{subequations} \begin{empheq}[box=\widefbox]{align*}
\max_{p_k \in \mathcal{V}, \forall k} \Pi(\mathbf{p}) &= \sum_{k=1}^{K} \beta_k \pi_k(p_k)\\ \text{subject to, }
|p_i - p_j| &\leq \alpha d(x_i, x_j), \forall i \neq j \\
& p_i \geq 0, \forall i \in [K] \end{empheq} \end{subequations} Let $\mathbf{p^{\star}}$ be a solution to the above problem. \if 0 And the optimal fair price vector $\mathbf{p^{\star}}$ is given by: \begin{equation*}
\mathbf{p^{\star}} = \underset{\mathbf{p} \in \mathbb{R}^{K}_{+} }{\operatorname{argmax}} \ \sum_{k=1}^{K} \beta_k \pi_k(p_k) \end{equation*} under the constraints mentioned above. The convex program can be solved using standard solvers. \ga{I dont think we need the paragraph above. We can simply write "Let $\mathbf{p}^{\star}$ be a solution of the above linear program. }\sg{yes...} \fi
\subsection{\textsc{LinP-FFP}\ and \textsc{CoF}\ Analysis} \label{sec:ourffp} Let $D_i := \min_{j\neq i} d_{ij}$. With the following proposition, we propose a class of $\alpha$-fair pricing strategies. \begin{restatable}{proposition}{ClaimOne} For a given $m\in[\underline{v},\overline{v}]$, if the price function satisfies
$|p_i - m| \leq \frac{\alpha}{2} D_i$ for all $i \in [K]$ then it satisfies \emph{$\alpha$-\textsf{fairness }}. \label{prop:linp} \end{restatable} \begin{proof} From triangle inequality, we have
$ |p_i - p_j| \leq |p_i - m| + |p_j - m|
\leq \frac{\alpha}{2} D_i + \frac{\alpha}{2} D_j \leq \alpha d_{ij}$. The last inequality results from the fact that $ D_i = \min_{k\neq i}d_{ik} \leq d_{ij} $ and $ D_j = \min_{k\neq j}d_{ik} \leq d_{ji} = d_{ij} $. \end{proof}
In other words, for ensuring that the prices for different segments are not too different, it is enough to ensure that the pricing for each segment is not too different from some common point $m$. The pricing for all the segments would hence be around this point and could be determined with respect to this point. We term this point as {\em pivot}. We now present the second FFP model, an $\alpha$-fair pricing strategy that is pivot-based and satisfies the condition in \cref{prop:linp}, with access to only $\widehat{p}_i$ for a given $m$. \begin{equation}
p_i = \begin{cases} m + \alpha D_i/2 & \text{ if } \ \widehat{p}_{i} - m \geq \alpha D_i/2 \\
m - \alpha D_i/2 & \text{ if } m - \widehat{p}_{i} \geq \alpha D_i/2 \\
\widehat{p}_i & \text{ otherwise} \end{cases}
\label{eq:fairPricing} \end{equation} We call this pricing scheme \textsc{LinP-FFP}. It is easy to see that the above pricing strategy is $\alpha$-fair. We now present the \textsc{CoF}\ bound for \textsc{LinP-FFP}.
\begin{theorem} \label{thm:cofbound} The Cost of Fairness for optimal fair price discrimination with concave revenue functions satisfies $$\textsc{CoF}\ \leq \frac{2}{1+ \min \left\{ \alpha \frac{\min_i D_i}{\Bar{v} - \underline{v}} , 1 \right\}} $$ \end{theorem} \begin{proof} We prove that the above \textsc{CoF}\ is satisfied by \textsc{LinP-FFP}\ and hence the theorem. Let $m \in [\underline{v}, \Bar{v}]$ be a pivot point (See Figure \ref{fig:my_label}). Let \begin{equation}\label{eq:gamma}
\gamma_i:= \begin{cases}
\frac{(m-\underline{v}) + \alpha D_i/2}{\widehat{p}_i - \underline{v}} & \text{if } \widehat{p}_{i} - m \geq \alpha D_i/2 \\
\frac{ (\Bar{v}-m) + \alpha D_i/2 }{\Bar{v} - \widehat{p}_i} & \text{if } m - \widehat{p}_{i} \geq \alpha D_i/2 \\
1 & \text{ otherwise} \end{cases}\end{equation} Let $\widehat{\pi}_{i}$ be the expected revenue generated from the $i^{\text{th}}$ segment under $\widehat{\mathbf{p}}$. We now show the following supporting lemma.
\begin{figure}
\caption{Concave revenue function $\pi_i(\cdot)$ and its linear approximation $L_i(\cdot)$ (arrows show equations for $L_i(\cdot)$). Figure represents the case $\widehat{p}_{i} - m \geq \alpha D_i/2$ for which \textsc{LinP-FFP}\ assigns $p_i = m + \alpha D_i/2$. The case $m - \widehat{p}_{i} \geq \alpha D_i/2$ is similar.}
\label{im:linapprox}
\label{fig:my_label}
\end{figure}
\begin{restatable}{lemma}{CoF} The pricing strategy given in \cref{eq:fairPricing} guarantees at-least $\gamma_i$ fraction of optimal revenue from segment $i$, i.e., $\pi_{i} \geq \gamma_i\widehat{\pi}_{i}$. \label{lem:approx} \end{restatable} \begin{proof} A lower bound to the concave revenue functions $\pi_i(\cdot)$ for any segment $i$ is the piecewise linear approximation $L_i$, given by (see Figure \ref{fig:my_label}): \begin{equation}
L_i(p) = \begin{cases} \frac{\widehat{\pi}_i}{\widehat{p}_i-\underline{v}} (p-\underline{v}), & p \leq \widehat{p}_i \\
\frac{-\widehat{\pi}_i}{\Bar{v}-\widehat{p}_i}(p-\Bar{v}), & p > \widehat{p}_i
\end{cases} \end{equation} So, for each consumer segment $i$ we have, \begin{equation*}\label{eq:linapprox} L_i(p) \leq \pi_i(p), \ \forall p \in [\underline{v},\Bar{v}] \end{equation*}
Expected revenues generated per consumer in segment $i$ by pricing rule in Eq. \eqref{eq:fairPricing} for $\widehat{p}_{i} - m \geq \alpha D_i/2$, $m - \widehat{p}_{i} \geq \alpha D_i/2$, and remaining cases are given below in the respective order \begin{align*} \pi_i(p_i)& \geq L_i(p_i) = \frac{\widehat{\pi}_i}{\widehat{p}_i - \underline{v}}(m+\alpha D_i/2-\underline{v}) = \widehat{\pi}_i \gamma_i \\ \pi_i(p_i)& \geq L_i(p_i) = \frac{-\widehat{\pi}_i}{\Bar{v} - \widehat{p}_i}(m - \alpha D_i/2 - \Bar{v} ) = \widehat{\pi}_i \gamma_i \\ \pi_i(p_i)& = L_i(\widehat{p}_i) = \widehat{\pi}_i \end{align*} This proves the lemma. \end{proof}
Let $\pi_{i}^{\star}$ denote the expected revenue generated from the $i^{\text{th}}$ segment by $\mathbf{p}^{\star}$. So, \textsc{CoF}\ for optimal FPP is given by: \begin{subequations} \begin{align*}
\textsc{CoF} &= \frac{\sum \limits_{i \in [K]} \beta_{i} \widehat{\pi}_{i}}{\sum \limits_{i \in [K]} \beta_{i} \pi_{i}^{\star}} \leq \frac{\sum \limits_{i \in [K]} \beta_{i} \widehat{\pi}_{i}}{\sum \limits_{i \in [K]} \beta_{i} \pi_{i}} \tag{Optimality of $\pi_i^{\star}$}\\
& \leq \frac{\sum \limits_{i \in [K]} \beta_{i} \widehat{\pi}_{i} }{\sum \limits_{i \in [K]} \beta_{i} \gamma_i \widehat{\pi}_{i}} \tag{\cref{lem:approx}}
\end{align*} \end{subequations}
In order to prove the said \textsc{CoF}\ bound, it suffices to show that there exists an $m$ (and hence a corresponding pricing strategy using \cref{eq:fairPricing}) for which the said \textsc{CoF}\ bound is satisfied. It can be seen that for $m = (\underline{v} + \Bar{v})/2$, and replacing denominators in \cref{eq:gamma} by $\Bar{v}-\underline{v}$, we have that \begin{subequations} \begin{align*}
\textsc{CoF} & \leq \frac{\sum_{i \in [K]} \beta_{i} \widehat{\pi}_{i}}{\sum_{i \in [K]} \beta_{i} \widehat{\pi}_{i}\left(\frac{1}{2} + \min\{ \frac{\alpha D_i}{2 (\Bar{v} - \underline{v})} , 1 \}\right)} \\
& \leq \frac{\sum_{i \in [K]} \beta_{i} \widehat{\pi}_{i}}{\left(\sum_{i \in [K]} \beta_{i} \widehat{\pi}_{i}\right) \left(\frac{1}{2} + \min\{ \frac{\alpha \min_j D_j}{2 (\Bar{v} - \underline{v})} , 1 \}\right)} \\
& = \frac{2}{ 1 + \min \left\{ \alpha \frac{\min_j D_j}{\Bar{v} - \underline{v}} , 1 \right\}} \end{align*} \end{subequations}\end{proof}
It is worth noting here that the cost of fairness does not depend on the number of the segments and the distribution of the population among these segments. So, if the segments are well separated in terms of the distance between features of consumers across segments the number of segments as well as the distribution of consumer population in these segments do not affect revenue guarantee. Also, if the admissible prices are supported over a large interval, the fairness guarantee becomes weaker. This insight discourages pricing schemes with wildly varying prices across segments. Finally, if $\alpha = 0$, i.e., without any fairness constraints, we recover the bound of 2 proved in \cite{bergemann2021thirddegree}.
We emphasize that the bound is strictly less than $2$ because, under fairness constraints, $\alpha \neq 0$ and typically the consumer types are well separated in the feature space according to the metric $d$ else, the consumer types are indistinguishable for the seller hence, $d_{ij} \neq 0$ for all $i, j \in [K]$. This is an improvement of the \textsc{CoF}\ bound given in \cite{bergemann2021thirddegree}.
\paragraph{Tightness of \textsc{CoF}\ bound:} We claim that the \textsc{CoF}\ bound presented above is tight. In the following example, equality holds and proves the tightness of the bound. \begin{example}[Tightness of the \textsc{CoF}\ bound] Consider $K=2$ where $\beta_1=\beta_2=\frac{1}{2}$. Consider $\mathcal{F}_i$ be such that $\pi_{i}(\cdot) = L_i(\cdot)$ with $\widehat{p}_1 = \underline{v}+\epsilon, \widehat{p}_2 = \Bar{v}-\epsilon,$ where $\epsilon \rightarrow 0$, and $\widehat{\pi}_1 = \widehat{\pi}_2$. It can be seen that if $\alpha$ is such that $\alpha d_{12} < \Bar{v}-\underline{v}$, any FP satisfying $p_2-p_1 = \alpha d_{12}$ and $p_1,p_2 \in [\widehat{p}_1, \widehat{p}_2]$ is an optimal FFP (fair FP), and the corresponding $\textsc{CoF} = \frac{2}{1+\frac{\alpha d_{12}}{\Bar{v}-\underline{v}}}$. If $\alpha d_{12} \geq \Bar{v}-\underline{v}$, the optimal FP is $\alpha$-fair and so, $\textsc{CoF} = 1$. Hence, for this example, $\textsc{CoF} = \frac{2}{1 + \min \left\{ \alpha \frac{d_{12}}{\Bar{v} - \underline{v}} , 1 \right\}}$. This shows the tightness of the \textsc{CoF}\ bound derived in \cref{thm:cofbound}. \end{example}
We now present an algorithm, OPT-\textsc{LinP-FFP}, to find the optimal pivot $m^{\star}$ in the above \textsc{LinP-FFP}\ strategy when only $\widehat{p}$ and $\widehat{\pi_i}$s are known.
\subsection{Proposed Algorithm} \label{sec:algorithm}
As \textsc{LinP-FFP}\ satisfies $\alpha$-\textsf{fairness } (\cref{prop:linp}), and also achieves \textsc{CoF}\ bounds in ~\cref{thm:cofbound}, we look for a pricing strategy optimal within class of \textsc{LinP-FFP}. It reduces to finding an optimal pivot that maximizes revenue. In this section, we propose a binary-search-based algorithm for the same. For pricing $\mathbf{p}$, the expected revenue generated per consumer is given by $ \Pi(\mathbf{p}) = \sum_{i=1}^{K} \beta_i \pi_i(p_i) $. Let $\tau_i := \frac{\alpha}{2} D_i$. Observe from \cref{lem:approx} that $\Pi(\mathbf{p})$ is lower bounded as:
\begin{subequations}\label{eq:fairrev} \begin{align*}
\Pi &(\mathbf{p}) \geq \Pi_m(\mathbf{L}) = \sum_{i=1}^{K} \beta_i \gamma_i \widehat{\pi}_i = \sum \limits_{\substack{ i:|\widehat{p}_i - m |< \tau_i}}\beta_i \widehat{\pi}_i \ \ + \\& \sum_{\substack{ i:\widehat{p}_i - m \geq \tau_i}}\beta_i \widehat{\pi}_i \frac{m+ \tau_i-\underline{v} }{\widehat{p}_i - \underline{v}} + \sum_{\substack{ i:m-\widehat{p}_i \geq \tau_i}} \beta_i \widehat{\pi}_i \frac{ \Bar{v}-m + \tau_i }{\Bar{v} - \widehat{p}_i} \tag{\ref{eq:fairrev}} \end{align*} \end{subequations}
\subsubsection*{Determining Optimal Pivot $m$}
As we can see, the revenue generated by \textsc{LinP-FFP}\ is lower bounded by a piecewise linear function in $m$. With the aim of achieving a better lower bound, we now address the problem of determining an optimal pivot $m^{\star} \in \underset{m \in [\underline{v}, \Bar{v}]}{\operatorname{argmax}} \ \Pi_m(\mathbf{L})$.
\subsubsection*{Pricing Algorithm}
In what follows, we call the candidate points $m$ for optimal pivot, i.e., for maximizing $\Pi_m(\mathbf{L})$, as \emph{critical points}. We denote the set of these critical points as $\mathcal{M}$.
\begin{lemma}\label{lem:unimodal} $\Pi_m(\mathbf{L})$ as a function of $m$ is concave and piecewise linear with the set of critical points $\mathcal{M} = \left( \{\widehat{p}_i - \frac{\alpha}{2} D_i , \widehat{p}_i + \frac{\alpha}{2} D_i \}_{i \in [K]} \cap [\underline{v}, \Bar{v}] \right) \cup \{\underline{v}, \Bar{v}\} $. \end{lemma} \begin{proof} It is easy to see that for a segment $i$, $\gamma_i$ as a function of $m$ is continuous and piecewise linear with breakpoints (i.e., points at which piecewise linear function changes slope): $\widehat{p}_i - \frac{\alpha}{2} D_i$ and $\widehat{p}_i + \frac{\alpha}{2} D_i$ provided they are in the range $[\underline{v}, \Bar{v}]$. The set of breakpoints is hence $\{ \widehat{p}_i - \frac{\alpha}{2} D_i, \widehat{p}_i + \frac{\alpha}{2} D_i\} \cap [\underline{v}, \Bar{v}]$. Also, the slope monotonically decreases at the breakpoints, i.e., $\gamma_i$ is a concave function of $m$.
From \cref{eq:fairrev}, we can see that $\Pi_m(\mathbf{L})$ is a weighted sum over all segments, of $\gamma_i$'s with constant weights $\beta_i \widehat{\pi}_i$. So, $\Pi_m(\mathbf{L})$ as a function of $m$ is concave and piecewise linear with breakpoints belonging to the following set: $\{\widehat{p}_i - \frac{\alpha}{2} D_i , \widehat{p}_i + \frac{\alpha}{2} D_i \}_{i \in [K]} \cap [\underline{v}, \Bar{v}]$. Hence, a point $m$ that maximizes $\Pi_m(\mathbf{L})$ belongs to either the aforementioned set of breakpoints, or the set of its boundary points $\{\underline{v}, \Bar{v}\}$. Thus, the set of critical points $\mathcal{M} = \left( \{\widehat{p}_i - \frac{\alpha}{2} D_i , \widehat{p}_i + \frac{\alpha}{2} D_i \}_{i \in [K]} \cap [\underline{v}, \Bar{v}] \right) \cup \{\underline{v}, \Bar{v}\} $. \end{proof}
Our algorithm OPT-\textsc{LinP-FFP}\ (Optimal Linearized Pivot-based Fair Feature-based Pricing) which determines an optimal pivot $m^{\star}$ and provides an $\alpha$-fair pricing strategy ($\widetilde{\mathbf{p}}$) is presented in \cref{alg:ouralgo}.
\begin{algorithm}[!ht] \caption{OPT-\textsc{LinP-FFP}\ } \label{alg:ouralgo} \DontPrintSemicolon
\KwInput{$\alpha, \mathbf{\widehat{p}}, (\widehat{\pi}_1,\ldots,\widehat{\pi}_K), (\beta_1,\ldots,\beta_K), (D_1, \ldots, D_K)$}
\KwOutput{$m^{\star}, \widetilde{\mathbf{p}} $ }
\tcc{Creating and sorting the set of critical points} $\mathcal{M} \gets \{\underline{v}, \Bar{v}\}$ \; \For{$i \in [K]$}{ $\tau_i \gets \frac{\alpha}{2} D_i$ \; \If{$\widehat{p}_i - \tau_i > \underline{v}$}{ $\mathcal{M} \gets \mathcal{M} \cup \{\widehat{p}_i - \tau_i\}$ \; } \If{$\widehat{p}_i + \tau_i < \Bar{v}$}{ $\mathcal{M} \gets \mathcal{M} \cup \{\widehat{p}_i + \tau_i\}$ \; } } sort($\mathcal{M}$) \;
\tcc{Binary search for optimal pivot}
$\ell \gets 0$, $r \gets |\mathcal{M}|-1$ \; \While{$\ell \leq r$} { $z \gets \lfloor\! \frac{\ell+r}{2} \!\rfloor$ \tcp*{$\!\mathcal{M}[z]\!$ is the current pivot} \tcc{Computing the expression in \cref{eq:fairrev} at current and adjacent critical points} $\Pi_{\mathcal{M}[z-1]} \gets 0$, $\Pi_{\mathcal{M}[z]} \gets 0$, $\Pi_{\mathcal{M}[z+1]} \gets 0$ \;
\For{$y \gets \{z-1, z, z+1\}$}
{
\For{$i \gets 1$ to $K$}
{
\uIf{$\widehat{p}_{i} \geq \mathcal{M}[y] + \tau_i$}{
$\gamma_i \gets \frac{\mathcal{M}[y] -\underline{v} + \tau_i }{\widehat{p}_i - \underline{v}}$
}
\uElseIf{$\widehat{p}_{i} \leq \mathcal{M}[y] - \tau_i$}{
$\gamma_i \gets \frac{ \Bar{v}- \mathcal{M}[y] + \tau_i }{\Bar{v} - \widehat{p}_i}$
}
\Else{
$\gamma_i \gets 1$
}
$\Pi_{\mathcal{M}[y]} \gets \Pi_{\mathcal{M}[y]} + \beta_i \gamma_i \widehat{\pi}_i$ \;
} } \uIf{$\Pi_{\mathcal{M}[z-1]} \leq \Pi_{\mathcal{M}[z]} \leq \Pi_{\mathcal{M}[z+1]}$} { $\ell \gets z+1$ \; } \uElseIf{$\Pi_{\mathcal{M}[z-1]} \geq \Pi_{\mathcal{M}[z]} \geq \Pi_{\mathcal{M}[z+1]}$} { $r \gets z-1$ } \Else{ $m^{\star} \gets \mathcal{M}[z]$ \; \textbf{break} \; } } \tcc{Pricing for the different segments} \For{$i \in [K]$}{ \uIf{$\widehat{p}_{i} \geq m^{\star} + \tau_i$}{ $\widetilde{p}_i \gets m^{\star} + \tau_i$ } \uElseIf{$\widehat{p}_{i} \leq m^{\star} - \tau_i$}{ $\widetilde{p}_i \gets m^{\star} - \tau_i$ } \Else{ $\widetilde{p}_i \gets \widehat{p}_i$ } } \end{algorithm}
\begin{comment} \begin{algorithm}[!ht] \caption{OPT-\textsc{LinP-FFP}\ } \label{alg:ouralgo} \DontPrintSemicolon
\KwInput{$\mathbf{\widehat{p}}$ (optimal FP), $\alpha$ (fairness parameter), $(d_{ij})_{i,j \in [K]}$ (distance matrix)}
\KwOutput{$m^{\star}$ (optimal pivot), $\widetilde{\mathbf{p}} $ (pricing vector)} \tcc{Creating and sorting the set of critical points} $\mathcal{M} \gets \{\underline{v}, \Bar{v}\}$ \; \For{$i \in [K]$}{ \If{$\widehat{p}_i - \frac{\alpha}{2}D_i > \underline{v}$}{ $\mathcal{M} \gets \mathcal{M} \cup \{\widehat{p}_i - \frac{\alpha}{2}D_i\}$ \; } \If{$\widehat{p}_i + \frac{\alpha}{2}D_i < \Bar{v}$}{ $\mathcal{M} \gets \mathcal{M} \cup \{\widehat{p}_i + \frac{\alpha}{2}D_i\}$ \; } } \sd{Getting $D_i \forall i$ may need $K^2$ time, right? Should we assume that they are given?}\sg{yes..best to say algorithm is $K^2$} \; sort($\mathcal{M}$) \; $m^{\star} \gets 0$ \; \sd{Here, $m$ is optimal index and not optimal pivot, right?} \;
$\ell = 0$, $r = |\mathcal{M}|-1$ \;
\tcc{Binary search for optimal pivot} \While{$\ell \geq r$ \sd{$\leq$, right?}} { $m \gets \lfloor\! \frac{\ell+r}{2} \!\rfloor$ \tcp*{$\!\mathcal{M}[m]\!$ is the current pivot} $\Pi_{\mathcal{M}[m-1]} \gets 0$, $\Pi_{\mathcal{M}[m]} \gets 0$, $\Pi_{\mathcal{M}[m+1]} \gets 0$ \; \tcc{Computing expected revenues at current \& adjacent critical points} \sd{What are $p_i$ and $L_{\mathcal{M}[m]}(p_i)$ below? Is it clear?} \;
\For{$i \gets 1$ to $K$}
{ $\Pi_{\mathcal{M}[m-1]} \gets \Pi_{\mathcal{M}[m-1]} + \beta_i L_{\mathcal{M}[m-1]}(p_i)$ \; $\Pi_{\mathcal{M}[m]} \gets \Pi_{\mathcal{M}[m]} + \beta_i L_{\mathcal{M}[m]}(p_i)$ \;
$\Pi_{\mathcal{M}[m+1]} \gets \Pi_{\mathcal{M}[m+1]} + \beta_i L_{\mathcal{M}[m+1]}(p_i)$ \; } \uIf{$\Pi_{\mathcal{M}[m-1]} \leq \Pi_{\mathcal{M}[m]} \leq \Pi_{\mathcal{M}[m+1]}$} { $\ell \gets m+1$ \; } \uElseIf{$\Pi_{\mathcal{M}[m-1]} \geq \Pi_{\mathcal{M}[m]} \geq \Pi_{\mathcal{M}[m+1]}$} { $r \gets m-1$ } \Else{ $m^{\star} \gets m$ \; \textbf{break} \; } } \tcc{Pricing for the different segments} \For{$i \in [K]$}{ \uIf{$\widehat{p}_{i} \geq m^{\star} + \frac{\alpha}{2} D_i$}{ $\widetilde{p}_i = m^{\star} + \frac{\alpha}{2} D_i$ } \uElseIf{$\widehat{p}_{i} \leq m^{\star} - \frac{\alpha}{2} D_i$}{ $\widetilde{p}_i = m^{\star} - \frac{\alpha}{2} D_i$ } \Else{ $\widetilde{p}_i = \widehat{p}_i$ } } \end{algorithm} \end{comment}
\begin{theorem} The \emph{OPT}-\textsc{LinP-FFP}\ algorithm (a) returns optimal pivot point $m^{\star}$ and runs in $\mathcal{O}(K \log(K))$ time, and (b) achieves the \textsc{CoF}\ bound given in \cref{thm:cofbound}. \label{thm:opt_cofbound} \end{theorem} \begin{proof}
(a) The first module is the creation and sorting of the set of critical points $\mathcal{M}$, which takes $\mathcal{O}(K \log(K))$ time. Owing to \cref{lem:unimodal}, we can find an optimal pivot $m^{\star}$ using binary search over $\mathcal{M}$. Here, the number of critical points are at most $2K+2$, i.e., $|\mathcal{M}| \leq 2K+2$. So, in the second module that finds an optimal pivot, the binary search in the outer (\emph{while}) loop runs for $\mathcal{O}(\log(|\mathcal{M}|))$ iterations, and the inner (\emph{for}) loops run for $\mathcal{O}(K)$ iterations overall. Thus, the running time of the second module is $\mathcal{O}(K \log(K))$. The third module that computes pricing for the different segments runs in $\mathcal{O}(K)$ time. So, the total running time of Algorithm \ref{alg:ouralgo} is $\mathcal{O}(K \log(K))$.
(b) From \cref{thm:cofbound}, for $m=(\underline{v}+\Bar{v})/2$, the \textsc{CoF}\ bound holds. Also, $\Pi_{m^{\star}}(\mathbf{L}) \geq \Pi_m(\mathbf{L})$ for all $m \neq m^{\star}$. We have: $$ \textsc{CoF} = \frac{\Pi(\widehat{\mathbf{p}})}{\Pi(\widetilde{\mathbf{p}})} \leq \frac{\Pi(\widehat{\mathbf{p}})}{\Pi_{m^{\star}}(\mathbf{L})} \leq \frac{\Pi(\widehat{\mathbf{p}})}{\Pi_m(\mathbf{L})} $$ This completes the proof of the theorem. \end{proof}
\section{Discussion} This paper built a foundation for the design of fair feature-based pricing by proposing a new fairness notion called $\alpha$-\textsf{fairness }$\!\!$. Our impossibility result on the discrete valuation setting restricted us from attaining a finite cost of fairness (\textsc{CoF}) in general settings. Interestingly, in the continuous valuation setting with concave revenue functions, we showed that a family of pricing schemes, \textsc{LinP-FFP}, provided a \textsc{CoF}\ strictly less than $2$. Finally, we proposed an algorithm, OPT-\textsc{LinP-FFP}, which gave us an optimal pricing strategy within this family. It is worth noting that the algorithm does not require a complete distribution function; peaks of revenue distributions are sufficient statistics for computing optimal fair feature-based pricing.
We leave the problem of finding an optimal segmentation (optimal value of $K$ and corresponding $K$-partition of the market) as interesting future work. We assumed a monopoly market. It will be interesting to study optimal fair pricing in the face of competition and other constraints such as finite supply, non-linear production cost, and variable demand.
\end{document} | arXiv |
OSA Publishing > Optics Express > Volume 28 > Issue 19 > Page 27808
Enantioselective optical trapping of chiral nanoparticles using a transverse optical needle field with a transverse spin
Ying Li, Guanghao Rui, Sichao Zhou, Bing Gu, Yanzhong Yu, Yiping Cui, and Qiwen Zhan
Ying Li,1 Guanghao Rui,1,5 Sichao Zhou,2 Bing Gu,1,3 Yanzhong Yu,4 Yiping Cui,1 and Qiwen Zhan2,6
1Advanced Photonics Center, Southeast University, Nanjing 210096, China
2Department of Electro-Optics and Photonics, University of Dayton, Dayton, Ohio 45469, USA
3Collaborative Innovation Center of Light Manipulations and Applications, Shandong Normal University, Jinan 250358, China
4College of Physics & Information Engineering, Quanzhou Normal University, Fujian 362000, China
[email protected]
[email protected]
Guanghao Rui https://orcid.org/0000-0003-3527-6309
Bing Gu https://orcid.org/0000-0002-7060-6046
Yanzhong Yu https://orcid.org/0000-0002-8381-5787
Y Li
G Rui
S Zhou
B Gu
Y Yu
Y Cui
Q Zhan
•https://doi.org/10.1364/OE.403556
Ying Li, Guanghao Rui, Sichao Zhou, Bing Gu, Yanzhong Yu, Yiping Cui, and Qiwen Zhan, "Enantioselective optical trapping of chiral nanoparticles using a transverse optical needle field with a transverse spin," Opt. Express 28, 27808-27822 (2020)
Enantioselective optical trapping of chiral nanoparticles by tightly focused vector beams (JOSAB)
Probing the optical chiral response of single nanoparticles with optical tweezers (JOSAB)
Design of an optical conveyor for selective separation of a mixture of enantiomers (OE)
Nanophotonics, Metamaterials, and Photonic Crystals
High numerical aperture optics
Liquid crystal modulators
Optical angular momentum
Optical fields
Optical force
Original Manuscript: July 27, 2020
Revised Manuscript: August 27, 2020
Manuscript Accepted: August 27, 2020
Construction of a transverse optical needle field with a transverse spin
Optical force and torque under dipole approximation
Since the fundamental building blocks of life are built of chiral amino acids and chiral sugar, enantiomer separation is of great interest in plenty of chemical syntheses. Light-chiral material interaction leads to a unique chiral optical force, which possesses opposite directions for specimens with different handedness. However, usually the enantioselective sorting is challenging in optical tweezers due to the dominating achiral force. In this work, we propose an optical technique to sort chiral specimens by use of a transverse optical needle field with a transverse spin (TONFTS), which is constructed through reversing the radiation patterns from an array of paired orthogonal electric dipoles located in the focal plane of a 4Pi microscopy and experimentally generated with a home-built vectorial optical field generator. It is demonstrated that the transverse component of the photonic spin gives rise to the chiral optical force perpendicular to the direction of the light's propagation, while the transverse achiral gradient force would be dramatically diminished by the uniform intensity profile of the optical needle field. Consequently, chiral nanoparticles with different handedness would be laterally sorted by the TONFTS and trapped at different locations along the optical needle field, providing a feasible route toward all-optical enantiopure chemical syntheses and enantiomer separations in pharmaceuticals.
Chirality is a geometrical property where an object cannot be superposed onto its mirror image via either a translational or a rotational operation [1]. The mirror images of a chiral structure are enantiomers, and individual enantiomer are often designated as either right or left-handedness. In fact, this type of symmetry is much harder to be maintained than to be broken, consequently chirality exists widely in various macroscopic and microscopic structures. For example, proteins and nucleic acids are built of chiral amino acids and chiral sugar. In addition, DNA double helix, sugar, quartz, cholesteric liquid crystals and biomolecules are also chiral structures. Although molecules with different handedness have the same chemical construction, usually they would possess distinct chemical behaviors. A chiral biomolecule can be inactive or toxic to cells if its original handedness is varied, which causes many diseases such as phocomelia, Parkinson's, Alzheimer's, type II diabetes and Huntington's [2]. Consequently, the sensitive detection and separation of substances by chirality is therefore of high demand in the fields of pharmacology, toxicology and pharmacodynamics.
The optical field can also be chiral. When illuminated by circularly polarized light, most molecules behave distinct optical responses in terms of refractive index, absorption and Raman scattering [3], which are named circular dichroism, optical rotation and Raman optical activity, providing noninvasive spectroscopic and efficient tool to unscramble the structural, kinetic, and thermodynamic information of molecules [4–7]. However, the chiroptical response in natural chiral materials is relatively weak due to the small electromagnetic interaction volume. Besides, the current spectroscopy is based on the measurement of the average far-field radiation, which suffers from low resolution structural details and considerable amount of sample, making it impossible to detect the structured chirality at nanoscale with high precision.
Different from conventional materials, the chirality of the electromagnetic material induces cross-polarization between electric and magnetic fields, leading to unique optical force effects, which derives from the momentum transfer associated with bending light when tightly focused laser beam interacts with the particles. Since Ashkin and his colleagues reported the first stable three-dimensional optical trapping (or optical tweezers) created using radiation pressure from a single focused laser beam, this technique has evolved from simple manipulation to the application of calibrated forces on, and the measurement of nanometer-level displacement of optically trapped objects [8–10]. In the early days, optical trapping has been mainly implemented in two size regimes: the sub-nanometer (e.g., cooling of atoms, ions and molecules) and micrometer scale (such as cells). Recently, new approaches have been developed to stably trap and manipulate mesoscopic objects, including metallic nanoparticles [11,12], carbon nanotubes [13,14] and quantum dots [15,16]. The continuous development of optical tweezers has revolutionized the experimental study of small particles and become an important tool for research in biology, physical chemistry and soft matter physics. Different from achiral metallic and dielectric materials, the optical response of chiral particle depends on not only the electric and magnetic polarizability, but also has close connection to the electromagnetic/magnetoelectric polarizability (also known as chiral polarizability), which provides additional degree of freedom to tailor the optical force and develop novel optical tweezers techniques. In recent years, plenty of chirality induced optical force effects have been reported (e.g. lateral force [17,18], optical pulling/repulsive force [19–21] and azimuthal/longitudinal optical torque [22,23]), enabling the effective manipulation of chiral particle with controllable dynamic behavior [24–28]. It has been reported that the chiral optical force would be generated along the direction of the photonic spin. In particular, the transverse component of spin angular momentum of light would give rise to a lateral optical force on chiral particles placed on top of a surface, and the direction of the force is determined by the handedness of the specimens [1,29–33]. Brasselet firstly anticipated and realized the optical sorting of particles with different handedness using chiral discriminatory force [34,35]. However, with the decrease of the particle size, the chiral optical force would be gradually suppressed by the achiral gradient force, making it difficult to realize the sorting of chiral nanoparticle.
With the rapid development of the optical engineering, various laser beam shaping system have been developed to tailor the spatial distribution of the light field at will [32,33]. Besides, the inverse design of the complete shaping of the optical focal field with the prescribed features has also been proposed [36–40]. These great progress in optical engineering enable us to synthesize optical focal field with prescribed intensity and three-dimensional polarization state, which are feasible to realize the optical force with unconventional features and are helpful to enhance the performance of the optical tweezers system. For example, the dynamic behaviors of metallic/dielectric nanoparticles with different shapes can be flexibly manipulated by adjusting the spatial distribution of the optical field in terms of phase, amplitude and polarization [41–45]. In this work, we proposed a novel strategy to discriminate and sort chiral nanoparticles by using the lateral optical force. The required incident field is calculated by reversing the radiated field from an array of paired orthogonal electric dipoles located in the focal plane of a 4Pi microscopy. When strongly focused by a high numerical aperture (NA) objective lens, a transverse optical needle field with transverse spin (TONFTS) is obtained in the focal volume, which is different from the previous optical needle fields that has long depth of focus along the optical axis. Deriving from uniform intensity distribution of the optical needle field and the transverse orientation of the photonic spin, this sculptural vectorial optical focal field offers both diminished achiral gradient force and handedness-dependent chiral gradient force. Consequently, the nanoparticles with different handedness can be spatially separated and trapped at different locations along the optical needled field, leading to the lateral deflection of chiral particles in opposite directions.
2. Construction of a transverse optical needle field with a transverse spin
In order to create an optical field that possesses the capability to laterally sort chiral nanoparticles, two requirements need to be satisfied: Firstly, the trapping light must have a transverse component of the spin angular momentum, which would product chiral optical force normal to the propagation direction of the light beam. Secondly, the achiral optical force caused by the interaction between the chiral nanoparticle and the optical field must be conquered by its chiral counterpart. As for the transverse spin angular momentum (e.g. focal field with photonic spin along y axis), it can be synthesized by two electric dipoles oscillating along x and z axis in the x-z plane with the same strength and phase difference of ±π/2, which are located at the focal point of a high NA lens. By changing the oscillating direction and the sign of the phase difference of the dipoles, both the orientation and the handedness of the photonic spin can be adjusted accordingly [39,43]. However, the optical field generated with this method is highly limited to a focal volume that is smaller than diffraction limit, leading to the dominating achiral gradient force that would pull the enantiomer towards the center of the focus. To sort enantiomers with chiral optical force, an optical needle field with transverse spin angular momentum and long uniform intensity along the spin axis is required to alleviate the dominating effect of the achiral gradient force. Consequently, TONFTS has great potentials in chiral optical tweezers, which can be constructed by pattering an array of paired orthogonal dipoles in the focal volume. As shown in Fig. 1(a), two pairs of orthogonal electric dipoles with the same oscillating direction are located mirror-symmetric with respect to the x-z plane. With the antenna pattern synthesis method for discrete linear dipole array, the radiation pattern of the dipoles collected by a high NA lens in the pupil plane can be calculated by the linear superposition of the emitted field of these infinitesimal dipoles:
(1)$${\boldsymbol{E}_{\boldsymbol{0}}}(\theta ,\varphi ) = ({E_\theta }{\boldsymbol{a}_{\boldsymbol{\theta}}} + {E_\varphi }{\boldsymbol{a}_{\boldsymbol{\varphi}}}){F_N},$$
where Eθ and Eϕ are the radial and azimuthal components of the incident field, respectively. Similar methods have been demonstrated to generate optical needle field carrying different polarizations [46–49]. However, the optical needle fields reported previously have extended depth of focus along the optical axis, in which cases the radiation field from longitudinal discrete dipole array would have a rotational symmetry with respect to the pupil plane of the high NA lens. As for the linear dipole array in the transverse plane, the array factor FN relates to not only the phase delay caused by the spacing distance dn and initial phase difference βn for each pair of the dipoles, but also depends on the spatial position of the observation point P(θ, ϕ):
(2)$${F_N} = \sum\limits_{n = 1}^N {{A_n}({e^{j\frac{{k{d_n}\sin \theta \sin \varphi + {\beta _n}}}{2}}}} + {e^{ - j\frac{{k{d_n}\sin \theta \sin \varphi + {\beta _n}}}{2}}}),$$
where k is the wave-vector in the medium, N is the total number of paired orthogonal dipole elements, An is the ratio of the radiation amplitude between the n-th dipole pair and the standard dipole pair with normalized amplitude.
Fig. 1. Diagram of the proposed optical tweezers setup. (a) The incident light is tailored by the vectorial optical field generator (VOF-Gen) to vectorial light with complex spatial distribution and then highly focused by the 4Pi focusing system to generate TONFTS. The orientation of the optical needle and the photonic spin is obtained through coherent superposition of the radiation patterns from an array of paired orthogonal electric dipoles. The intensity distribution superimposed with the polarization map of the (b) right-propagating and (c) left-propagating illumination.
For a conventional objective lens obeys sine condition, the required incident pupil field is:
(3)$${E_{ri}}({r,\varphi } )= \frac{1}{{\sqrt {\cos \theta } }}{F_N}({X_{ri}} \cdot {\boldsymbol{e}_{\boldsymbol{x}}} + {Y_{ri}} \cdot {\boldsymbol{e}_{\boldsymbol{y}}}),$$
(4)$${X_{ri}} = {e^{i\frac{\pi }{2}}}\sin \theta \cos \varphi - \cos \theta {\cos ^2}\varphi - {\sin ^2}\varphi ,$$
(5)$${Y_{ri}} = {e^{i\frac{\pi }{2}}}\sin \theta \sin \varphi - \cos \theta \cos \varphi \sin \varphi + \sin \varphi \cos \varphi ,$$
where r = f·sinθ, f is the focal length of the objective lens, ex and ey are the unit vectors along x and y axis, respectively. After being focused by a high NA lens, the electric field in the vicinity of the focus can be calculated by using Richard–Wolf vectorial diffraction theory:
(6)$${\boldsymbol{E}_{rf}}({r_p},\phi ,{z_p}) = {C_s}\int_0^{{\theta _{\max }}} {\int_0^{2\pi } {({X_{rf}} \cdot {\boldsymbol{e}_{\boldsymbol{x}}} + {Y_{rf}} \cdot {\boldsymbol{e}_{\boldsymbol{y}}} + {Z_{rf}} \cdot {\boldsymbol{e}_{\boldsymbol{z}}})} } \times {e^{jk{r_p}\sin \theta \cos (\varphi - \phi ) + j{z_p}\cos \theta }}\sin \theta {F_N}d\theta d\varphi ,$$
where ${C_s} = \pi {\lambda ^{ - 1}}\sqrt {{{2P} / {({{n_m}{\varepsilon_0}c\pi N{A^2}} )}}}$, P is the power of the illumination, nm is the refractive index of the surrounding medium, λ is the wavelength in medium, θmax is the maximal focusing angle determined by NA of the lens, rp = (x2 + y2)1/2 and ϕ = tan−1(y/x) are the polar coordinates in the focal volume. The electric field components Xrf, Yrf, and Zrf can be expressed as:
(7)$${X_{rf}} = {e^{i\phi }}\sin \theta \cos \theta \cos \varphi - {\cos ^2}\theta {\cos ^2}\varphi - {\sin ^2}\varphi ,$$
(8)$${Y_{rf}} = {e^{i\phi }}\sin \theta \cos \theta \sin \varphi + {\sin ^2}\theta \sin \varphi \cos \varphi ,$$
(9)$${Z_{rf}} = {e^{i\phi }}{\sin ^2}\theta - \cos \theta \sin \theta \cos \varphi .$$
A 4Pi focusing system is applied to correct the asymmetric radiation patterns from the off-axis electric dipoles, which is consisted of two lens with effective NA of 0.74 and two counter-propagating beams, and the space in-between the lenses is filled with water (nm = 1.33) [33]. Compared with the right-propagating light beam that described by Eqs. (3)–(5), the Ex component of the left propagating field Eli would be out of phase, which is necessary to maintain the symmetry of the optical needle field about x axis:
(10)$${E_{li}}({r,\varphi } )= \frac{1}{{\sqrt {\cos \theta } }}{F_N}({X_{li}} \cdot {\boldsymbol{e}_{\boldsymbol{x}}} + {Y_{li}} \cdot {\boldsymbol{e}_{\boldsymbol{y}}}),$$
(11)$${X_{li}} ={-} {e^{i\frac{\pi }{2}}}\sin \theta \cos \varphi - \cos \theta {\cos ^2}\varphi - {\sin ^2}\varphi ,$$
(12)$${Y_{li}} = {e^{i\frac{\pi }{2}}}\sin \theta \sin \varphi + \cos \theta \cos \varphi \sin \varphi - \sin \varphi \cos \varphi .$$
After being focused by a 4Pi focusing system, the electric field in the vicinity of the focus can be described as:
$$\boldsymbol{E}({r_p},\phi ,{z_p}) = {\boldsymbol{E}_{rf}}({r_p},\phi ,{z_p})\textrm{ + }{\boldsymbol{E}_{lf}}({r_p}, - \phi , - {z_p})\textrm{ = }{C_s}\int_0^{{\theta _{\max }}} {\int_0^{2\pi } {({X_f} \cdot {\boldsymbol{e}_{\boldsymbol{x}}} + {Y_f} \cdot {\boldsymbol{e}_{\boldsymbol{y}}} + {Z_f} \cdot {\boldsymbol{e}_{\boldsymbol{z}}})} } \times$$
(13)$$({e^{jk{r_p}\sin \theta \cos (\varphi - \phi ) + j{z_p}\cos \theta }} + {e^{jk{r_p}\sin \theta \cos (\varphi + \phi ) - j{z_p}\cos \theta }})\sin \theta {F_N}d\theta d\varphi ,$$
(14)$${X_f} = {e^{i\phi }}\sin \theta \cos \theta \cos \varphi - {\cos ^2}\theta {\cos ^2}\varphi - {\sin ^2}\varphi ,$$
(15)$${Y_f} = {e^{i\phi }}\sin \theta \cos \theta \sin \varphi + {\sin ^2}\theta \sin \varphi \cos \varphi ,$$
(16)$${Z_f} = {e^{i\phi }}{\sin ^2}\theta - \cos \theta \sin \theta \cos \varphi .$$
Besides, in order to achieve a TONFTS along y axis, the parameters of each paired orthogonal dipoles in the linear array in terms of spacing distance dn, radiation amplitude An and phase difference βn need to be carefully chosen. Generally, large dipole spacing dn brings large modulation of the interacted dipole field which makes it impossible to realize uniform intensity, while small dn leads to optical needle field with limited length. Radiation amplitude An controls the intensity near the positions of the dipole elements, making sure the peak intensities around each dipole position are approximately equal. Phase difference βn is used to control the overall flatness profile of the transversal intensity distribution. In this work, particle swarm algorithm is utilized with multi-objective optimization (length and flatness of the optical needle field). Particle swarm optimization is a method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. In this case, the radiation amplitude An, spacing distance dn and phase difference βn were chosen as the parameter group. Based on the reasonable value range, one hundred groups of parameters were randomly selected at the initialization stage. Besides, length and flatness of the optical needle in the y-direction and the intensity symmetry in the x-z plane are set as the figure of merit. The globally optimized parameters are summarized in Table 1. In addition, the corresponding distribution of the required incident field for generating TONFTS is calculated with Eqs. (3)–(5) and presented in Figs. 1(b) and 1(c). Besides amplitude, the state of polarization (SOP) at the beam cross-section is also indicated by the polarization ellipses, where green and blue colors represent left- and right-handedness respectively. It can be clearly seen that both the amplitude and the polarization distributions of the illumination are neither uniform nor symmetric. For the right-propagating illumination, most of the input light is in the lower half, the polarization of which have opposite handedness compared with that of the top half. Besides, the SOP of the incident light would change its ellipticity from circular to linear along the vertical direction, while its elevation angle would also vary along the horizontal direction. As for the left-propagating illumination, the SOP distribution is the mirror image of the case shown in Fig. 1(b) due to the phase difference between the Ex components.
Table 1. Optimized parameters of the TONFTS
View Table | View all tables in this article
To illustrate the feasibility of the proposed strategy to generate TONFTS, the characteristics of the optical focal field are numerically explored with Eqs. (6)–(9) and presented in Fig. 2, with the assumption that the optical wavelength of the illumination is 632 nm and incident power is 150 mW. From the three-dimensional intensity distribution (Fig. 2(a)) and the corresponding line-scans (Fig. 2(b)), it can be clearly seen that a uniform and symmetric optical needle field along y axis is obtained. The length of the optical needle is measured to be 1.61λ, which is defined as the transversal full width of above 80% maximum intensity. Besides, the full-width-half-maximum of the focal field along x and z axes are found to be 0.3985λ and 0.3377λ, respectively. It is worthy of noting that the needle length can be further extended by adding more elements in the dipole array. To examine the polarization of the optical needle field, intensity distribution superimposed with SOP in three longitudinal slices (y = −0.8λ, 0, and 0.8λ) are shown in Figs. 2(c)–2(e). One may find from the polarization ellipse that the ellipticity within the main lobe keeps close to 1, indicating the optical needle field maintains circular polarization in the x-z plane. To better study the polarization characteristics of the focal field, both the full Stokes parameters S and the spin density D (D ∝ Im(E*×E)) of focused beam in the x-z plane (y = 0) are calculated and illustrated in Figs. 2(f)–2(k). It can be clearly that both the normalized S3 and Dy are nearly unit in the main lobe, indicating that the spin axis of the optical needle field is mainly in the positive y axis. To quantitively evaluate the quality of the TONFTS, the beam purity η = ΦRHC/(ΦRHC+ΦLHC) is calculated to be 0.8359, which is defined as intensity percentage of the right-hand circular polarization component in the x-z plane throughout the optical needle field: ${\Phi _{RHC({LHC} )}} = \int\limits_{ - {x_l}}^{{x_h}} {\int\limits_{ - {y_l}}^{{y_h}} {\int\limits_{ - {z_l}}^{{z_h}} {{{|{{E_{RHC({LHC} )}}({x,y,z} )} |}^2}dxdydz} } } ,$ where the integration area is restricted by xl/yl/zl and xh/yh/zh (marked on Fig. 2(b)).
Fig. 2. Synthesized TONFTS in the focal volume of the 4Pi focusing system. (a) Intensity distribution of the TONFTS in three-dimensional space. (b) Intensity line-scans of the TONFTS along different axes. Intensity distribution superimposed with SOP in x-z plane at (c) y = −0.8λ, (d) 0 and (e) 0.8λ. (f-h) Stokes images and (i-k) spin density distribution in the vicinity of the focus.
To demonstrate that the complexity of the required incident light is still within the current optical engineering technology, the vectorial optical field is experimentally generated and compared with the theoretical design. Figure 3(a) illustrates the diagram of the vectorial optical field generator (VOF-Gen) shown in Fig. 1, which is the key system to create an arbitrary optical field with independent control of phase, amplitude, and polarization distribution on the pixel level. Taking advantage of the high definition television format of the Holoeye HEO 1080P reflective phase-only liquid crystal spatial light modulator, full control of all the degrees of freedom to create an arbitrary complex optical field is enabled [37]. The experimental results of the input beam designed for generating TONFTS is captured by a charge coupled device camera and shown in Fig. 3(f). Besides, the full Stokes parameter measurement (shown in Figs. 3(f)–3(i)) is also performed, showing very good agreement between the theoretical predications (shown in Figs. 3(b)–3(e)) and the experimental behavior. In order to quantitatively evaluate the synthesized input light in terms of the overall quality of the SOP, the cumulative normalized Stokes parameters are introduced:
(17)$${P_i} = \sqrt {{{\sum {S_i^2({{x_0},{y_0}} )} } / {\sum {S_0^2({{x_0},{y_0}} )} }}} ,\quad \quad \quad i = 1,2,3$$
where (x0, y0) are the indexes of the pixels of the Stokes image. Table 2 shows the theoretical and experimental values of P1, P2 and P3 of the input light presented in Figs. 3(b)–3(e) and 3(f)–3(i). The average error is calculated to be about 15%, demonstrating that the VOF-Gen is capable of sculpturing the complex light field and realizing TONFTS with high quality.
Fig. 3. Experimentally generated complex optical field for synthesizing TONFTS. (a) Diagram of the experimental setup. HWP, half-wave plate; P, polarizer; BS, beam splitter; L, lens; M, mirror; SF, spatial filter. (b-e) Theoretical and (f-i) experimental Stokes images of the incident light.
Table 2. Theoretical and experimental Pi values for the incident light presented in Fig. 3(b) and Fig. 3(f).
3. Optical force and torque under dipole approximation
To demonstrate the unique mechanical response using this TONFTS, we then analyze the optical interaction by introducing enantiomers into the focal volume of the 4Pi focusing system. The scattering behavior of the particle can be rigorously solved by the Mie theory, and the corresponding Mie scattering coefficients ($a_n^{(1)},a_n^{(2)},b_n^{(1)},b_n^{(2)}$) govern the relationship between the incident and scattering fields [23,50–53]:
(18)$$\begin{aligned} &a_n^{(1)} = [A_n^{(2)}V_n^{(1)} + A_n^{(1)}V_n^{(2)}]{Q_n}, \;\;\; a_n^{(2)} = [A_n^{(1)}W_n^{(2)} - A_n^{(2)}W_n^{(1)}]{Q_n},\\ &b_n^{(1)} = [B_n^{(1)}W_n^{(2)} + B_n^{(2)}W_n^{(1)}]{Q_n},\;\; b_n^{(2)} = a_n^{(2)}, \end{aligned}$$
(19)$$\begin{aligned} &A_n^{(j)} = {Z_S}D_n^{(1)}({x_j}) - D_n^{(1)}({x_0}), \;\;\; B_n^{(j)} = D_n^{(1)}({x_j}) - {Z_S}D_n^{(1)}({x_0}),\\ &W_n^{(j)} = {Z_S}D_n^{(1)}({x_j}) - D_n^{(3)}({x_0}), \;\; V_n^{(j)} = D_n^{(1)}({x_j}) - {Z_S}D_n^{(3)}({x_0}),\\ &{Q_n} = \frac{{{{{\psi _n}({x_0})} / {{\xi _n}({x_0})}}}}{{V_n^{(1)}W_n^{(2)} + V_n^{(2)}W_n^{(1)}}}, \end{aligned}$$
where x0 = krs, x1 = k1rs, and x2 = k2rs with rs represents the radius of particle and wave number ${k_1} = k(\sqrt {{\varepsilon _r}{\mu _r}} - \kappa )\textrm{/}\sqrt {{\varepsilon _m}} , {k_2} = k(\sqrt {{\varepsilon _r}{\mu _r}} + \kappa )\textrm{/}\sqrt {{\varepsilon _m}} ,$ with εr and μr represents the relative permittivity and permeability of the chiral medium, and εm represents the relative permittivity of the immersion medium. ${Z_S} = \sqrt {{{{\mu _r}{\varepsilon _m}} / {{\varepsilon _r}}}}$, while ψn(x) and ξn(x) are the Riccati-Bessel functions of the first and third kinds and $D_n^{(1)}(x) = {\psi ^{\prime}_n}(x)/{\psi _n}(x),$ $D_n^{(3)}(x) = {\xi ^{\prime}_n}(x)/{\xi _n}(x)$ are the corresponding derivatives.
Considering a spherical chiral particle that is much smaller than the incident wavelength, it can be treated in quasi-static limit and represented by point polarizability. Within the dipole approximation, the induced electric dipole moment p and magnetic dipole moment m can be expressed as p = αeeE+αemB and m = −αemE+αmmB, where E and B are the electric field and magnetic induction of the incident optical fields, αee, αmm and αem describe the electric, magnetic and electromagnetic polarizability of the sample. Note that αem is related to the chirality parameter κ of the material that the particle is made of, and the imaginary part of αem will change its sign if the handedness of the chiral sample changes. The polarizability elements of the dipolar chiral particle relate to the scattering coefficients [23,47]:
(20)$${\alpha _{ee}} = \frac{{i6\pi {\varepsilon _0}{\varepsilon _m}}}{{{k^3}}}a_1^{(1)},{\alpha _{mm}} = \frac{{i6\pi }}{{{\mu _0}{k^3}}}b_1^{(1)},{\alpha _{em}} ={-} \frac{{6\pi }}{{{Z_0}{k^3}}}a_1^{(2)},$$
where ${Z_0} = \sqrt {{\mu _0}{\varepsilon _m}/{\varepsilon _0}}$ is the wave impedance in vacuum and ε0 and μ0 are the vacuum permittivity and permeability. It can be proved that the Mie coefficients satisfy a1(1)(−κ) = a1(1)(κ), b1(1)(−κ) = b1(1)(κ), but a1(2)(−κ) = −a1(2)(κ), consequently only the electromagnetic polarizability will change its sign if the handedness of the chiral sample changes.
With the counter-propagating incident fields with expressions given in Eqs. (3)–(6) and Eqs. (10)–(12), the optical force exerted on a chiral nanoparticle in the TONFTS can be derived in terms of the electric and magnetic dipoles [32,54,55]:
(21)$$\left\langle \boldsymbol{F} \right\rangle = \frac{1}{2}{\rm{Re}} [(\nabla {\boldsymbol{E}^{\ast }}) \cdot \boldsymbol{p} + (\nabla {\boldsymbol{B}^{\ast}}) \cdot \boldsymbol{m} - \frac{{{Z_0}{k^4}}}{{6\pi }}(\boldsymbol{p} \times {\boldsymbol{m}^{\ast}})].$$
The force expression can be written as follows:
(22)$$\begin{aligned} \left\langle \boldsymbol{F} \right\rangle ={-} &\nabla \left\langle U \right\rangle + \frac{{n{}_m}}{c}({{C_{ext}} + {C_{recoil}}} )\left\langle \boldsymbol{S} \right\rangle + {\mu _0}\nabla \times {\rm{Re}} [{{\alpha_{em}}} ]\left\langle \boldsymbol{S} \right\rangle \textrm{ + }\nabla \times \left\{ {{C_p}\frac{c}{{{n_m}}}\left\langle {{\boldsymbol{L}_{\boldsymbol{p}}}} \right\rangle + {C_m}\frac{c}{{{n_m}}}\left\langle {{\boldsymbol{L}_{\boldsymbol{m}}}} \right\rangle } \right\}\\ &+ \left\{ {2{\omega^2}{\mu_0}{\rm{Re}} [{{\alpha_{em}}} ]- \frac{{{k^5}}}{{3\pi \varepsilon_0^2\varepsilon_m^2}}{\mathop{\rm Im}\nolimits} [{{\alpha_{ee}}\alpha_{em}^ \ast } ]} \right\}\left\langle {{\boldsymbol{L}_{\boldsymbol{p}}}} \right\rangle + \left\{ {2{\omega^2}{\mu_0}{\rm{Re}} [{{\alpha_{em}}} ]- \frac{{{k^5}{\mu_0}}}{{3\pi {\varepsilon_0}{\varepsilon_m}}}{\mathop{\rm Im}\nolimits} [{{\alpha_{mm}}\alpha_{em}^ \ast } ]} \right\}\left\langle {{\boldsymbol{L}_{\boldsymbol{m}}}} \right\rangle \\ &+ \frac{{c{k^4}\mu _0^2}}{{12\pi {n_m}}}{\mathop{\rm Im}\nolimits} [{{\alpha_{ee}}\alpha_{mm}^ \ast } ]{\mathop{\rm Im}\nolimits} [{\boldsymbol{E} \times {\boldsymbol{H}^{\ast}}} ]. \end{aligned}$$
where c is the speed of light in vacuum, 〈S〉 = Re[E × H*]/2 denotes the time-averaged Poynting vector, 〈U〉 = −1/4(Re[αee]|E|2 + Re[αmm]|B|2 − 2Im[αem]Im[B·E*]) is an energy term due to the interaction of the dipolar chiral particle with the TONFTS, 〈Lp〉 = Re[ε0εm/(4iω)E × E*] and 〈Lm〉 = Re[μ0/(4iω)H × H*] represent the electric and magnetic part of the time-averaged spin angular momentum densities respectively, ω is the angular frequency. Cext = Cp + Cm = k(Im[αee]/ (ε0εm) + μ0Im[αmm]) is a sum of contribution from the electric and magnetic dipole channel. Crecoil = −k4μ0/(6πε0εm)Re[αeeα*mm] − k4μ0/(6πε0εm)|αem|2 describes the recoil force and is related to the asymmetry parameter. The first and second terms in Eq. (22) correspond to the gradient force and the radiation pressure, respectively. The third term is a vortex force determined by the energy flow vortex around the particle and the optical activity, while the fourth term represents the scattering force associated with the curl of spin angular momentum densities. The fifth and sixth terms are referred to the spin density force that is related to the spin angular momentum densities. The particle chirality makes no explicit contribution to the last term, which is due to the alternating flow of the stored energy. Consequently, the movement of a chiral particle immersed in the optical focal field is subject to the induced time-averaged optical force, arising from the transfer of the linear momentum between the light and the material.
Assuming a lossy chiral nanoparticle with radius of 5 nm located near the center of the TONFTS, the optical force is calculated with Eq. (22) and the corresponding force distributions in different planes are illustrated in Fig. 4. Note that the magnetic field B is calculated from the spatial distributions of the electric fields in the focal volume as B = 1/(iω)∇×E. The parameters of the chiral material are set to be (ε, κ) = (1.62+0.04i, −0.5 + 0.02i). The chiral particles can be a conventional electromagnetic material with chiral shape, or a spherical particle made of chiral material. By controlling the concentration of the lossy constituent added, the chiral particle with controllable absorption can be precisely fabricated. Figures 4(a) and 4(b) shows the x- and z-component of the optical force in the x-z plane (y = 0), indicating that the particles in the focal area would be trapped at x = 0 and z = 0. Besides, from the distribution of the y-component of the optical force Fy in the x-y plane (shown in Fig. 4(c)) it can be clearly seen that there is an equilibrium position at y = 0. To explore the physical mechanism of the induced force, the total optical force has been divided into different parts and the corresponding line-scans are presented in Figs. 4(d)–4(f). Firstly, it can be seen that the scattering force which contains the second, third, fourth, fifth and sixth terms in Eq. (22) is maximized along the direction of the photonic spin, but still is difficult to compete with the gradient force. Secondly, the gradient force can be decomposed into the combination of chiral and achiral gradient force, which would change/not change its sign for particles with opposite chirality. Specifically, achiral gradient force refer to Fachiral = −1/4(Re[αee]|E|2 + Re[αmm]|B|2) and chiral gradient force refer to Fchiral = 1/2Im[αem]Im[B·E*]. Thirdly, the achiral gradient force is dominating in the plane perpendicular to the orientation of the photonic spin, which confines the enantiomers at the center of the focus. Lastly, due to the uniform intensity distribution of the optical needle field, the achiral gradient force would become negligible along y-axis in the vicinity of the focus, while the movement of the particle is primarily determined by the chiral gradient force. As a particle with refractive index larger than the ambient environment, it would be trapped at the location possessing zero net optical force and negative slope of the optical force distribution. The stability of the optical trapping can be demonstrated by the potential depth U, which is estimated as the work done by optical forces along the axes ${U_n} = - \int {F_n} \cdot dn (n = x,y,z)$. It can reach 1.658 kBT and 7.5 kBT for the equilibrium positions at the center and the ends of the optical needle field respectively, where kB is the Boltzmann constant and T = 300 K is the absolute temperature of the ambient. Traditionally an optical trap with potential depth U larger than kBT can be considered as stable [11]. Consequently, a chiral nanoparticle with chirality parameter (κ = −0.5 + 0.02i) would be stably trapped at the central position (x = y = z = 0) of the TONFTS. In addition, a chiral nanoparticle with opposite handedness (κ = 0.5 + 0.02i) is also considered and the corresponding optical force distributions are presented in Figs. 4(g)–4(l). In this case, the particle is still centrally confined in the x-z plane by the achiral gradient force, while it will be pushed out of the focus by the chiral gradient force along y axis and finally be stably trapped at the ends of the TONFTS, and the separation distance between the enantiomers should be around 0.5λ or 0.6λ. Therefore, we demonstrated that the TONFTS is capable to realize laterally sorting of chiral nanoparticle, in which the enantiomers with different handedness would be trapped at the center and two ends of the optical needle field, respectively. It is worthy of noting that the equilibrium positions of the enantiomers can be easily switched, which is realized by changing the handedness of the orthogonal dipoles (Eqs. (4) and (5)).
Fig. 4. Calculated optical force on 5 nm (radius) chiral absorbing nanoparticle at wavelength of 632 nm using the proposed strategy with TONFTS. Distribution of the (a) x-, (b) z-, and (c) y-component of the optical force exerted on an chiral nanoparticle with (ε, κ) = (1.62+0.04i, −0.5 + 0.02i) located near the center of the TONFTS, and the corresponding components of the optical force along (d) x- (e) z- and (f) y-axis. Distribution of the (g) x-, (h) z-, and (i) y-component of the optical force exerted on the chiral nanoparticle with opposite handedness (κ = 0.5 + 0.02i) and the components of the optical force along (j) x- (k) z- and (l) y-axis.
For comparison, the optical force effects of chiral nanoparticle interacting with the optical focal field with pure transverse spin (along y axis) is also considered, which can be designed by reversing the radiation field from two orthogonal electric dipoles with phase difference of π/2 at the focus of a high NA lens. Note that in this case the intensity distribution of the focal field along y axis is not uniform anymore. As shown in Figs. 5(a)–5(c), although the particle can still be trapped at the center of the x-z plane, the difference in optical force which is mainly determined by the chiral gradient force becomes negligible, therefore the enantiomers cannot be separated by the achiral gradient force in three-dimensional space. From Eq. (22) one can find that the chiral gradient force is proportional to the gradient of the optical chirality density Im[B·E*]. Compared with the TONFTS, the optical focal field with purely transverse spin angular momentum carries relatively weak Ey, Hx and Hz, leading to the small optical chirality density. As is shown in Figs. 5(d)–5(g), the optical chirality of purely transverse spin filed in y-z plane and x-y plane is an obviously smaller than TONFTS, and the low chirality gradient leads to negligible chiral gradient force. Consequently, the TONFTS we proposed in this work has the advantage of providing negligible achiral gradient force and large optical chiral density in the transversal direction. Due to the enhanced chiral gradient force provided by the TONFTS, it is feasible to stably trap and laterally sort natural /artificial chiral materials with smaller chirality parameter by increasing the laser power, or constructing complex optical focal field that consists of more pair of electric and magnetic dipoles, which would boost the efficiency and sensitivity of the techniques involving the enantiomers sorting and detecting.
Fig. 5. The optical force effect of optical field with transverse spin exerted on 5 nm (radius) chiral absorbing nanoparticle at the wavelength of 632 nm. Optical force along (a) x-, (b) y-, and (c) z-axis exerted on chiral nanoparticles with (ε, κ) = (1.62+0.04i, ±0.5 + 0.02i) located near the focus of an optical focal field with transverse spin along y axis. Optical chirality distribution of the optical focal field with pure transverse spin and TONFTS in the (d, f) y-z plane and (e, g) x-y plane, respectively.
Moreover, due to the spin angular momentum transfer from the TONFTS, the chiral particle would experience an intrinsic torque Γ and spin around its center of mass:
(23)$$\begin{aligned} &\left\langle {\boldsymbol{\varGamma}} \right\rangle = [ - 2{\mu _0}{\rm{Re}} ({\alpha _{em}}) + \frac{{{\mu _0}{k^3}}}{{3\pi {\varepsilon _0}{\varepsilon _m}}}{\mathop{\rm Im}\nolimits} ({\alpha _{ee}}\alpha _{em}^ \ast ) + \frac{{\mu _0^2{k^3}}}{{3\pi }}{\mathop{\rm Im}\nolimits} ({\alpha _{mm}}\alpha _{em}^ \ast )]\left\langle \boldsymbol{S} \right\rangle \\ &+ [\frac{{{\mu _0}{k^3}}}{{6\pi {\varepsilon _0}{\varepsilon _m}}}{\rm{Re}} ({\alpha _{ee}}\alpha _{em}^ \ast ) - \frac{{\mu _0^2{k^3}}}{{6\pi }}{\rm{Re}} ({\alpha _{mm}}\alpha _{em}^ \ast )]{\mathop{\rm Im}\nolimits} (\boldsymbol{E} \times {\boldsymbol{H}^{\ast}})\\ &+ [\frac{{2\omega }}{{{\varepsilon _0}{\varepsilon _m}}}{\mathop{\rm Im}\nolimits} ({\alpha _{ee}}) - \frac{{\omega {k^3}}}{{3\pi \varepsilon _0^2\varepsilon _m^2}}{\alpha _{ee}}\alpha _{ee}^ \ast{-} \frac{{\omega {\mu _0}{k^3}}}{{3\pi {\varepsilon _0}{\varepsilon _m}}}{\alpha _{em}}\alpha _{em}^ \ast ]\left\langle {{\boldsymbol{L}_{\boldsymbol{p}}}} \right\rangle \\ &+ [2\omega {\mu _0}{\mathop{\rm Im}\nolimits} ({\alpha _{mm}}) - \frac{{\omega \mu _0^2{k^3}}}{{3\pi }}{\alpha _{mm}}\alpha _{mm}^ \ast{-} \frac{{\omega {\mu _0}{k^3}}}{{3\pi {\varepsilon _0}{\varepsilon _m}}}{\alpha _{em}}\alpha _{em}^ \ast ]\left\langle {{\boldsymbol{L}_{\boldsymbol{m}}}} \right\rangle . \end{aligned}$$
To intuitively illustrate the activity of the trapped enantiomers, the distributions of the optical torque in three x-z planes containing the equilibrium locations (y = −0.6λ, 0, 0.5λ) are calculated and shown in Fig. 6. It can be clearly seen that the chiral particle would rotate around y axis. The rotation orientation of the particle is determined by that of the photonic spin and it is not related to the chirality of the material. Besides, the rotation frequencies ${\Omega }p = - {\Gamma }/8\pi \xi {r_s}^{3}$ at the equilibrium positions are calculated to be 7.3259×105 Hz (viscosity of water ξ is about 0.001 N⋅s/m2 at 20°C)) for particles with κ = −0.5 + 0.02i and 4.0034×105 for particles with κ = 0.5 + 0.02i. Note that the chiral particle trapped at the center of the TONFTS would experience a faster rotation frequency compared to its counterpart with the opposite handedness trapped at the ends of the optical needle field, providing another feasible way to distinguish the enantiomers by their dynamic behaviors.
Fig. 6. Calculated optical torque distribution at equilibrium positions. Distribution of the y-component of the optical torque in the z-x plane at (a) y = −0.6λ, (b) 0 and (c) 0.5λ. The x- and z-components of the optical torque is indicated by the arrows.
In conclusion, we propose a method to lateral sort chiral nanoparticles with different handedness by using the transverse optical needle field with transverse spin. To achieve controllable orientation of both the photonic spin and the optical needle field, the required pupil field of the illumination is analytically derived through reversing the radiation patterns from an array of paired orthogonal electric dipoles located in the focal plane of a 4Pi microscopy and experimentally generated with a home-built VOF-Gen. Furthermore, the scattering behavior of the chiral particle is evaluated by the Mie scattering coefficients and the induced optical force from the interaction between the chiral nanoparticle and the TONFTS is calculated under dipole approximation. Numerical results demonstrated that the particles experience a transverse chiral optical force with direction determined by the handedness of the chiral material. Besides, the transverse achiral gradient force is negligible along the optical needle field, giving rise the possibility to lateral sort and trap chiral nanoparticle with different handedness at different locations. Moreover, we demonstrated that the intrinsic torque leads to the rotation of the chiral nanoparticle with handedness-dependent rotation frequency around the spin axis. This versatile trapping method may open new avenues for all-optical enantiopure chemical syntheses and enantiomer separations in pharmaceuticals.
National Natural Science Foundation of China (11504049, 11774055).
G. R. acknowledged the support by the Zhishan Young Scholar Program of Southeast University.
1. A. Hayat, J. P. B. Mueller, and F. Capasso, "Lateral chirality-sorting optical forces," Proc. Natl. Acad. Sci. U. S. A. 112(43), 13190–13194 (2015). [CrossRef]
2. J. Lekner, "Optical properties of isotropic chiral media," J. Opt. 5(4), 417–443 (1996). [CrossRef]
3. C. M. Dobson, "Protein folding and misfolding," Nature 426(6968), 884–890 (2003). [CrossRef]
4. L. D. Barron and A. D. Buckingham, "Rayleigh and Raman scattering from optically active molecules," Mol. Phys. 20(6), 1111–1119 (1971). [CrossRef]
5. W. Hug and H. Surbeck, "Vibrational Raman optical activity spectra recorded in perpendicular polarization," Chem. Phys. Lett. 60(2), 186–192 (1979). [CrossRef]
6. S. F. Mason, Optical Activity and Chiral Discrimination (Springer, 1979).
7. L. D. Barron, E. Blanch, and L. Hecht, "Unfolded proteins studied by Raman optical activity," Adv. Protein Chem. 62, 51–90 (2002). [CrossRef]
8. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, "Observation of a single–beam gradient force optical trap for dielectric particles," Opt. Lett. 11(5), 288–290 (1986). [CrossRef]
9. D. G. Grier, "A revolution in optical manipulation," Nature 424(6950), 810–816 (2003). [CrossRef]
10. O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, "Optical trapping and manipulation of nanostructures," Nat. Nanotechnol. 8(11), 807–819 (2013). [CrossRef]
11. K. Svoboda and S. M. Block, "Optical trapping of metallic Rayleigh particles," Opt. Lett. 19(13), 930–932 (1994). [CrossRef]
12. K. C. Toussaint, M. Liu, M. Pelton, J. Pesic, M. J. Guffey, P. Guyot-Sionnest, and N. F. Scherer, "Plasmon resonance–based optical trapping of single and multiple Au nanoparticles," Opt. Express 15(19), 12017–12029 (2007). [CrossRef]
13. S. Tan, H. A. Lopez, C. W. Cai, and Y. Zhang, "Optical trapping of single–walled carbon nanotubes," Nano Lett. 4(8), 1415–1419 (2004). [CrossRef]
14. T. Rodgers, S. Shoji, Z. Sekkat, and S. Kawata, "Selective aggregation of single–walled carbon nanotubes using the large optical field gradient of a focused laser beam," Phys. Rev. Lett. 101(12), 127402 (2008). [CrossRef]
15. L. Jauffred, A. C. Richardson, and L. B. Oddershede, "Three–dimensional optical control of individual quantum dots," Nano Lett. 8(10), 3376–3380 (2008). [CrossRef]
16. Y. F. Chen, X. Serey, R. Sarkar, P. Chen, and D. Erickson, "Controlled photonic manipulation of proteins and other nanomaterials," Nano Lett. 12(3), 1633–1637 (2012). [CrossRef]
17. H. Chen, C. Liang, S. Liu, and Z. Lin, "Chirality sorting using two–wave–interference–induced lateral optical force," Phys. Rev. A 93(5), 053833 (2016). [CrossRef]
18. H. Chen, Y. Jiang, N. Wang, W. Lu, S. Liu, and Z. Lin, "Lateral optical force on paired chiral nanoparticles in linearly polarized plane waves," Opt. Lett. 40(23), 5530–5533 (2015). [CrossRef]
19. G. Cipparrone, A. Mazzulla, A. Pane, R. J. Hernandez, and R. Bartolino, "Chiral self–assembled solid microspheres: A novel multifunctional microphotonic device," Adv. Mater. 23(48), 5773–5778 (2011). [CrossRef]
20. R. J. Hernández, A. Mazzulla, A. Pane, K. Volke-Sepúlveda, and G. Cipparrone, "Attractive–repulsive dynamics on light–responsive chiral microparticles induced by polarized tweezers," Lab Chip 13(3), 459–467 (2013). [CrossRef]
21. K. Ding, J. Ng, L. Zhou, and C. T. Chan, "Realization of optical pulling forces using chirality," Phys. Rev. A 89(6), 063825 (2014). [CrossRef]
22. H. Chen, W. Lu, X. Yu, C. Xue, S. Liu, and Z. Lin, "Optical torque on small chiral particles in generic optical fields," Opt. Express 25(26), 32867–32878 (2017). [CrossRef]
23. H. Chen, N. Wang, W. Lu, S. Liu, and Z. Lin, "Tailoring azimuthal optical force on lossy chiral particles in Bessel beams," Phys. Rev. A 90(4), 043850 (2014). [CrossRef]
24. Y. Yang, P. D. Brimicombe, N. W. Roberts, M. R. Dickinson, M. Osipov, and H. F. Gleeson, "Continuously rotating chiral liquid crystal droplets in a linearly polarized laser trap," Opt. Express 16(10), 6877–6882 (2008). [CrossRef]
25. M. Mosallaeipour, Y. Hatwalne, N. V. Madhusudana, and S. Ananthamurthy, "Laser induced rotation of trapped chiral and achiral nematic droplets," J. Mod. Opt. 57(5), 395–399 (2010). [CrossRef]
26. D. Smith, C. Woods, A. Seddon, and H. Hoerber, "Photophoretic separation of single–walled carbon nanotubes: a novel approach to selective chiral sorting," Phys. Chem. Chem. Phys. 16(11), 5221–5228 (2014). [CrossRef]
27. A. Canaguier-Durand, J. A. Hutchison, C. Genet, and T. W. Ebbesen, "Mechanical separation of chiral dipoles by chiral light," New J. Phys. 15(12), 123037 (2013). [CrossRef]
28. R. P. Cameron, S. M. Barnett, and A. M. Yao, "Discriminatory optical force for chiral molecules," New J. Phys. 16(1), 013020 (2014). [CrossRef]
29. S. B. Wang and C. T. Chan, "Lateral optical force on chiral particles near a surface," Nat. Commun. 5(1), 3307 (2014). [CrossRef]
30. H. A. Arredondo Champi, R. H. Bustamante, and W. J. Salcedo, "Optical enantioseparation of chiral molecules using asymmetric plasmonic nanoapertures," Opt. Mater. Express 9(4), 1763–1775 (2019). [CrossRef]
31. N. Kravets, A. Aleksanyan, H. Chraïbi, J. Leng, and E. Brasselet, "Optical Enantioseparation of Racemic Emulsions of Chiral Microparticles," Phys. Rev. Appl. 11(4), 044025 (2019). [CrossRef]
32. M. Li, S. Yan, Y. Zhang, P. Zhang, and B. Yao, "Enantioselective optical trapping of chiral nanoparticles by tightly focused vector beams," J. Opt. Soc. Am. B 36(8), 2099–2105 (2019). [CrossRef]
33. P. Acebal, L. Carretero, and S. Blaya, "Design of an optical conveyor for selective separation of a mixture of enantiomers," Opt. Express 25(26), 32290–32304 (2017). [CrossRef]
34. G. Tkachenko and E. Brasselet, "Optofluidic sorting of material chirality by chiral light," Nat. Commun. 5(1), 3577 (2014). [CrossRef]
35. G. Tkachenko and E. Brasselet, "Helicity–dependent three–dimensional optical trapping of chiral microparticles," Nat. Commun. 5(1), 4491 (2014). [CrossRef]
36. W. Han, Y. Yang, W. Cheng, and Q. Zhan, "Vectorial optical field generator for the creation of arbitrarily complex fields," Opt. Express 21(18), 20692–20706 (2013). [CrossRef]
37. I. Moreno, J. A. Davis, T. M. Hernandez, D. M. Cottrell, and D. Sand, "Complete polarization control of light from a liquid crystal spatial light modulator," Opt. Express 20(1), 364–376 (2012). [CrossRef]
38. J. Chen, C. Wan, L. Kong, and Q. Zhan, "Experimental generation of complex optical fields for diffraction limited optical focus with purely transverse spin angular momentum," Opt. Express 25(8), 8966–8974 (2017). [CrossRef]
39. J. Chen, C. Wan, L. J. Kong, and Q. Zhan, "Tightly focused optical field with controllable photonic spin orientation," Opt. Express 25(16), 19517–19528 (2017). [CrossRef]
40. W. Chen and Q. Zhan, "Diffraction limited focusing with controllable arbitrary three–dimensional polarization," J. Opt. 12(4), 045707 (2010). [CrossRef]
41. G. Rui and Q. Zhan, "Trapping of resonant metallic nanoparticles with engineered vectorial optical field," Nanophotonics 3(6), 351–361 (2014). [CrossRef]
42. G. Rui, X. Wang, B. Gu, Q. Zhan, and Y. Cui, "Manipulation metallic nanoparticle at resonant wavelength using engineered azimuthally polarized optical field," Opt. Express 24(7), 7212–7223 (2016). [CrossRef]
43. G. Rui, Y. Li, S. Zhou, Y. Wang, B. Gu, Y. Cui, and Q. Zhan, "Optically induced rotation of Rayleigh particles by arbitrary photonic spin," Photonics Res. 7(1), 69–79 (2019). [CrossRef]
44. G. Rui, X. Wang, and Y. Cui, "Manipulation of metallic nanoparticle with evanescent vortex Bessel beam," Opt. Express 23(20), 25707–25716 (2015). [CrossRef]
45. G. Rui, Y. Wang, X. Wang, B. Gu, and Y. Cui, "Trapping of low–refractive–index nanoparticles in a hollow dark spherical spot," J. Phys. Commun. 2(6), 065015 (2018). [CrossRef]
46. J. Wang, W. Chen, and Q. Zhan, "Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation," Opt. Express 18(21), 21965–21972 (2010). [CrossRef]
47. H. Hu, Q. Gan, and Q. Zhan, "Generation of a Nondiffracting Superchiral Optical Needle for Circular Dichroism Imaging of Sparse Subdiffraction Objects," Phys. Rev. Lett. 122(22), 223901 (2019). [CrossRef]
48. J. Luo, H. Zhang, S. Wang, L. Shi, Z. Zhu, B. Gu, X. Wang, and X. Li, "Three-dimensional magnetization needle arrays with controllable orientation," Opt. Lett. 44(4), 727–730 (2019). [CrossRef]
49. Z. Man, C. Min, L. Du, Y. Zhang, S. Zhu, and X. Yuan, "Sub-wavelength sized transversely polarized optical needle with exceptionally suppressed side-lobes," Opt. Express 24(2), 874–882 (2016). [CrossRef]
50. A. Lakhtakia and V. V. Varadan, Time–Harmonic Electromagnetic Fields in Chiral Media (Springer–Verlag, 1989).
51. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (JohnWiley & Sons, 1983).
52. Ø. Farsund and B. U. Felderhof, "Force, torque, and absorbed energy for a body of arbitrary shape and constitution in an electromagnetic radiation field," Phys. A (Amsterdam, Neth.) 227(1-2), 108–130 (1996). [CrossRef]
53. M. Li, S. Yan, Y. Zhang, Y. Liang, P. Zhang, and B. Yao, "Optical sorting of small chiral particles by tightly focused vector beams," Phys. Rev. A 99(3), 033825 (2019). [CrossRef]
54. J. Chen, J. Ng, Z. Lin, and C. T. Chan, "Optical pulling force," Nat. Photonics 5(9), 531–534 (2011). [CrossRef]
55. H. Tu, Y. Zhao, Y. Liu, and S. A. Boppart, "Static third–harmonic lines in widely variable fiber continuum generation," Phys. Rev. A 89(1), 013813 (2014). [CrossRef]
A. Hayat, J. P. B. Mueller, and F. Capasso, "Lateral chirality-sorting optical forces," Proc. Natl. Acad. Sci. U. S. A. 112(43), 13190–13194 (2015).
J. Lekner, "Optical properties of isotropic chiral media," J. Opt. 5(4), 417–443 (1996).
C. M. Dobson, "Protein folding and misfolding," Nature 426(6968), 884–890 (2003).
L. D. Barron and A. D. Buckingham, "Rayleigh and Raman scattering from optically active molecules," Mol. Phys. 20(6), 1111–1119 (1971).
W. Hug and H. Surbeck, "Vibrational Raman optical activity spectra recorded in perpendicular polarization," Chem. Phys. Lett. 60(2), 186–192 (1979).
S. F. Mason, Optical Activity and Chiral Discrimination (Springer, 1979).
L. D. Barron, E. Blanch, and L. Hecht, "Unfolded proteins studied by Raman optical activity," Adv. Protein Chem. 62, 51–90 (2002).
A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, "Observation of a single–beam gradient force optical trap for dielectric particles," Opt. Lett. 11(5), 288–290 (1986).
D. G. Grier, "A revolution in optical manipulation," Nature 424(6950), 810–816 (2003).
O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, "Optical trapping and manipulation of nanostructures," Nat. Nanotechnol. 8(11), 807–819 (2013).
K. Svoboda and S. M. Block, "Optical trapping of metallic Rayleigh particles," Opt. Lett. 19(13), 930–932 (1994).
K. C. Toussaint, M. Liu, M. Pelton, J. Pesic, M. J. Guffey, P. Guyot-Sionnest, and N. F. Scherer, "Plasmon resonance–based optical trapping of single and multiple Au nanoparticles," Opt. Express 15(19), 12017–12029 (2007).
S. Tan, H. A. Lopez, C. W. Cai, and Y. Zhang, "Optical trapping of single–walled carbon nanotubes," Nano Lett. 4(8), 1415–1419 (2004).
T. Rodgers, S. Shoji, Z. Sekkat, and S. Kawata, "Selective aggregation of single–walled carbon nanotubes using the large optical field gradient of a focused laser beam," Phys. Rev. Lett. 101(12), 127402 (2008).
L. Jauffred, A. C. Richardson, and L. B. Oddershede, "Three–dimensional optical control of individual quantum dots," Nano Lett. 8(10), 3376–3380 (2008).
Y. F. Chen, X. Serey, R. Sarkar, P. Chen, and D. Erickson, "Controlled photonic manipulation of proteins and other nanomaterials," Nano Lett. 12(3), 1633–1637 (2012).
H. Chen, C. Liang, S. Liu, and Z. Lin, "Chirality sorting using two–wave–interference–induced lateral optical force," Phys. Rev. A 93(5), 053833 (2016).
H. Chen, Y. Jiang, N. Wang, W. Lu, S. Liu, and Z. Lin, "Lateral optical force on paired chiral nanoparticles in linearly polarized plane waves," Opt. Lett. 40(23), 5530–5533 (2015).
G. Cipparrone, A. Mazzulla, A. Pane, R. J. Hernandez, and R. Bartolino, "Chiral self–assembled solid microspheres: A novel multifunctional microphotonic device," Adv. Mater. 23(48), 5773–5778 (2011).
R. J. Hernández, A. Mazzulla, A. Pane, K. Volke-Sepúlveda, and G. Cipparrone, "Attractive–repulsive dynamics on light–responsive chiral microparticles induced by polarized tweezers," Lab Chip 13(3), 459–467 (2013).
K. Ding, J. Ng, L. Zhou, and C. T. Chan, "Realization of optical pulling forces using chirality," Phys. Rev. A 89(6), 063825 (2014).
H. Chen, W. Lu, X. Yu, C. Xue, S. Liu, and Z. Lin, "Optical torque on small chiral particles in generic optical fields," Opt. Express 25(26), 32867–32878 (2017).
H. Chen, N. Wang, W. Lu, S. Liu, and Z. Lin, "Tailoring azimuthal optical force on lossy chiral particles in Bessel beams," Phys. Rev. A 90(4), 043850 (2014).
Y. Yang, P. D. Brimicombe, N. W. Roberts, M. R. Dickinson, M. Osipov, and H. F. Gleeson, "Continuously rotating chiral liquid crystal droplets in a linearly polarized laser trap," Opt. Express 16(10), 6877–6882 (2008).
M. Mosallaeipour, Y. Hatwalne, N. V. Madhusudana, and S. Ananthamurthy, "Laser induced rotation of trapped chiral and achiral nematic droplets," J. Mod. Opt. 57(5), 395–399 (2010).
D. Smith, C. Woods, A. Seddon, and H. Hoerber, "Photophoretic separation of single–walled carbon nanotubes: a novel approach to selective chiral sorting," Phys. Chem. Chem. Phys. 16(11), 5221–5228 (2014).
A. Canaguier-Durand, J. A. Hutchison, C. Genet, and T. W. Ebbesen, "Mechanical separation of chiral dipoles by chiral light," New J. Phys. 15(12), 123037 (2013).
R. P. Cameron, S. M. Barnett, and A. M. Yao, "Discriminatory optical force for chiral molecules," New J. Phys. 16(1), 013020 (2014).
S. B. Wang and C. T. Chan, "Lateral optical force on chiral particles near a surface," Nat. Commun. 5(1), 3307 (2014).
H. A. Arredondo Champi, R. H. Bustamante, and W. J. Salcedo, "Optical enantioseparation of chiral molecules using asymmetric plasmonic nanoapertures," Opt. Mater. Express 9(4), 1763–1775 (2019).
N. Kravets, A. Aleksanyan, H. Chraïbi, J. Leng, and E. Brasselet, "Optical Enantioseparation of Racemic Emulsions of Chiral Microparticles," Phys. Rev. Appl. 11(4), 044025 (2019).
M. Li, S. Yan, Y. Zhang, P. Zhang, and B. Yao, "Enantioselective optical trapping of chiral nanoparticles by tightly focused vector beams," J. Opt. Soc. Am. B 36(8), 2099–2105 (2019).
P. Acebal, L. Carretero, and S. Blaya, "Design of an optical conveyor for selective separation of a mixture of enantiomers," Opt. Express 25(26), 32290–32304 (2017).
G. Tkachenko and E. Brasselet, "Optofluidic sorting of material chirality by chiral light," Nat. Commun. 5(1), 3577 (2014).
G. Tkachenko and E. Brasselet, "Helicity–dependent three–dimensional optical trapping of chiral microparticles," Nat. Commun. 5(1), 4491 (2014).
W. Han, Y. Yang, W. Cheng, and Q. Zhan, "Vectorial optical field generator for the creation of arbitrarily complex fields," Opt. Express 21(18), 20692–20706 (2013).
I. Moreno, J. A. Davis, T. M. Hernandez, D. M. Cottrell, and D. Sand, "Complete polarization control of light from a liquid crystal spatial light modulator," Opt. Express 20(1), 364–376 (2012).
J. Chen, C. Wan, L. Kong, and Q. Zhan, "Experimental generation of complex optical fields for diffraction limited optical focus with purely transverse spin angular momentum," Opt. Express 25(8), 8966–8974 (2017).
J. Chen, C. Wan, L. J. Kong, and Q. Zhan, "Tightly focused optical field with controllable photonic spin orientation," Opt. Express 25(16), 19517–19528 (2017).
W. Chen and Q. Zhan, "Diffraction limited focusing with controllable arbitrary three–dimensional polarization," J. Opt. 12(4), 045707 (2010).
G. Rui and Q. Zhan, "Trapping of resonant metallic nanoparticles with engineered vectorial optical field," Nanophotonics 3(6), 351–361 (2014).
G. Rui, X. Wang, B. Gu, Q. Zhan, and Y. Cui, "Manipulation metallic nanoparticle at resonant wavelength using engineered azimuthally polarized optical field," Opt. Express 24(7), 7212–7223 (2016).
G. Rui, Y. Li, S. Zhou, Y. Wang, B. Gu, Y. Cui, and Q. Zhan, "Optically induced rotation of Rayleigh particles by arbitrary photonic spin," Photonics Res. 7(1), 69–79 (2019).
G. Rui, X. Wang, and Y. Cui, "Manipulation of metallic nanoparticle with evanescent vortex Bessel beam," Opt. Express 23(20), 25707–25716 (2015).
G. Rui, Y. Wang, X. Wang, B. Gu, and Y. Cui, "Trapping of low–refractive–index nanoparticles in a hollow dark spherical spot," J. Phys. Commun. 2(6), 065015 (2018).
J. Wang, W. Chen, and Q. Zhan, "Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation," Opt. Express 18(21), 21965–21972 (2010).
H. Hu, Q. Gan, and Q. Zhan, "Generation of a Nondiffracting Superchiral Optical Needle for Circular Dichroism Imaging of Sparse Subdiffraction Objects," Phys. Rev. Lett. 122(22), 223901 (2019).
J. Luo, H. Zhang, S. Wang, L. Shi, Z. Zhu, B. Gu, X. Wang, and X. Li, "Three-dimensional magnetization needle arrays with controllable orientation," Opt. Lett. 44(4), 727–730 (2019).
Z. Man, C. Min, L. Du, Y. Zhang, S. Zhu, and X. Yuan, "Sub-wavelength sized transversely polarized optical needle with exceptionally suppressed side-lobes," Opt. Express 24(2), 874–882 (2016).
A. Lakhtakia and V. V. Varadan, Time–Harmonic Electromagnetic Fields in Chiral Media (Springer–Verlag, 1989).
C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (JohnWiley & Sons, 1983).
Ø. Farsund and B. U. Felderhof, "Force, torque, and absorbed energy for a body of arbitrary shape and constitution in an electromagnetic radiation field," Phys. A (Amsterdam, Neth.) 227(1-2), 108–130 (1996).
M. Li, S. Yan, Y. Zhang, Y. Liang, P. Zhang, and B. Yao, "Optical sorting of small chiral particles by tightly focused vector beams," Phys. Rev. A 99(3), 033825 (2019).
J. Chen, J. Ng, Z. Lin, and C. T. Chan, "Optical pulling force," Nat. Photonics 5(9), 531–534 (2011).
H. Tu, Y. Zhao, Y. Liu, and S. A. Boppart, "Static third–harmonic lines in widely variable fiber continuum generation," Phys. Rev. A 89(1), 013813 (2014).
Acebal, P.
Aleksanyan, A.
Ananthamurthy, S.
Arredondo Champi, H. A.
Ashkin, A.
Barnett, S. M.
Barron, L. D.
Bartolino, R.
Bjorkholm, J. E.
Blanch, E.
Blaya, S.
Block, S. M.
Bohren, C. F.
Boppart, S. A.
Brasselet, E.
Brimicombe, P. D.
Buckingham, A. D.
Bustamante, R. H.
Cai, C. W.
Cameron, R. P.
Canaguier-Durand, A.
Capasso, F.
Carretero, L.
Chan, C. T.
Chen, H.
Chen, P.
Chen, W.
Chen, Y. F.
Cheng, W.
Chraïbi, H.
Chu, S.
Cipparrone, G.
Cottrell, D. M.
Cui, Y.
Davis, J. A.
Dickinson, M. R.
Ding, K.
Dobson, C. M.
Du, L.
Dziedzic, J. M.
Ebbesen, T. W.
Erickson, D.
Farsund, Ø.
Felderhof, B. U.
Ferrari, A. C.
Gan, Q.
Genet, C.
Gleeson, H. F.
Grier, D. G.
Gu, B.
Gucciardi, P. G.
Guffey, M. J.
Guyot-Sionnest, P.
Han, W.
Hatwalne, Y.
Hayat, A.
Hecht, L.
Hernandez, R. J.
Hernandez, T. M.
Hernández, R. J.
Hoerber, H.
Hu, H.
Huffman, D. R.
Hug, W.
Hutchison, J. A.
Jauffred, L.
Jiang, Y.
Jones, P. H.
Kawata, S.
Kong, L.
Kong, L. J.
Kravets, N.
Lakhtakia, A.
Lekner, J.
Leng, J.
Li, M.
Li, Y.
Liang, C.
Liang, Y.
Lin, Z.
Liu, S.
Liu, Y.
Lopez, H. A.
Lu, W.
Luo, J.
Madhusudana, N. V.
Man, Z.
Maragò, O. M.
Mason, S. F.
Mazzulla, A.
Min, C.
Moreno, I.
Mosallaeipour, M.
Mueller, J. P. B.
Ng, J.
Oddershede, L. B.
Osipov, M.
Pane, A.
Pelton, M.
Pesic, J.
Richardson, A. C.
Roberts, N. W.
Rodgers, T.
Rui, G.
Salcedo, W. J.
Sand, D.
Sarkar, R.
Scherer, N. F.
Seddon, A.
Sekkat, Z.
Serey, X.
Shi, L.
Shoji, S.
Surbeck, H.
Svoboda, K.
Tan, S.
Tkachenko, G.
Toussaint, K. C.
Tu, H.
Varadan, V. V.
Volke-Sepúlveda, K.
Volpe, G.
Wan, C.
Wang, N.
Wang, S.
Wang, S. B.
Wang, Y.
Woods, C.
Xue, C.
Yan, S.
Yang, Y.
Yao, A. M.
Yao, B.
Yu, X.
Yuan, X.
Zhan, Q.
Zhang, H.
Zhang, P.
Zhao, Y.
Zhou, L.
Zhou, S.
Zhu, S.
Zhu, Z.
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(1) E 0 ( θ , φ ) = ( E θ a θ + E φ a φ ) F N ,
(2) F N = ∑ n = 1 N A n ( e j k d n sin θ sin φ + β n 2 + e − j k d n sin θ sin φ + β n 2 ) ,
(3) E r i ( r , φ ) = 1 cos θ F N ( X r i ⋅ e x + Y r i ⋅ e y ) ,
(4) X r i = e i π 2 sin θ cos φ − cos θ cos 2 φ − sin 2 φ ,
(5) Y r i = e i π 2 sin θ sin φ − cos θ cos φ sin φ + sin φ cos φ ,
(6) E r f ( r p , ϕ , z p ) = C s ∫ 0 θ max ∫ 0 2 π ( X r f ⋅ e x + Y r f ⋅ e y + Z r f ⋅ e z ) × e j k r p sin θ cos ( φ − ϕ ) + j z p cos θ sin θ F N d θ d φ ,
(7) X r f = e i ϕ sin θ cos θ cos φ − cos 2 θ cos 2 φ − sin 2 φ ,
(8) Y r f = e i ϕ sin θ cos θ sin φ + sin 2 θ sin φ cos φ ,
(9) Z r f = e i ϕ sin 2 θ − cos θ sin θ cos φ .
(10) E l i ( r , φ ) = 1 cos θ F N ( X l i ⋅ e x + Y l i ⋅ e y ) ,
(11) X l i = − e i π 2 sin θ cos φ − cos θ cos 2 φ − sin 2 φ ,
(12) Y l i = e i π 2 sin θ sin φ + cos θ cos φ sin φ − sin φ cos φ .
(13) E ( r p , ϕ , z p ) = E r f ( r p , ϕ , z p ) + E l f ( r p , − ϕ , − z p ) = C s ∫ 0 θ max ∫ 0 2 π ( X f ⋅ e x + Y f ⋅ e y + Z f ⋅ e z ) ×
(13) ( e j k r p sin θ cos ( φ − ϕ ) + j z p cos θ + e j k r p sin θ cos ( φ + ϕ ) − j z p cos θ ) sin θ F N d θ d φ ,
(14) X f = e i ϕ sin θ cos θ cos φ − cos 2 θ cos 2 φ − sin 2 φ ,
(15) Y f = e i ϕ sin θ cos θ sin φ + sin 2 θ sin φ cos φ ,
(16) Z f = e i ϕ sin 2 θ − cos θ sin θ cos φ .
(17) P i = ∑ S i 2 ( x 0 , y 0 ) / ∑ S 0 2 ( x 0 , y 0 ) , i = 1 , 2 , 3
(18) a n ( 1 ) = [ A n ( 2 ) V n ( 1 ) + A n ( 1 ) V n ( 2 ) ] Q n , a n ( 2 ) = [ A n ( 1 ) W n ( 2 ) − A n ( 2 ) W n ( 1 ) ] Q n , b n ( 1 ) = [ B n ( 1 ) W n ( 2 ) + B n ( 2 ) W n ( 1 ) ] Q n , b n ( 2 ) = a n ( 2 ) ,
(19) A n ( j ) = Z S D n ( 1 ) ( x j ) − D n ( 1 ) ( x 0 ) , B n ( j ) = D n ( 1 ) ( x j ) − Z S D n ( 1 ) ( x 0 ) , W n ( j ) = Z S D n ( 1 ) ( x j ) − D n ( 3 ) ( x 0 ) , V n ( j ) = D n ( 1 ) ( x j ) − Z S D n ( 3 ) ( x 0 ) , Q n = ψ n ( x 0 ) / ξ n ( x 0 ) V n ( 1 ) W n ( 2 ) + V n ( 2 ) W n ( 1 ) ,
(20) α e e = i 6 π ε 0 ε m k 3 a 1 ( 1 ) , α m m = i 6 π μ 0 k 3 b 1 ( 1 ) , α e m = − 6 π Z 0 k 3 a 1 ( 2 ) ,
(21) ⟨ F ⟩ = 1 2 R e [ ( ∇ E ∗ ) ⋅ p + ( ∇ B ∗ ) ⋅ m − Z 0 k 4 6 π ( p × m ∗ ) ] .
(22) ⟨ F ⟩ = − ∇ ⟨ U ⟩ + n m c ( C e x t + C r e c o i l ) ⟨ S ⟩ + μ 0 ∇ × R e [ α e m ] ⟨ S ⟩ + ∇ × { C p c n m ⟨ L p ⟩ + C m c n m ⟨ L m ⟩ } + { 2 ω 2 μ 0 R e [ α e m ] − k 5 3 π ε 0 2 ε m 2 Im [ α e e α e m ∗ ] } ⟨ L p ⟩ + { 2 ω 2 μ 0 R e [ α e m ] − k 5 μ 0 3 π ε 0 ε m Im [ α m m α e m ∗ ] } ⟨ L m ⟩ + c k 4 μ 0 2 12 π n m Im [ α e e α m m ∗ ] Im [ E × H ∗ ] .
(23) ⟨ Γ ⟩ = [ − 2 μ 0 R e ( α e m ) + μ 0 k 3 3 π ε 0 ε m Im ( α e e α e m ∗ ) + μ 0 2 k 3 3 π Im ( α m m α e m ∗ ) ] ⟨ S ⟩ + [ μ 0 k 3 6 π ε 0 ε m R e ( α e e α e m ∗ ) − μ 0 2 k 3 6 π R e ( α m m α e m ∗ ) ] Im ( E × H ∗ ) + [ 2 ω ε 0 ε m Im ( α e e ) − ω k 3 3 π ε 0 2 ε m 2 α e e α e e ∗ − ω μ 0 k 3 3 π ε 0 ε m α e m α e m ∗ ] ⟨ L p ⟩ + [ 2 ω μ 0 Im ( α m m ) − ω μ 0 2 k 3 3 π α m m α m m ∗ − ω μ 0 k 3 3 π ε 0 ε m α e m α e m ∗ ] ⟨ L m ⟩ .
Optimized parameters of the TONFTS
An (radiation amplitude)
dn (spacing distance)
βn (phase difference)
Dipole-pair 1 1 0.6524λ 0.1273π
Dipole-pair 2 0.685 1.5038λ π
Theoretical and experimental Pi values for the incident light presented in Fig. 3(b) and Fig. 3(f).
Theoretical values 0.6621 0.1916 0.7245
Experimental values 0.6661 0.1920 0.7296 | CommonCrawl |
What problem in pure mathematics required solution techniques from the widest range of math sub-disciplines?
(This is a restatement of a question asked on the Mathematics.SE, where the solutions were a bit disappointing. I'm hoping that professional mathematicians here might have a better solution.)
What are some problems in pure mathematics that require(d) solution techniques from the broadest and most disparate range of sub-disciplines of mathematics? The difficulty or importance or real-world application of the problem is not my concern, but instead the breadth of the range of sub-disciplines needed for its solution. The ideal answer would be a problem that required, for instance, number theory, group theory, set theory, formal logic, homotopy theory, graph theory, combinatorics, geometry, and so forth.
Of course, most sub-branches of mathematics overlap with other sub-branches, so just to be clear, in this case you should consider two sub-branches as separate if they have separate listings (numbers) in the Mathematics Subject Classification at the time of the result. (Later, and possibly in response to such a result, the Subject Classifications might be modified slightly.)
One of the reasons I'm interested in this problem is by analogy to technology. More and more problems in technology require a range of disciplines, e.g., electrical engineering, materials science, perceptual psychology, optics, thermal physics, and so forth. Is this also the case in research mathematics?
I'm not asking for an opinion—this question is fact-based, or at minimum a summary of the quantification of the expert views of research mathematicians, mathematics journal editors, mathematics textbook authors, and so forth. The issue can minimize the reliance on opinion by casting it as an objectively verifiable question (at least in principle):
What research mathematics paper, theorem or result has been classified at the time of the result with the largest number of Mathematics Subject Classification numbers?
Moreover, as pointed out in a comment, the divisions (and hence Subject Classification numbers) are set by experts analyzing the current state of mathematics, especially its foundations.
The ideal answer would point to a particular paper, a result, a theorem, where one can identify objectively the range of sub-branches that were brought to bear on the proof or result (as, for instance, might be documented in the Mathematics Subject Classification or appearance in textbooks from disparate fields). Perhaps one can point to particular mathematicians from disparate sub-fields who collaborated on the result.
11 revs
David G. Stork
$\begingroup$ I think a more genuine, sophisticated, professional version of "number theory" may use/require the greatest range of other bits of mathematics for substantial success. (Part of the point is that an entry-level or elementary notion of "number theory" is typically 200 years out of date, or based on inaccurate if popular premises... seeming to make the subject a special case of elementary abstract algebra and elementary combinatorics... which will not get anyone very much farther than Euler 250 years ago...) Is such a response of interest? $\endgroup$ – paul garrett Jun 10 '17 at 0:58
$\begingroup$ I disagree with the statement that this question has any objective meaning: all possible answers will be based on an opinion. $\endgroup$ – Alexandre Eremenko Jun 10 '17 at 7:28
$\begingroup$ @alexandreeremenko: A perfectly reasonable objective interpretation of this question is "What math paper's citation list has the largest total number of distinct arxiv subject tags?" I don't expect anyone will answer the question this way, but it is rooted in something factual. $\endgroup$ – Paul Siegel Jun 10 '17 at 7:43
$\begingroup$ I disagree with the claim that this question is fact-based in any meaningful way, since quantitative measures would rely on rather arbitrarily chosen divisions between mathematical areas. On the other hand, we could just use it as an opportunity to list our favorite theorems whose proofs involve a large number of unexpected techniques. For this purpose, I am imposing "community wiki" mode. $\endgroup$ – S. Carnahan♦ Jun 10 '17 at 9:01
$\begingroup$ I think this is a good question and should stay open, but it would be much better if it were formulated without reference to "the most msc or arxiv subject tags" (which, taken literally, seems unlikely to be an interesting or useful measure of mathematical breadth). $\endgroup$ – John Pardon Jun 11 '17 at 18:56
The proof of the Ramanujan conjecture by Deligne. It uses:
algebraic geometry
representation theory
commutative algebra
Libli
$\begingroup$ Didn't the Wiles Theorem require all of this too? $\endgroup$ – Deane Yang Jun 10 '17 at 12:27
$\begingroup$ Probably. So another exmple to single out. $\endgroup$ – Libli Jun 10 '17 at 17:31
$\begingroup$ Is there a real world example of the usage of Ramanujan Conjecture in practical applications? Wiki states application as for the construction of "Ramanujan Graphs" but no real world example. $\endgroup$ – I should change my Username Jun 12 '17 at 5:14
$\begingroup$ Ramanujan graphs are special classes of expander graphs (googling will give you some "real world applications" of these). More specifically, you can apply Ramanujan graphs in cryptography, see e.g. here: whitman.edu/Documents/Academics/Mathematics/2014/maricqaj.pdf $\endgroup$ – Lennart Meier Jun 12 '17 at 7:56
$\begingroup$ I want to stress that the application to "real-world" problems or "practical applications" is irrelevant to the question. $\endgroup$ – David G. Stork Jun 12 '17 at 17:21
I'm not qualified to certify optimality, but I've always thought that the Mostow rigidity theorem is a good candidate. The theorem says that every isomorphism between the fundamental groups of two finite volume hyperbolic manifolds of dimension at least 3 is induced by a unique isometry. Mostow's original proof (for the compact case) used:
Geometric group theory
Ergodic theory
A dash of number theory
For generalizations to symmetric spaces you need algebraic geometry and more serious number theory as well.
Paul Siegel
$\begingroup$ Why number theory and representation theory? $\endgroup$ – ThiKu Jun 10 '17 at 20:02
$\begingroup$ @ThiKu It's hard to give a lot of detail without really getting into the proof, but roughly speaking one lifts the fundamental group of a hyperbolic manifold to a lattice in a Lie group (viewing hyperbolic space as a Riemannian symmetric space). The theorem (and especially its generalizations) uses arithmetic properties of this lattice and the representation theory of the Lie group. $\endgroup$ – Paul Siegel Jun 10 '17 at 23:52
$\begingroup$ There is a number of different proofs of Mostow rigidity now. This master thesis: wiki.epfl.ch/grtr/documents/lucker2010.pdf reviews proofs by Thurston (via analysis and ergodic theory), by Gromov (via homological methods and geometry of hyperbolic simplices) and by Besson-Courtois-Gallot (via Riemannian geometry and entropy). $\endgroup$ – Lennart Meier Jun 12 '17 at 7:41
$\begingroup$ The first proof is actually Mostow's even though it appears in Thurston's lecture notes. $\endgroup$ – ThiKu Jun 12 '17 at 17:58
One relatively recent result that comes to mind is the Kadison-Singer problem, which was originally formulated in 1959 as a question in $C^*$-algebra theory, but was successively reduced to more tractable and accessible questions in other fields by several mathematicians (including the MathOverflow member Nik Weaver). It was solved in 2013 by the computer scientists Marcus, Spielman and Srivastava using properties of random polynomials.
S. Carnahan
The Smith conjecture; Morgan and Bass could write in 1984 that "the Smith conjecture stands in the first rank of mathematical problems when measured by the amount and depth of new mathematics required to solve it." See https://en.wikipedia.org/wiki/Smith_conjecture for the statement. You get convinced of the huge variety of techniques used in the proof, just by looking at the table of contents of the book: Morgan, J. W. and Bass, H. (Eds.). The Smith Conjecture (Papers Presented at the Symposium Held at Columbia University, New York, 1979. Orlando, FL: Academic Press, 1984)
Alain Valette
$\begingroup$ I must say that this is the best justified answer so far, including reference to a book that touts the wide range of techniques. I will study the Smith Conjecture now to appreciate the disciplinary breadth. $\endgroup$ – David G. Stork Jun 12 '17 at 17:53
The Banach-Ruciewicz problem: Is the Lebesgue measure the only finitely additive measure on the Lebesgue sets in $S^n$ that is invariant under the rotation action by $O(n+1)$ and has total measure $1$?
The answer was shown to be negative for $n=1$ by Banach. While this is ostensibly a problem in measure theory, the case of $n\geq 4$ was affirmatively solved by Margulis and Sullivan using mainly infinite-dimensional representation theory (property T), but also a bit of number theory and algebraic group theory. The case of $n=2,3$ was affirmatively solved by Drinfeld. His use of representation theory was more hardcore and actually used crucially Deligne's solution of Ramanujan's conjecture and the Jacquet-Langlands correspondence. In particular, all the topics entering in Deligne's solution of the Ramanujan conjecture (which uses in turn the Weil conjectures) also enter into the solution of the Banach-Ruciewicz problem.
It should be said that the techniques are very similar to those to construct expander graphs and Ramanujan graphs. See Lubotzky's book Discrete Groups, Expanding Graphs and Invariant Measures.
Lennart Meier
Kronheimer, P.B.; Mrowka, T.S., Witten's conjecture and Property P, Geom. Topol. 8, 295-310 (2004). ZBL1072.57005.
This paper is rather deep in the field of 3-manifold topology, using most of the major developments of low-dimensional topology from the previous 30 years. The main theorem states that a homotopy 3-sphere cannot occur as non-trivial surgery on a knot. Of course, this also follows now from the geometrization theorem and the knot complement problem. However, at the time of publication the geometrization theorem had not been vetted or published. Tracing back the proofs of theorems that this relies on involves the fields of
Riemannian geometry (e.g. used in instanton homology)
Algebraic geometry (featuring heavily in the proof of the cyclic surgery theorem)
Complex analysis (used in Thurston's proof of geometrization of Haken 3-manifolds, as well as pseudo-holomorphic curves I suppose)
Dynamics (used in Thurston's proof again, e.g. in Sullivan rigidity, a generalization of Mostow rigidity)
Analysis and PDEs (for gauge theory)
Mathematical Physics, in the guise of gauge theory, but specifically the work on Witten's conjecture of the equivalence between Seiberg-Witten and Donaldson invariants. This conjecture was motivated by ideas from string theory, so is not rigorous mathematics.
and of course Topology, with quite a few specialties involved (foliations, symplectic and contact structures, 3- and 4-dimensional manifolds, Kleinian groups, Morse Theory).
I should also comment that there are now shorter proofs of this and related theorems independent of the Poincaré conjecture available that don't use quite as much gauge theory. And one can substitute Perelman's proof of geometrization for Thurston's, which substitutes Riemannian geometry and PDEs for complex analysis and dynamics.
Ian Agol
$\begingroup$ Excellent answer. Thanks. I'll look up the paper and see how much I can understand before I award the reputation points. $\endgroup$ – David G. Stork Jun 13 '17 at 5:13
In a sense this question is ill-posed. Every instance of coordinated use of several topics to prove a single result gives an evidence that these topics are strongly interconnected, so if these topics have been classified as separate, the classification must be revised to take into account these interconnections.
The mathematical subject classification can never be finalized I think. After all, in ancient times there was even no clear distinction between music, physics and mathematics.
მამუკა ჯიბლაძე
$\begingroup$ What, objective, then would you say is the difference between a result such as Gödel's Incompleteness Theorem (which relied nearly entirely on formal logic), and the Ramanujan conjecture by Deligne, noted above? How would you modify or edit the question to capture such differences and avoid it being "ill-posed"? What if the question restricted consideration to a single point in time, admitting that later the fields thought to be rather disparate were in fact closer than originally thought? What about a question forcusing on the greatest consolidation of fields previous thought disparate? $\endgroup$ – David G. Stork Jun 12 '17 at 17:08
$\begingroup$ This should be a comment imo. That's not to say I disagree with you. $\endgroup$ – Wojowu Jun 12 '17 at 17:15
$\begingroup$ @DavidG.Stork I honestly don't know how to formulate such question, although I must say your suggestions are very interesting for me. Seems like most mathematicians switch between two opposite regimes. Me myself I am like this - I can try to avoid other topics for years, when digging for an answer to some particular deep question. And then for years I might try to broaden my understanding, driven by strong feeling that the whole mathematics is one unified body of knowledge that we are challenged to grasp in total. The truth is more likely somewhere in between, as it usually happens... $\endgroup$ – მამუკა ჯიბლაძე Jun 12 '17 at 17:15
$\begingroup$ @Wojowu I agree that definitely it is not an answer. On the other hand, it contains a conceptual viewpoint on the content of the question, so maybe it is not a comment either. Besides, this is cw anyway :D $\endgroup$ – მამუკა ჯიბლაძე Jun 12 '17 at 17:17
$\begingroup$ @DavidG.Stork let me also add that if I created impression of negative attitude, I regret it. The aim was rather to bring in some paradoxical aspects inherent in your question. Something like what happens in quantum mechanics where observing implies altering the state of the observed, which makes the observation obsolete, so that one needs some new non-classical approaches because of that. $\endgroup$ – მამუკა ჯიბლაძე Jun 12 '17 at 17:22
The Beilinson regulator is $\frac{1}{2}$ times the Borel regulator.
The Beilinson regulator is a map from algebraic K-theory to Deligne cohomology and the above equality generalizes Borel's theorem on the algebraic K-theory of number rings, which in turn generalizes the class number formula from algebraic number theory. A complete proof is contained in this book. To get an impression of the range of involved fields one may just look at its Table of Content, from which I copy the names of chapters:
Simplicial and Cosimplicial Objects
H-spaces and Hopf Algebras
The Cohomology of the General Linear Group
Lie Algebra Cohomology and the Weil Algebra
Group Cohomology and the van Est Isomorphism
Small Cosimplicial Algebras
Higher Diagonals and Differential Forms
Borel's regulator
Beilinson's Regulator
ThiKu
The classical statistical mechanics (on lattices) already feels as big as the entire mathematics. It relates to analysis, measure theory, algebra (especially the commutative algebra), combinatorics, ... There is a promise in it that should connect to algebraic geometry and number theory. No wonder that the technique of the classical statistical mechanics has contributed to a breakthrough in the knot theory (geometric topology strongly connected to the general groups which as a rule are not abelian).
A huge problem is the theory of phase transitions in the temperatures which are neither high nor low. The high temperature case is easy, while it took a very long time to basically solve the low temperatures for the ferromagnetic systems (and nearly ferromagnetic); actually, there is still a lot of thong to do there. The problem by definition has an analytical character. In the case of low temperatures, everything got reduced to algebra--you may say that the low temperatures froze analysis into algebra. However the intermediate temperatures present a huge problem which will require analysis (including dynamic systems and ergodicity considerations), algebra, combinatorics, ... Even the non-ferromagnetic systems in low temperatures still present a challenge.
The quantum statistical mechanics is still much richer then the classical. However one infinity versus two infinities... The classical case is already overwhelming.
PS. It's been long years since I was active in this topic (the classical case). I am sure that there were some breakthroughs during that time. But I am equally sure that it is still very far from fully meeting the challenge which I have mentioned above).
Wlod AA
If you are not working in pure math and is concerned about some real world problem which is connected with broad knowledge of mathematics, you might be interested in computational complex geometry, which involves not only numerical algorithms but also advanced math knowledge including complex geometry, geometric analysis, algebraic topology, quasi-conformal mapping and Teichmuller space. I'm not an expert in this field but it's really interesting. Applications of computational complex geometry includes computer graphics, medical diagnosis, and machine learning. For more information you may refer to Xianfeng Gu, webpage http://www3.cs.stonybrook.edu/~gu/.
Yan He
$\begingroup$ This is not an answer to the question. $\endgroup$ – j.c. Sep 12 '17 at 23:28
$\begingroup$ Indeed.... this is not an answer to the question, which concerns "pure mathematics." Of course one can craft all types of complex, compound physics or engineering problems that require many branches of mathematics, but that is not what is at issue, for precisely the reasons I stated. $\endgroup$ – David G. Stork Sep 13 '17 at 0:16
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Lie algebroid
In mathematics, a Lie algebroid is a vector bundle $A\rightarrow M$ together with a Lie bracket on its space of sections $\Gamma (A)$ and a vector bundle morphism $\rho :A\rightarrow TM$, satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.
Lie algebroids play a similar same role in the theory of Lie groupoids that Lie algebras play in the theory of Lie groups: reducing global problems to infinitesimal ones. Indeed, any Lie groupoid gives rise to a Lie algebroid, which is the vertical bundle of the source map restricted at the units. However, unlike Lie algebras, not every Lie algebroid arises from a Lie groupoid.
Lie algebroids were introduced in 1967 by Jean Pradines.[1]
Definition and basic concepts
A Lie algebroid is a triple $(A,[\cdot ,\cdot ],\rho )$ consisting of
• a vector bundle $A$ over a manifold $M$
• a Lie bracket $[\cdot ,\cdot ]$ on its space of sections $\Gamma (A)$
• a morphism of vector bundles $\rho :A\rightarrow TM$, called the anchor, where $TM$ is the tangent bundle of $M$
such that the anchor and the bracket satisfy the following Leibniz rule:
$[X,fY]=\rho (X)f\cdot Y+f[X,Y]$
where $X,Y\in \Gamma (A),f\in C^{\infty }(M)$ and $\rho (X)f$ is the derivative of $f$ along the vector field $\rho (X)$.
One often writes $A\to M$ when the bracket and the anchor are clear from the context; some authors denote Lie algebroids by $A\Rightarrow M$, suggesting a "limit" of a Lie groupoids when the arrows denoting source and target become "infinitesimally close".[2]
First properties
It follows from the definition that
• for every $x\in M$, the kernel ${\mathfrak {g}}_{x}(A)=\ker(\rho _{x})$ is a Lie algebra, called the isotropy Lie algebra at $x$
• the kernel ${\mathfrak {g}}(A)=\ker(\rho )$ is a (not necessarily locally trivial) bundle of Lie algebras, called the isotropy Lie algebra bundle
• the image $\mathrm {Im} (\rho )\subseteq TM$ is a singular distribution which is integrable, i.e. its admits maximal immersed submanifolds ${\mathcal {O}}\subseteq M$, called the orbits, satisfying $\mathrm {Im} (\rho _{x})=T_{x}{\mathcal {O}}$ for every $x\in {\mathcal {O}}$. Equivalently, orbits can be explicitly described as the sets of points which are joined by A-paths, i.e. pairs $(a:I\to A,\gamma :I\to M)$ of paths in $A$ and in $M$ such that $a(t)\in A_{\gamma (t)}$ and $\rho (a(t))=\gamma '(t)$
• the anchor map $\rho $ descends to a map between sections $\rho :\Gamma (A)\rightarrow {\mathfrak {X}}(M)$ :\Gamma (A)\rightarrow {\mathfrak {X}}(M)} which is a Lie algebra morphism, i.e.
$\rho ([X,Y])=[\rho (X),\rho (Y)]$
for all $X,Y\in \Gamma (A)$.
The property that $\rho $ induces a Lie algebra morphism was taken as an axiom in the original definition of Lie algebroid.[1] Such redundancy, despite being known from an algebraic point of view already before Pradine's definition,[3] was noticed only much later.[4][5]
Subalgebroids and ideals
A Lie subalgebroid of a Lie algebroid $(A,[\cdot ,\cdot ],\rho )$ is a vector subbundle $A'\to M'$ of the restriction $A_{\mid M'}\to M'$ such that $\rho _{\mid A'}$ takes values in $TM'$ and $\Gamma (A,A'):=\{\alpha \in \Gamma (A)\mid \alpha _{\mid M'}\in \Gamma (A')\}$ is a Lie subalgebra of $\Gamma (A)$. Clearly, $A'\to M'$ admits a unique Lie algebroid structure such that $\Gamma (A,A')\to \Gamma (A')$ is a Lie algebra morphism. With the language introduced below, the inclusion $A'\hookrightarrow A$ is a Lie algebroid morphism.
A Lie subalgebroid is called wide if $M'=M$. In analogy to the standard definition for Lie algebra, an ideal of a Lie algebroid is wide Lie subalgebroid $I\subseteq A$ such that $\Gamma (I)\subseteq \Gamma (A)$ is a Lie ideal. Such notion proved to be very restrictive, since $I$ is forced to be inside the isotropy bundle $\ker(\rho )$. For this reason, the more flexible notion of infinitesimal ideal system has been introduced.[6]
Morphisms
A Lie algebroid morphism between two Lie algebroids $(A_{1},[\cdot ,\cdot ]_{A_{1}},\rho _{1})$ and $(A_{2},[\cdot ,\cdot ]_{A_{2}},\rho _{2})$ with the same base $M$ is a vector bundle morphism $\phi :A_{1}\to A_{2}$ which is compatible with the Lie brackets, i.e. $\phi ([\alpha ,\beta ]_{A_{1}})=[\phi (\alpha ),\phi (\beta )]_{A_{2}}$ for every $\alpha ,\beta \in \Gamma (A_{1})$, and with the anchors, i.e. $\rho _{2}\circ \phi =\rho _{1}$.
A similar notion can be formulated for morphisms with different bases, but the compatibility with the Lie brackets becomes more involved.[7] Equivalently, one can ask that the graph of $\phi :A_{1}\to A_{2}$ to be a subalgebroid of the direct product $A_{1}\times A_{2}$ (introduced below).[8]
Lie algebroids together with their morphisms form a category.
Examples
Trivial and extreme cases
• Given any manifold $M$, its tangent Lie algebroid is the tangent bundle $TM\to M$ together with the Lie bracket of vector fields and the identity of $TM$ as an anchor.
• Given any manifold $M$, the zero vector bundle $M\times 0\to M$ is a Lie algebroid with zero bracket and anchor.
• Lie algebroids $A\to \{*\}$ over a point are the same thing as Lie algebras.
• More generally, any bundles of Lie algebras is Lie algebroid with zero anchor and Lie bracket defined pointwise.
Examples from differential geometry
• Given a foliation ${\mathcal {F}}$ on $M$, its foliation algebroid is the associated involutive subbundle ${\mathcal {F}}\subseteq TM$, with brackets and anchor induced from the tangent Lie algebroid.
• Given the action of a Lie algebra ${\mathfrak {g}}$ on a manifold $M$, its action algebroid is the trivial vector bundle ${\mathfrak {g}}\times M\to M$, with anchor given by the Lie algebra action and brackets uniquely determined by the bracket of ${\mathfrak {g}}$ on constant sections $M\to {\mathfrak {g}}$ and by the Leibniz identity.
• Given a principal G-bundle $P$ over a manifold $M$, its Atiyah algebroid is the Lie algebroid $A=TP/G$ fitting in the following short exact sequence:
$0\to \ker(\rho )\to TP/G\xrightarrow {\rho } TM\to 0.$
The space of sections of the Atiyah algebroid is the Lie algebra of $G$-invariant vector fields on $P$, its isotropy Lie algebra bundle is isomorphic to the adjoint vector bundle $P\times _{G}{\mathfrak {g}}$, and the right splittings of the sequence above are principal connections on $P$.
• Given a vector bundle $E\to M$, its general linear algebroid, denoted by ${\mathfrak {gl}}(E)$ or $\mathrm {Der} (E)$, is the vector bundle whose sections are derivations of $E$, i.e. first-order differential operators $\Gamma (E)\to \Gamma (E)$ admitting a vector field $\rho (D)\in {\mathfrak {X}}(M)$ such that $D(f\sigma )=fD(\sigma )+\rho (D)(f)\sigma $ for every $f\in {\mathcal {C}}^{\infty }(M),\sigma \in \Gamma (E)$. The anchor is simply the assignment $D\mapsto \rho (D)$ and the Lie bracket is given by the commutator of differential operators.
• Given a Poisson manifold $(M,\pi )$, its cotangent algebroid is the cotangent vector bundle $A=T^{*}M$, with Lie bracket $[\alpha ,\beta ]:={\mathcal {L}}_{\pi ^{\sharp }(\alpha )}(\beta )-{\mathcal {L}}_{\pi ^{\sharp }(\beta )}(\alpha )-d\pi (\alpha ,\beta )$ and anchor map $\pi ^{\sharp }:T^{*}M\to TM,\alpha \mapsto \pi (\alpha ,\cdot )$.
• Given a closed 2-form $\omega \in \Omega ^{2}(M)$, the vector bundle $A_{\omega }:=TM\times \mathbb {R} \to M$ is a Lie algebroid with anchor the projection on the first component and Lie bracket
$[(X,f),(Y,g)]:={\Big (}[X,Y],{\mathcal {L}}_{X}(g)-{\mathcal {L}}_{Y}(f)-\omega (X,Y){\Big )}.$
Actually, the bracket above can be defined for any 2-form $\omega $, but $A_{\omega }$ is a Lie algebroid if and only if $\omega $ is closed.
Constructions from other Lie algebroids
• Given any Lie algebroid $(A\to M,[\cdot ,\cdot ],\rho )$, there is a Lie algebroid $(TA\to TM,[\cdot ,\cdot ],\rho )$, called its tangent algebroid, obtained by considering the tangent bundle of $A$ and $M$ and the differential of the anchor.
• Given any Lie algebroid $(A\to M,[\cdot ,\cdot ]_{A},\rho _{A})$, there is a Lie algebroid $(J^{k}A\to M,[\cdot ,\cdot ],\rho )$, called its k-jet algebroid, obtained by considering the k-jet bundle of $A\to M$, with Lie bracket uniquely defined by $[j^{k}\alpha ,j^{k}\beta ]:=j^{k}[\alpha ,\beta ]_{A}$ and anchor $\rho (j_{x}^{k}\alpha ):=\rho _{A}(\alpha (x))$.
• Given two Lie algebroids $A_{1}\to M_{1}$ and $A_{2}\to M_{2}$, their direct product is the unique Lie algebroid $A_{1}\times A_{2}\to M_{1}\times M_{2}$ with anchor $(\alpha _{1},\alpha _{2})\mapsto \rho _{1}(\alpha _{1})\oplus \rho _{2}(\alpha _{2})\in TM_{1}\oplus TM_{2}\cong T(M_{1}\times M_{2}),$ and such that $\Gamma (A_{1})\oplus \Gamma (A_{2})\to \Gamma (A_{1}\times A_{2}),\alpha _{1}\oplus \alpha _{2}\mapsto \mathrm {pr} _{M_{1}}^{*}\alpha _{1}+\mathrm {pr} _{M_{2}}^{*}\alpha _{2}$ is a Lie algebra morphism.
• Given a Lie algebroid $(A\to M,[\cdot ,\cdot ]_{A},\rho _{A})$ and a map $f:M'\to M$ whose differential is transverse to the anchor map $\rho :A\to TM$ (for instance, it is enough for $f$ to be a surjective submersion), the pullback algebroid is the unique Lie algebroid $f^{!}A\to M'$, with $f^{!}A:=TM'\times _{TM}A\to M'$ the pullback vector bundle, and $\rho _{f^{!}A}:f^{!}A\to TM'$ the projection on the first component, such that $f^{!}A\to A$ is a Lie algebroid morphism.
Important classes of Lie algebroids
Totally intransitive Lie algebroids
A Lie algebroid is called totally intransitive if the anchor map $\rho :A\to TM$ is zero.
Bundle of Lie algebras (hence also Lie algebras) are totally intransitive. This actually exhaust completely the list of totally intransitive Lie algebroids: indeed, if $A$ is totally intransitive, it must coincide with its isotropy Lie algebra bundle.
Transitive Lie algebroids
A Lie algebroid is called transitive if the anchor map $\rho :A\to TM$ is surjective. As a consequence:
• there is a short exact sequence
$0\to \ker(\rho )\to A\xrightarrow {\rho } TM\to 0;$
• right-splitting of $\rho $ defines a principal bundle connections on $\ker(\rho )$;
• the isotropy bundle $\ker(\rho )$ is locally trivial (as bundle of Lie algebras);
• the pullback of $A$ exist for every $f:M'\to M$.
The prototypical examples of transitive Lie algebroids are Atiyah algebroids. For instance:
• tangent algebroids $TM$ are trivially transitive (indeed, they are Atiyah algebroid of the principal $\{e\}$-bundle $M\to M$)
• Lie algebras ${\mathfrak {g}}$ are trivially transitive (indeed, they are Atiyah algebroid of the principal $G$-bundle $G\to *$, for $G$ an integration of ${\mathfrak {g}}$)
• general linear algebroids ${\mathfrak {gl}}(E)$ are transitive (indeed, they are Atiyah algebroids of the frame bundle $Fr(E)\to M$)
In analogy to Atiyah algebroids, an arbitrary transitive Lie algebroid is also called abstract Atiyah sequence, and its isotropy algebra bundle $\ker(\rho )$ is also called adjoint bundle. However, it is important to stress that not every transitive Lie algebroid is an Atiyah algebroid. For instance:
• pullbacks of transitive algebroids are transitive
• cotangent algebroids $T^{*}M$ associated to Poisson manifolds $(M,\pi )$ are transitive if and only if the Poisson structure $\pi $ is non-degenerate
• Lie algebroids $A_{\omega }$ defined by closed 2-forms are transitive
These examples are very relevant in the theory of integration of Lie algebroid (see below): while any Atiyah algebroid is integrable (to a gauge groupoid), not every transitive Lie algebroid is integrable.
Regular Lie algebroids
A Lie algebroid is called regular if the anchor map $\rho :A\to TM$ is of constant rank. As a consequence
• the image of $\rho $ defines a regular foliation on $M$;
• the restriction of $A$ over each leaf ${\mathcal {O}}\subseteq M$ is a transitive Lie algebroid.
For instance:
• any transitive Lie algebroid is regular (the anchor has maximal rank);
• any totally intransitive Lie algebroids is transitive (the anchor has zero rank);
• foliation algebroids are always regular;
• cotangent algebroids $T^{*}M$ associated to Poisson manifolds $(M,\pi )$ are regular if and only if the Poisson structure $\pi $ is regular.
Further related concepts
Actions
An action of a Lie algebroid $A\to M$ on a manifold P along a smooth map $\mu :P\to M$ consists of a Lie algebra morphism
$a:\Gamma (A)\to {\mathfrak {X}}(P)$
such that, for every $p\in P,X\in \Gamma (A),f\in {\mathcal {C}}^{\infty }(M)$,
$d_{p}\mu (a(X_{p}))=\rho _{\mu (p)}(X_{\mu (p)}),\quad a(f\cdot X)=(f\circ \mu )\cdot a(X).$
Of course, when $A={\mathfrak {g}}$, both the anchor $A\to \{*\}$ and the map $P\to \{*\}$ must be trivial, therefore both conditions are empty, and we recover the standard notion of action of a Lie algebra on a manifold.
Connections
Given a Lie algebroid $A\to M$, an A-connection on a vector bundle $E\to M$ consists of an $\mathbb {R} $-bilinear map
$\nabla :\Gamma (A)\times \Gamma (E)\to \Gamma (E),\quad (\alpha ,s)\mapsto \nabla _{\alpha }(s)$ :\Gamma (A)\times \Gamma (E)\to \Gamma (E),\quad (\alpha ,s)\mapsto \nabla _{\alpha }(s)}
which is ${\mathcal {C}}^{\infty }(M)$-linear in the first factor and satisfies the following Leibniz rule:
$\nabla _{\alpha }(fs)=f\nabla _{\alpha }(s)+{\mathcal {L}}_{\rho (\alpha )}(f)s$
for every $\alpha \in \Gamma (A),s\in \Gamma (E),f\in {\mathcal {C}}^{\infty }(M)$, where ${\mathcal {L}}_{\rho (\alpha )}$ denotes the Lie derivative with respect to the vector field $\rho (\alpha )$. The curvature of an A-connection $\nabla $ is the ${\mathcal {C}}^{\infty }(M)$-bilinear map
$R_{\nabla }:\Gamma (A)\times \Gamma (A)\to \mathrm {Hom} (E,E),\quad (\alpha ,\beta )\mapsto \nabla _{\alpha }\nabla _{\beta }-\nabla _{\beta }\nabla _{\alpha }-\nabla _{[\alpha ,\beta ]},$
and $\nabla $ is called flat if $R_{\nabla }=0$.
Of course, when $A=TM$, we recover the standard notion of connection on a vector bundle, as well as those of curvature and flatness.
Representations
A representation of a Lie algebroid $A\to M$ is a vector bundle $E\to M$ together with a flat A-connection $\nabla $. Equivalently, a representation $(E,\nabla )$ is a Lie algebroid morphism $A\to {\mathfrak {gl}}(E)$.
The set $\mathrm {Rep} (A)$ of isomorphism classes of representations of a Lie algebroid $A\to M$ has a natural structure of semiring, with direct sums and tensor products of vector bundles.
Examples include the following:
• When $A={\mathfrak {g}}$, an $A$-connection simplifies to a linear map ${\mathfrak {g}}\to {\mathfrak {gl}}(V)$ and the flatness condition makes it into a Lie algebra morphism, therefore we recover the standard notion of representation of a Lie algebra.
• When $A={\mathfrak {g}}\times M\to M$ and $V$ is a representation the Lie algebra ${\mathfrak {g}}$, the trivial vector bundle $V\times M\to M$ is automatically a representation of $A$
• Representations of the tangent algebroid $A=TM$ are vector bundles endowed with flat connections
• Every Lie algebroid $A\to M$ has a natural representation on the line bundle $Q_{A}:=\wedge ^{top}A\otimes \wedge ^{top}T^{*}M\to M$, i.e. the tensor product between the determinant line bundles of $A$ and of $T^{*}M$. One can associate a cohomology class in $H^{1}(A,Q_{A})$ (see below) known as the modular class of the Lie algebroid.[9] For the cotangent algebroid $T^{*}M\to M$ associated to a Poisson manifold $(M,\pi )$ one recovers the modular class of $\pi $.[10]
Note that there an arbitrary Lie groupoid does not have a canonical representation on its Lie algebroid, playing the role of the adjoint representation of Lie groups on their Lie algebras. However, this becomes possible if one allows the more general notion of representation up to homotopy.
Lie algebroid cohomology
Consider a Lie algebroid $A\to M$ and a representation $(E,\nabla )$. Denoting by $\Omega ^{n}(A,E):=\Gamma (\wedge ^{n}A^{*}\otimes E)$ the space of $n$-differential forms on $A$ with values in the vector bundle $E$, one can define a differential $d^{n}:\Omega ^{n}(A,E)\to \Omega ^{n+1}(A,E)$ with the following Koszul-like formula:
$d(\omega (\alpha _{0},\ldots ,\alpha _{n}):=\sum _{i=1}^{n}(-1)^{i}\nabla _{\alpha _{i}}{\big (}\omega (\alpha _{0},\ldots ,{\widehat {\alpha _{i}}},\ldots ,\alpha _{n}){\big )}-\sum _{i<j}^{n}(-1)^{i+j+1}\omega ([\alpha _{i},\alpha _{j}],\alpha _{0},\ldots ,{\widehat {\alpha _{i}}},\ldots ,{\widehat {\alpha _{j}}},\ldots ,\alpha _{n})$
Thanks to the flatness of $\nabla $, $(\Omega ^{n}(A,E),d^{n})$ becomes a cochain complex and its cohomology, denoted by $H^{*}(A,E)$, is called the Lie algebroid cohomology of $A$ with coefficients in the representation $(E,\nabla )$.
This general definition recovers well-known cohomology theories:
• The cohomology of a Lie algebroid ${\mathfrak {g}}\to \{*\}$ coincides with the Chevalley-Eilenberg cohomology of ${\mathfrak {g}}$ as a Lie algebra.
• The cohomology of a tangent Lie algebroid $TM\to M$ coincides with the de Rham cohomology of $M$.
• The cohomology of a foliation Lie algebroid ${\mathcal {F}}\to M$ coincides with the leafwise cohomology of the foliation ${\mathcal {F}}$.
• The cohomology of the cotangent Lie algebroid $T^{*}M$ associated to a Poisson structure $\pi $ coincides with the Poisson cohomology of $\pi $.
Lie groupoid-Lie algebroid correspondence
The standard construction which associates a Lie algebra to a Lie group generalises to this setting: to every Lie groupoid $G\rightrightarrows M$ one can canonically associate a Lie algebroid $\mathrm {Lie} (G)$ defined as follows:
• the vector bundle is $\mathrm {Lie} (G)=A:=u^{*}T^{s}G$, where $T^{s}G\subseteq TG$ is the vertical bundle of the source fibre $s:G\to M$ and $u:M\to G$ is the groupoid unit map;
• the sections of $A$ are identified with the right-invariant vector fields on $G$, so that $\Gamma (A)$ inherits a Lie bracket;
• the anchor map is the differential $\rho :=dt_{\mid A}:A\to TM$ :=dt_{\mid A}:A\to TM} of the target map $t:G\to M$.
Of course, a symmetric construction arises when swapping the role of the source and the target maps, and replacing right- with left-invariant vector fields; an isomorphism between the two resulting Lie algebroids will be given by the differential of the inverse map $i:G\to G$.
The flow of a section $\alpha \in \Gamma (A)$ is the 1-parameter bisection $\phi _{\alpha }^{\epsilon }\in \mathrm {Bis} (G)$, defined by $\phi _{\alpha }^{\epsilon }(x):=\phi _{\tilde {\alpha }}^{\epsilon }(1_{x})$, where $\phi _{\tilde {\alpha }}^{\epsilon }\in \mathrm {Diff} (G)$ is the flow of the corresponding right-invariant vector field ${\tilde {\alpha }}\in {\mathfrak {X}}(G)$. This allows one to defined the analogue of the exponential map for Lie groups as $\exp :\Gamma (A)\to \mathrm {Bis} (G),\exp(\alpha )(x):=\phi _{\alpha }^{1}(x)$ :\Gamma (A)\to \mathrm {Bis} (G),\exp(\alpha )(x):=\phi _{\alpha }^{1}(x)} .
Lie functor
The mapping $G\mapsto \mathrm {Lie} (G)$ sending a Lie groupoid to a Lie algebroid is actually part of a categorical construction. Indeed, any Lie groupoid morphism $\phi :G_{1}\to G_{2}$ can be differentiated to a morphism $d\phi _{\mid \mathrm {Lie} (G_{1})}:\mathrm {Lie} (G_{1})\to \mathrm {Lie} (G_{2})$ between the associated Lie algebroids.
This construction defines a functor from the category of Lie groupoids and their morphisms to the category of Lie algebroids and their morphisms, called the Lie functor.
Structures and properties induced from groupoids to algebroids
Let $G\rightrightarrows M$ be a Lie groupoid and $(A\to M,[\cdot ,\cdot ],\rho )$ its associated Lie algebroid. Then
• The isotropy algebras ${\mathfrak {g}}_{x}(A)$ are the Lie algebras of the isotropy groups $G_{x}$
• The orbits of $G$ coincides with the orbits of $A$
• $G$ is transitive and $(s,t):G\to M\times M$ is a submersion if and only if $A$ is transitive
• an action $m:G\times _{M}P\to P$ of $G$ on $P\to M$ induces an action $a:\Gamma (A)\to {\mathfrak {X}}(P)$ of $A$ (called infinitesimal action), defined by
$a(\alpha )_{p}:=d_{1_{\mu (p)}}m(\cdot ,p)(\alpha _{\mu (p)})=d_{(1_{\mu (p)},p)}m(\alpha _{\mu (p)},0)$
• a representation of $G$ on a vector bundle $E\to M$ induces a representation $\nabla $ of $A$ on $E\to M$, defined by
$\nabla _{\alpha }\sigma (x):={\frac {d}{d\epsilon }}_{\mid \epsilon =0}{\Big (}\phi _{\alpha }^{\epsilon }(x){\Big )}^{-1}\cdot \sigma {\Big (}t(\phi _{\alpha }^{\epsilon }(x)){\Big )}$
Moreover, there is a morphism of semirings $\mathrm {Rep} (G)\to \mathrm {Rep} (A)$, which becomes an isomorphism if $G$ is source-simply connected.
• there is a morphism $VE^{k}:H_{d}^{k}(G,E)\to H^{k}(A,E)$, called Van Est morphism, from the differentiable cohomology of $G$ with coefficients in some representation on $E$ to the cohomology of $A$ with coefficients in the induced representation on $E$. Moreover, if the $s$-fibres of $G$ are homologically $n$-connected, then $VE^{k}$ is an isomorphism for $k\leq n$, and is injective for $k=n+1$.[11]
Examples
• The Lie algebroid of a Lie group $G\rightrightarrows \{*\}$ is the Lie algebra ${\mathfrak {g}}\to \{*\}$
• The Lie algebroid of both the pair groupoid $M\times M\rightrightarrows M$ and the fundamental groupoid $\Pi _{1}(M)\rightrightarrows M$ is the tangent algebroid $TM\to M$
• The Lie algebroid of the unit groupoid $u(M)\rightrightarrows M$ is the zero algebroid $M\times 0\to M$
• The Lie algebroid of a Lie group bundle $G\rightrightarrows M$ is the Lie algebra bundle $A\to M$
• The Lie algebroid of an action groupoid $G\times M\rightrightarrows M$ is the action algebroid ${\mathfrak {g}}\times M\to M$
• The Lie algebroid of a gauge groupoid $(P\times P)/G\rightrightarrows M$ is the Atiyah algebroid $TP/G\to M$
• The Lie algebroid of a general linear groupoid $GL(E)\rightrightarrows M$ is the general linear algebroid ${\mathfrak {gl}}(E)\to M$
• The Lie algebroid of both the holonomy groupoid $\mathrm {Hol} ({\mathcal {F}})\rightrightarrows M$ and the monodromy groupoid $\Pi _{1}({\mathcal {F}})\rightrightarrows M$ is the foliation algebroid ${\mathcal {F}}\to M$
• The Lie algebroid of a tangent groupoid $TG\rightrightarrows TM$ is the tangent algebroid $TA\to TM$, for $A=\mathrm {Lie} (G)$
• The Lie algebroid of a jet groupoid $J^{k}G\rightrightarrows M$ is the jet algebroid $J^{k}A\to M$, for $A=\mathrm {Lie} (G)$
Detailed example 1
Let us describe the Lie algebroid associated to the pair groupoid $G:=M\times M$. Since the source map is $s:G\to M:(p,q)\mapsto q$, the $s$-fibers are of the kind $M\times \{q\}$, so that the vertical space is $T^{s}G=\bigcup _{q\in M}TM\times \{q\}\subset TM\times TM$. Using the unit map $u:M\to G:q\mapsto (q,q)$, one obtain the vector bundle $A:=u^{*}T^{s}G=\bigcup _{q\in M}T_{q}M=TM$.
The extension of sections $X\in \Gamma (A)$ to right-invariant vector fields ${\tilde {X}}\in {\mathfrak {X}}(G)$ is simply ${\tilde {X}}(p,q)=X(p)\oplus 0$ and the extension of a smooth function $f$ from $M$ to a right-invariant function on $G$ is ${\tilde {f}}(p,q)=f(q)$. Therefore, the bracket on $A$ is just the Lie bracket of tangent vector fields and the anchor map is just the identity.
Detailed example 2
Consider the (action) Lie groupoid
$\mathbb {R} ^{2}\times U(1)\rightrightarrows \mathbb {R} ^{2}$
where the target map (i.e. the right action of $U(1)$ on $\mathbb {R} ^{2}$) is
$((x,y),e^{i\theta })\mapsto {\begin{bmatrix}\cos(\theta )&-\sin(\theta )\\\sin(\theta )&\cos(\theta )\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}.$
The $s$-fibre over a point $p=(x,y)$ are all copies of $U(1)$, so that $u^{*}(T^{s}(\mathbb {R} ^{2}\times U(1)))$ is the trivial vector bundle $\mathbb {R} ^{2}\times U(1)\to \mathbb {R} ^{2}$.
Since its anchor map $\rho :\mathbb {R} ^{2}\times U(1)\to T\mathbb {R} ^{2}$ :\mathbb {R} ^{2}\times U(1)\to T\mathbb {R} ^{2}} is given by the differential of the target map, there are two cases for the isotropy Lie algebras, corresponding to the fibers of $T^{t}(\mathbb {R} ^{2}\times U(1))$:
${\begin{aligned}t^{-1}(0)\cong &U(1)\\t^{-1}(p)\cong &\{(a,u)\in \mathbb {R} ^{2}\times U(1):ua=p\}\end{aligned}}$
This demonstrates that the isotropy over the origin is $U(1)$, while everywhere else is zero.
Integration of a Lie algebroid
Lie theorems
A Lie algebroid is called integrable if it is isomorphic to $\mathrm {Lie} (G)$ for some Lie groupoid $G\rightrightarrows M$. The analogue of the classical Lie I theorem states that:[12]
if $A$ is an integrable Lie algebroid, then there exists a unique (up to isomorphism) $s$-simply connected Lie groupoid $G$ integrating $A$.
Similarly, a morphism $F:A_{1}\to A_{2}$ between integrable Lie algebroids is called integrable if it is the differential $F=d\phi _{\mid A}$ for some morphism $\phi :G_{1}\to G_{2}$ between two integrations of $A_{1}$ and $A_{2}$. The analogue of the classical Lie II theorem states that:[13]
if $F:\mathrm {Lie} (G_{1})\to \mathrm {Lie} (G_{2})$ is a morphism of integrable Lie algebroids, and $G_{1}$ is $s$-simply connected, then there exists a unique morphism of Lie groupoids $\phi :G_{1}\to G_{2}$ integrating $F$.
In particular, by choosing as $G_{2}$ the general linear groupoid $GL(E)$ of a vector bundle $E$, it follows that any representation of an integrable Lie algebroid integrates to a representation of its $s$-simply connected integrating Lie groupoid.
On the other hand, there is no analogue of the classical Lie III theorem, i.e. going back from any Lie algebroid to a Lie groupoid is not always possible. Pradines claimed that such a statement hold,[14] and the first explicit example of non-integrable Lie algebroids, coming for instance from foliation theory, appeared only several years later.[15] Despite several partial results, including a complete solution in the transitive case,[16] the general obstructions for an arbitrary Lie algebroid to be integrable have been discovered only in 2003 by Crainic and Fernandes.[17] Adopting a more general approach, one can see that every Lie algebroid integrates to a stacky Lie groupoid.[18][19]
Weinstein groupoid
Given any Lie algebroid $A$, the natural candidate for an integration is given by the Weinstein groupoid $G(A):=P(A)/\sim $, where $P(A)$ denotes the space of $A$-paths and $\sim $ the relation of $A$-homotopy between them. Indeed, one can show that $G(A)$ is an $s$-simply connected topological groupoid, with the multiplication induced by the concatenation of paths. Moreover, if $A$ is integrable, $G(A)$ admits a smooth structure such that it coincides with the unique $s$-simply connected Lie groupoid integrating $A$.
Accordingly, the only obstruction to integrability lies in the smoothness of $G(A)$. This approach led to the introduction of objects called monodromy groups, associated to any Lie algebroid, and to the following fundamental result:[17]
A Lie algebroid is integrable if and only if its monodromy groups are uniformly discrete.
Such statement simplifies in the transitive case:
A transitive Lie algebroid is integrable if and only if its monodromy groups are discrete.
The results above show also that every Lie algebroid admits an integration to a local Lie groupoid (roughly speaking, a Lie groupoid where the multiplication is defined only in a neighbourhood around the identity elements).
Integrable examples
• Lie algebras are always integrable (by Lie III theorem)
• Atiyah algebroids of a principal bundle are always integrable (to the gauge groupoid of that principal bundle)
• Lie algebroids with injective anchor (hence foliation algebroids) are alway integrable (by Frobenius theorem)
• Lie algebra bundle are always integrable[20]
• Action Lie algebroids are always integrable (but the integration is not necessarily an action Lie groupoid)[21]
• Any Lie subalgebroid of an integrable Lie algebroid is integrable.[12]
A non-integrable example
Consider the Lie algebroid $A_{\omega }=TM\times \mathbb {R} \to M$ associated to a closed 2-form $\omega \in \Omega ^{2}(M)$ and the group of spherical periods associated to $\omega $, i.e. the image $\Lambda :=\mathrm {Im} (\Phi )\subseteq \mathbb {R} $ :=\mathrm {Im} (\Phi )\subseteq \mathbb {R} } of the following group homomorphism from the second homotopy group of $M$
$\Phi :\pi _{2}(M)\to \mathbb {R} :\quad [f]\mapsto \int _{S^{2}}f^{*}\omega .$ :\pi _{2}(M)\to \mathbb {R} :\quad [f]\mapsto \int _{S^{2}}f^{*}\omega .}
Since $A_{\omega }$ is transitive, it is integrable if and only if it is the Atyah algebroid of some principal bundle; a careful analysis shows that this happens if and only if the subgroup $\Lambda \subseteq \mathbb {R} $ is a lattice, i.e. it is discrete. An explicit example where such condition fails is given by taking $M=S^{2}\times S^{2}$ and $\omega =\mathrm {pr} _{1}^{*}\sigma +{\sqrt {2}}\mathrm {pr} _{2}^{*}\sigma \in \Omega ^{2}(M)$ for $\sigma \in \Omega ^{2}(S^{2})$ the area form. Here $\Lambda $ turns out to be $\mathbb {Z} +{\sqrt {2}}\mathbb {Z} $, which is dense in $\mathbb {R} $.
See also
• R-algebroid
• Lie bialgebroid
References
1. Pradines, Jean (1967). "Théorie de Lie pour les groupoïdes dif́férentiables. Calcul différentiel dans la caté́gorie des groupoïdes infinitésimaux". C. R. Acad. Sci. Paris (in French). 264: 245–248.
2. Meinrenken, Eckhard (2021-05-08). "On the integration of transitive Lie algebroids". arXiv:2007.07120 [math.DG].
3. J. C., Herz (1953). "Pseudo-algèbres de Lie". C. R. Acad. Sci. Paris (in French). 236: 1935–1937.
4. Kosmann-Schwarzbach, Yvette; Magri, Franco (1990). "Poisson-Nijenhuis structures". Annales de l'Institut Henri Poincaré A. 53 (1): 35–81.
5. Grabowski, Janusz (2003-12-01). "Quasi-derivations and QD-algebroids". Reports on Mathematical Physics. 52 (3): 445–451. arXiv:math/0301234. Bibcode:2003RpMP...52..445G. doi:10.1016/S0034-4877(03)80041-1. ISSN 0034-4877. S2CID 119580956.
6. Jotz Lean, M.; Ortiz, C. (2014-10-01). "Foliated groupoids and infinitesimal ideal systems". Indagationes Mathematicae. 25 (5): 1019–1053. doi:10.1016/j.indag.2014.07.009. ISSN 0019-3577. S2CID 121209093.
7. Mackenzie, Kirill C. H. (2005). General Theory of Lie Groupoids and Lie Algebroids. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press. doi:10.1017/cbo9781107325883. ISBN 978-0-521-49928-6.
8. Eckhard Meinrenken, Lie groupoids and Lie algebroids, Lecture notes, fall 2017
9. Evens, S; Lu, J-H; Weinstein, A (1999-12-01). "Transverse measures, the modular class and a cohomology pairing for Lie algebroids". The Quarterly Journal of Mathematics. 50 (200): 417–436. doi:10.1093/qjmath/50.200.417. ISSN 0033-5606.
10. Weinstein, Alan (1997). "The modular automorphism group of a Poisson manifold". Journal of Geometry and Physics. 23 (3–4): 379–394. Bibcode:1997JGP....23..379W. doi:10.1016/S0393-0440(97)80011-3.
11. Crainic, Marius (2003-12-31). "Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes". Commentarii Mathematici Helvetici. 78 (4): 681–721. arXiv:math/0008064. doi:10.1007/s00014-001-0766-9. ISSN 0010-2571. S2CID 6392715.
12. Moerdijk, Ieke; Mrcun, Janez (2002). "On integrability of infinitesimal actions" (PDF). American Journal of Mathematics. 124 (3): 567–593. arXiv:math/0006042. doi:10.1353/ajm.2002.0019. ISSN 1080-6377. S2CID 53622428.
13. Mackenzie, Kirill; Xu, Ping (2000-05-01). "Integration of Lie bialgebroids". Topology. 39 (3): 445–467. arXiv:dg-ga/9712012. doi:10.1016/S0040-9383(98)00069-X. ISSN 0040-9383. S2CID 119594174.
14. Pradines, Jean (1968). "Troisieme théorème de Lie pour les groupoides différentiables". Comptes Rendus de l'Académie des Sciences, Série A (in French). 267: 21–23.
15. Almeida, Rui; Molino, Pierre (1985). "Suites d'Atiyah et feuilletages transversalement complets". Comptes Rendus de l'Académie des Sciences, Série I (in French). 300: 13–15.
16. Mackenzie, K. (1987). Lie Groupoids and Lie Algebroids in Differential Geometry. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press. doi:10.1017/cbo9780511661839. ISBN 978-0-521-34882-9.
17. Crainic, Marius; Fernandes, Rui L. (2003). "Integrability of Lie brackets". Ann. of Math. 2. 157 (2): 575–620. arXiv:math/0105033. doi:10.4007/annals.2003.157.575. S2CID 6992408.
18. Hsian-Hua Tseng; Chenchang Zhu (2006). "Integrating Lie algebroids via stacks". Compositio Mathematica. 142 (1): 251–270. arXiv:math/0405003. doi:10.1112/S0010437X05001752. S2CID 119572919.
19. Chenchang Zhu (2006). "Lie II theorem for Lie algebroids via stacky Lie groupoids". arXiv:math/0701024.
20. Douady, Adrien; Lazard, Michel (1966-06-01). "Espaces fibrés en algèbres de Lie et en groupes". Inventiones Mathematicae (in French). 1 (2): 133–151. Bibcode:1966InMat...1..133D. doi:10.1007/BF01389725. ISSN 1432-1297. S2CID 121480154.
21. Dazord, Pierre (1997-01-01). "Groupoïde d'holonomie et géométrie globale". Comptes Rendus de l'Académie des Sciences, Série I. 324 (1): 77–80. doi:10.1016/S0764-4442(97)80107-3. ISSN 0764-4442.
Books and lecture notes
• Alan Weinstein, Groupoids: unifying internal and external symmetry, AMS Notices, 43 (1996), 744-752. Also available at arXiv:math/9602220.
• Kirill Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge U. Press, 1987.
• Kirill Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge U. Press, 2005.
• Marius Crainic, Rui Loja Fernandes, Lectures on Integrability of Lie Brackets, Geometry&Topology Monographs 17 (2011) 1–107, available at arXiv:math/0611259.
• Eckhard Meinrenken, Lecture notes on Lie groupoids and Lie algebroids, available at http://www.math.toronto.edu/mein/teaching/MAT1341_LieGroupoids/Groupoids.pdf.
• Ieke Moerdijk, Janez Mrčun, Introduction to Foliations and Lie Groupoids, Cambridge U. Press, 2010.
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| Wikipedia |
How many positive multiples of nine are two-digit numbers?
If we divide 99 (the largest two digit number) by 9, we get 11. So, there are 11 positive multiples of 9 that are less than or equal to 99. However, we must eliminate any that are not two digit numbers. The first multiple of 9 is $9\cdot1=9$ and the second is $9\cdot2=18$ . So, only one positive multiple of nine is not at least a two digit number and there are $11-1=\boxed{10}$ two-digit multiples of 9. | Math Dataset |
\begin{document}
\sloppy
\title{On the center of a Coxeter group} \author{Tetsuya Hosaka} \address{Department of Mathematics, Utsunomiya University, Utsunomiya, 321-8505, Japan} \date{September 9, 2005} \email{[email protected]} \keywords{the center of a Coxeter group, a splitting theorem for a Coxeter group} \subjclass[2000]{20F55} \thanks{ Partly supported by the Grant-in-Aid for Young Scientists (B), The Ministry of Education, Culture, Sports, Science and Technology, Japan. (No.~15740029).} \maketitle
\begin{abstract} In this paper, we show that the center of every Coxeter group is finite and isomorphic to $({\Bbb Z}_2)^n$ for some $n\ge 0$. Moreover, for a Coxeter system $(W,S)$, we prove that $Z(W)=Z(W_{S\setminus\tilde{S}})$ and $Z(W_{\tilde{S}})=1$, where $Z(W)$ is the center of the Coxeter group $W$ and $\tilde{S}$ is the subset of $S$ such that the parabolic subgroup $W_{\tilde{S}}$ is the {\it essential parabolic subgroup} of $(W,S)$ (i.e.\ $W_{\tilde{S}}$ is the minimum parabolic subgroup of finite index in $(W,S)$). The finiteness of the center of a Coxeter group implies that a splitting theorem holds for Coxeter groups. \end{abstract}
\section{Introduction and preliminaries}
In this paper, we investigate the center of a Coxeter group. A {\it Coxeter group} is a group $W$ having a presentation
$$\langle \,S \, | \, (st)^{m(s,t)}=1 \ \text{for}\ s,t \in S \, \rangle,$$ where $S$ is a finite set and $m:S \times S \rightarrow {\Bbb N} \cup \{\infty\}$ is a function satisfying the following conditions: \begin{enumerate} \item[(1)] $m(s,t)=m(t,s)$ for each $s,t \in S$, \item[(2)] $m(s,s)=1$ for each $s \in S$, and \item[(3)] $m(s,t) \ge 2$ for each $s,t \in S$ such that $s\neq t$. \end{enumerate} The pair $(W,S)$ is called a {\it Coxeter system}. Let $(W,S)$ be a Coxeter system. For a subset $T \subset S$, $W_T$ is defined as the subgroup of $W$ generated by $T$, and called a {\it parabolic subgroup}. It is known that $(W_T,T)$ is also a Coxeter system (cf.\ \cite{Bo} and \cite{Hu}). A subset $T\subset S$ is called a {\it spherical subset} of $S$, if the parabolic subgroup $W_T$ is finite.
The purpose of this paper is to prove the following theorems.
\begin{Theorem}\label{Thm1} The center of every Coxeter group is finite and isomorphic to $({\Bbb Z}_2)^n$ for some $n\ge 0$. \end{Theorem}
A Coxeter system $(W,S)$ is said to be {\it irreducible}, if for any nonempty and proper subset $T$ of $S$, $W$ does not decompose as the direct product of $W_T$ and $W_{S \setminus T}$.
Let $(W,S)$ be a Coxeter system. Then there exists a unique decomposition $\{S_1,\ldots,S_r\}$ of $S$ such that $W$ is the direct product of the parabolic subgroups $W_{S_1},\ldots,W_{S_r}$ and
each Coxeter system $(W_{S_i},S_i)$ is irreducible (cf.\ \cite{Bo} and \cite{Hu}). We define $\tilde{S}=\bigcup \{S_i \,|\, W_{S_i} \ \text{is infinite} \}$, and the parabolic subgroup $W_{\tilde{S}}$ is called the {\it essential parabolic subgroup} of $(W,S)$. We note that $W=W_{\tilde{S}}\times W_{S\setminus \tilde{S}}$ and $W_{S\setminus \tilde{S}}$ is finite. In \cite{Ho2}, it was proved that the essential parabolic subgroup $W_{\tilde{S}}$ is the minimum parabolic subgroup of finite index in $(W,S)$.
We denote the center of a group $G$ by $Z(G)$.
We also prove the following theorem in Section~2.
\begin{Theorem}\label{Thm2} For a Coxeter system $(W,S)$, $Z(W)=Z(W_{S\setminus\tilde{S}})$ and $Z(W_{\tilde{S}})=1$. \end{Theorem}
For an irreducible Coxeter system $(W,S)$, if $W$ is infinite, then $W=W_{\tilde{S}}$. Hence Theorem~\ref{Thm2} implies the following.
\begin{Corollary}\label{Cor1} For an irreducible Coxeter system $(W,S)$, if the Coxeter group $W$ is infinite, then the center of $W$ is trivial. \end{Corollary}
In \cite{Ho4}, we have obtained some splitting theorems for CAT(0) groups whose centers are finite. Theorem~\ref{Thm1} and \cite[Theorem~2]{Ho4} implies the following splitting theorem for Coxeter groups.
\begin{Corollary}\label{Cor2} Let $(W,S)$ be a Coxeter system and let $W=W_{S_1}\times W_{S_2}$. Suppose that the Coxeter group $W$ acts geometrically on a CAT(0) space $X$. Then there exists a closed, convex, $W$-invariant, quasi-dense subspace $X'\subset X$ such that $X'$ splits as a product $X_1 \times X_2$ and the action of $W=W_{S_1}\times W_{S_2}$ on $X'=X_1 \times X_2$ is the product action. \end{Corollary}
By \cite[Lemma~II.6.24]{BH}, we also can obtain the following corollary.
\begin{Corollary}\label{Cor3} Suppose that a Coxeter group $W=W_{S_1}\times W_{S_2}$ acts geometrically on a CAT(0) space $X$. Then $W_{S_1}$ and $W_{S_2}$ are convex-cocompact. \end{Corollary}
Here the definition and some properties of {\it convex-cocompactness} is found in \cite{Ho1} and \cite{Ho3}. We note that ``geometrically finiteness'' in \cite{Ho1} and ``convex-cocompactness'' in \cite{Ho3} coincide.
\section{Proof of the main theorems}
Let $(W,S)$ be a Coxeter system and $w\in W$. A representation $w=s_1\cdots s_l$ ($s_i \in S$) is said to be {\it reduced}, if $\ell(w)=l$, where $\ell(w)$ is the minimum length of word in $S$ which represents $w$.
The following lemmas are known.
\begin{Lemma}[\cite{Bo}]\label{lem1} Let $(W,S)$ be a Coxeter system. Suppose that $W$ is finite. Then there exists a unique element $w_0\in W$ of longest length, and for each $w\in W$, $\ell(w_0w)=\ell(w_0)-\ell(w)$. In particular, $w_0^2=1$. \end{Lemma}
\begin{Lemma}[{\cite{Bo}, \cite[Lemma~7.11]{D1}}]\label{lem2} Let $(W,S)$ be a Coxeter system, let $T\subset S$ and let $w\in W_T$. Then the following statements are equivalent. \begin{enumerate} \item[(1)] $W_T$ is finite and $w$ is the element of longest length in $W_T$. \item[(2)] $\ell(wt)<\ell(w)$ for each $t\in T$. \end{enumerate} \end{Lemma}
Using lemmas above, we prove the following main theorem.
\begin{Theorem} Let $(W,S)$ be a Coxeter system and let $Z(W)$ be the center of $W$. \begin{enumerate} \item[(1)] For each $w\in Z(W)$, there exists a spherical subset $T$ of $S$ such that $w$ is the element of longest length in $W_T$. \item[(2)] $w^2=1$ for any $w\in Z(W)$. \item[(3)] $Z(W)$ is finite. \item[(4)] $Z(W)$ is isomorphic to $({\Bbb Z}_2)^n$ for some $n\ge 0$. \item[(5)] $Z(W)=Z(W_{S\setminus\tilde{S}})$. \item[(6)] $Z(W_{\tilde{S}})$ is trivial. \end{enumerate} \end{Theorem}
\begin{proof} Let $(W,S)$ be a Coxeter system and let $Z(W)$ be the center of $W$.
(1) Let $w\in Z(W)$ and let $w=s_1\cdots s_l$ be a reduced representation. Then $ws_1=s_1w$, since $w\in Z(W)$. Hence $$ ws_1=s_1w=s_1(s_1s_2\cdots s_l)=s_2\cdots s_l.$$ Thus $\ell(ws_1)<\ell(w)$ and $w=(s_2\cdots s_l)s_1$ is reduced. Since $w\in Z(W)$, $ws_2=s_2w$. Hence $$ ws_2=s_2w=s_2((s_2s_3\cdots s_l)s_1)=(s_3\cdots s_l)s_1.$$ Thus $\ell(ws_2)<\ell(w)$ and $w=(s_3\cdots s_l)s_1s_2$ is reduced. By iterating the above argument, we obtain that $\ell(ws_i)<\ell(w)$ for each $i=1,\dots,l$. Let $T=\{s_1,\dots,s_l\}$. By Lemma~\ref{lem2}, $W_T$ is finite and $w$ is the element of longest length in $W_T$.
(2) By (1) and Lemma~\ref{lem1}, we have that $w^2=1$ for any $w\in Z(W)$.
(3) For a spherical subset $T$, let $w_T$ be the element of longest length in $W_T$. By (1),
$$Z(W)\subset \{w_T\,|\, \text{$T$ is a spherical subset of $S$}\}, $$ which is finite, since $S$ is finite. Hence the center $Z(W)$ is finite.
(4) We note that $w^2=1$ for any $w\in Z(W)$ by (2) and $vw=wv$ for any $v,w\in Z(W)$ because $Z(W)$ is the center. Thus $Z(W)$ is isomorphic to $({\Bbb Z}_2)^n$ for some $n\ge 0$.
(5) Let $w\in Z(W)$, let $w=s_1\cdots s_l$ be a reduced representation and let $T=\{s_1,\dots,s_l\}$. Then $w$ is the element of longest length in $W_T$ by (1). Let $s\in S\setminus T$. Then $sw=ws$, since $w\in Z(W)$. Hence $sws=w$ and $s(s_1\cdots s_l)s=s_1\cdots s_l$. Here we note that $s\not\in T=\{s_1,\cdots,s_l\}$. By Tits's theorem in \cite{T} and \cite[p.50]{Br}, $ss_i=s_is$ for each $i=1,\dots,l$. This means that $st=ts$ for any $t\in T$ and $s\in S\setminus T$. Hence $W$ splits as the product $W=W_T\times W_{S\setminus T}$. Since $W_T$ is finite, $T\subset S\setminus \tilde{S}$ by the definition of $\tilde{S}$. Hence $w\in W_T\subset W_{S\setminus \tilde{S}}$ for each $w\in Z(W)$. Thus $Z(W)\subset W_{S\setminus \tilde{S}}$. We note that $W=W_{\tilde{S}}\times W_{S\setminus \tilde{S}}$ and $Z(W)=Z(W_{\tilde{S}})\times Z(W_{S\setminus \tilde{S}})$. Therefore $Z(W)=Z(W_{S\setminus \tilde{S}})$.
(6) We can obtain that $Z(W_{\tilde{S}})$ is trivial from (5). \end{proof}
\end{document} | arXiv |
Boolean algebra (structure)
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution).
For an introduction to the subject, see Boolean algebra. For an alternative presentation, see Boolean algebras canonically defined.
Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle.[1]
History
The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English mathematician. He introduced the algebraic system initially in a small pamphlet, The Mathematical Analysis of Logic, published in 1847 in response to an ongoing public controversy between Augustus De Morgan and William Hamilton, and later as a more substantial book, The Laws of Thought, published in 1854. Boole's formulation differs from that described above in some important respects. For example, conjunction and disjunction in Boole were not a dual pair of operations. Boolean algebra emerged in the 1860s, in papers written by William Jevons and Charles Sanders Peirce. The first systematic presentation of Boolean algebra and distributive lattices is owed to the 1890 Vorlesungen of Ernst Schröder. The first extensive treatment of Boolean algebra in English is A. N. Whitehead's 1898 Universal Algebra. Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by Edward V. Huntington. Boolean algebra came of age as serious mathematics with the work of Marshall Stone in the 1930s, and with Garrett Birkhoff's 1940 Lattice Theory. In the 1960s, Paul Cohen, Dana Scott, and others found deep new results in mathematical logic and axiomatic set theory using offshoots of Boolean algebra, namely forcing and Boolean-valued models.
Definition
A Boolean algebra is a set A, equipped with two binary operations ∧ (called "meet" or "and"), ∨ (called "join" or "or"), a unary operation ¬ (called "complement" or "not") and two elements 0 and 1 in A (called "bottom" and "top", or "least" and "greatest" element, also denoted by the symbols ⊥ and ⊤, respectively), such that for all elements a, b and c of A, the following axioms hold:[2]
a ∨ (b ∨ c) = (a ∨ b) ∨ c a ∧ (b ∧ c) = (a ∧ b) ∧ c associativity
a ∨ b = b ∨ a a ∧ b = b ∧ a commutativity
a ∨ (a ∧ b) = a a ∧ (a ∨ b) = a absorption
a ∨ 0 = a a ∧ 1 = a identity
a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) distributivity
a ∨ ¬a = 1 a ∧ ¬a = 0 complements
Note, however, that the absorption law and even the associativity law can be excluded from the set of axioms as they can be derived from the other axioms (see Proven properties).
A Boolean algebra with only one element is called a trivial Boolean algebra or a degenerate Boolean algebra. (In older works, some authors required 0 and 1 to be distinct elements in order to exclude this case.)
It follows from the last three pairs of axioms above (identity, distributivity and complements), or from the absorption axiom, that
a = b ∧ a if and only if a ∨ b = b.
The relation ≤ defined by a ≤ b if these equivalent conditions hold, is a partial order with least element 0 and greatest element 1. The meet a ∧ b and the join a ∨ b of two elements coincide with their infimum and supremum, respectively, with respect to ≤.
The first four pairs of axioms constitute a definition of a bounded lattice.
It follows from the first five pairs of axioms that any complement is unique.
The set of axioms is self-dual in the sense that if one exchanges ∨ with ∧ and 0 with 1 in an axiom, the result is again an axiom. Therefore, by applying this operation to a Boolean algebra (or Boolean lattice), one obtains another Boolean algebra with the same elements; it is called its dual.[3]
Examples
• The simplest non-trivial Boolean algebra, the two-element Boolean algebra, has only two elements, 0 and 1, and is defined by the rules:
∧01
0 00
1 01
∨01
0 01
1 11
a01
¬a 10
• It has applications in logic, interpreting 0 as false, 1 as true, ∧ as and, ∨ as or, and ¬ as not. Expressions involving variables and the Boolean operations represent statement forms, and two such expressions can be shown to be equal using the above axioms if and only if the corresponding statement forms are logically equivalent.
• The two-element Boolean algebra is also used for circuit design in electrical engineering;[note 1] here 0 and 1 represent the two different states of one bit in a digital circuit, typically high and low voltage. Circuits are described by expressions containing variables, and two such expressions are equal for all values of the variables if and only if the corresponding circuits have the same input-output behavior. Furthermore, every possible input-output behavior can be modeled by a suitable Boolean expression.
• The two-element Boolean algebra is also important in the general theory of Boolean algebras, because an equation involving several variables is generally true in all Boolean algebras if and only if it is true in the two-element Boolean algebra (which can be checked by a trivial brute force algorithm for small numbers of variables). This can for example be used to show that the following laws (Consensus theorems) are generally valid in all Boolean algebras:
• (a ∨ b) ∧ (¬a ∨ c) ∧ (b ∨ c) ≡ (a ∨ b) ∧ (¬a ∨ c)
• (a ∧ b) ∨ (¬a ∧ c) ∨ (b ∧ c) ≡ (a ∧ b) ∨ (¬a ∧ c)
• The power set (set of all subsets) of any given nonempty set S forms a Boolean algebra, an algebra of sets, with the two operations ∨ := ∪ (union) and ∧ := ∩ (intersection). The smallest element 0 is the empty set and the largest element 1 is the set S itself.
• After the two-element Boolean algebra, the simplest Boolean algebra is that defined by the power set of two atoms:
∧0ab1
0 0000
a 0a0a
b 00bb
1 0ab1
∨0ab1
0 0ab1
a aa11
b b1b1
1 1111
x0ab1
¬x 1ba0
• The set $A$ of all subsets of $S$ that are either finite or cofinite is a Boolean algebra and an algebra of sets called the finite–cofinite algebra. If $S$ is infinite then the set of all cofinite subsets of $S,$ which is called the Fréchet filter, is a free ultrafilter on $A.$ However, the Fréchet filter is not an ultrafilter on the power set of $S.$
• Starting with the propositional calculus with κ sentence symbols, form the Lindenbaum algebra (that is, the set of sentences in the propositional calculus modulo logical equivalence). This construction yields a Boolean algebra. It is in fact the free Boolean algebra on κ generators. A truth assignment in propositional calculus is then a Boolean algebra homomorphism from this algebra to the two-element Boolean algebra.
• Given any linearly ordered set L with a least element, the interval algebra is the smallest algebra of subsets of L containing all of the half-open intervals [a, b) such that a is in L and b is either in L or equal to ∞. Interval algebras are useful in the study of Lindenbaum–Tarski algebras; every countable Boolean algebra is isomorphic to an interval algebra.
• For any natural number n, the set of all positive divisors of n, defining $a\leq b$ if a divides b, forms a distributive lattice. This lattice is a Boolean algebra if and only if n is square-free. The bottom and the top element of this Boolean algebra is the natural number 1 and n, respectively. The complement of a is given by n/a. The meet and the join of a and b is given by the greatest common divisor (gcd) and the least common multiple (lcm) of a and b, respectively. The ring addition a+b is given by lcm(a,b)/gcd(a,b). The picture shows an example for n = 30. As a counter-example, considering the non-square-free n=60, the greatest common divisor of 30 and its complement 2 would be 2, while it should be the bottom element 1.
• Other examples of Boolean algebras arise from topological spaces: if X is a topological space, then the collection of all subsets of X which are both open and closed forms a Boolean algebra with the operations ∨ := ∪ (union) and ∧ := ∩ (intersection).
• If $R$ is an arbitrary ring then its set of central idempotents, which is the set
$A=\left\{e\in R:e^{2}=e{\text{ and }}ex=xe\;{\text{ for all }}\;x\in R\right\},$
becomes a Boolean algebra when its operations are defined by $e\vee f:=e+f-ef$ and $e\wedge f:=ef.$
Homomorphisms and isomorphisms
A homomorphism between two Boolean algebras A and B is a function f : A → B such that for all a, b in A:
f(a ∨ b) = f(a) ∨ f(b),
f(a ∧ b) = f(a) ∧ f(b),
f(0) = 0,
f(1) = 1.
It then follows that f(¬a) = ¬f(a) for all a in A. The class of all Boolean algebras, together with this notion of morphism, forms a full subcategory of the category of lattices.
An isomorphism between two Boolean algebras A and B is a homomorphism f : A → B with an inverse homomorphism, that is, a homomorphism g : B → A such that the composition g ∘ f: A → A is the identity function on A, and the composition f ∘ g: B → B is the identity function on B. A homomorphism of Boolean algebras is an isomorphism if and only if it is bijective.
Boolean rings
Main article: Boolean ring
Every Boolean algebra (A, ∧, ∨) gives rise to a ring (A, +, ·) by defining a + b := (a ∧ ¬b) ∨ (b ∧ ¬a) = (a ∨ b) ∧ ¬(a ∧ b) (this operation is called symmetric difference in the case of sets and XOR in the case of logic) and a · b := a ∧ b. The zero element of this ring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the 1 of the Boolean algebra. This ring has the property that a · a = a for all a in A; rings with this property are called Boolean rings.
Conversely, if a Boolean ring A is given, we can turn it into a Boolean algebra by defining x ∨ y := x + y + (x · y) and x ∧ y := x · y.[4][5] Since these two constructions are inverses of each other, we can say that every Boolean ring arises from a Boolean algebra, and vice versa. Furthermore, a map f : A → B is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings. The categories of Boolean rings and Boolean algebras are equivalent.[6]
Hsiang (1985) gave a rule-based algorithm to check whether two arbitrary expressions denote the same value in every Boolean ring. More generally, Boudet, Jouannaud, and Schmidt-Schauß (1989) gave an algorithm to solve equations between arbitrary Boolean-ring expressions. Employing the similarity of Boolean rings and Boolean algebras, both algorithms have applications in automated theorem proving.
Ideals and filters
Main articles: Ideal (order theory) and Filter (mathematics)
An ideal of the Boolean algebra A is a subset I such that for all x, y in I we have x ∨ y in I and for all a in A we have a ∧ x in I. This notion of ideal coincides with the notion of ring ideal in the Boolean ring A. An ideal I of A is called prime if I ≠ A and if a ∧ b in I always implies a in I or b in I. Furthermore, for every a ∈ A we have that a ∧ −a = 0 ∈ I and then a ∈ I or −a ∈ I for every a ∈ A, if I is prime. An ideal I of A is called maximal if I ≠ A and if the only ideal properly containing I is A itself. For an ideal I, if a ∉ I and −a ∉ I, then I ∪ a or I ∪ {−a} is properly contained in another ideal J. Hence, that an I is not maximal and therefore the notions of prime ideal and maximal ideal are equivalent in Boolean algebras. Moreover, these notions coincide with ring theoretic ones of prime ideal and maximal ideal in the Boolean ring A.
The dual of an ideal is a filter. A filter of the Boolean algebra A is a subset p such that for all x, y in p we have x ∧ y in p and for all a in A we have a ∨ x in p. The dual of a maximal (or prime) ideal in a Boolean algebra is ultrafilter. Ultrafilters can alternatively be described as 2-valued morphisms from A to the two-element Boolean algebra. The statement every filter in a Boolean algebra can be extended to an ultrafilter is called the Ultrafilter Theorem and cannot be proven in ZF, if ZF is consistent. Within ZF, it is strictly weaker than the axiom of choice. The Ultrafilter Theorem has many equivalent formulations: every Boolean algebra has an ultrafilter, every ideal in a Boolean algebra can be extended to a prime ideal, etc.
Representations
It can be shown that every finite Boolean algebra is isomorphic to the Boolean algebra of all subsets of a finite set. Therefore, the number of elements of every finite Boolean algebra is a power of two.
Stone's celebrated representation theorem for Boolean algebras states that every Boolean algebra A is isomorphic to the Boolean algebra of all clopen sets in some (compact totally disconnected Hausdorff) topological space.
Axiomatics
Proven properties
UId1If x ∨ o = x for all x, then o = 0
Proof:If x ∨ o = x, then
0
=0 ∨ oby assumption
=o ∨ 0by Cmm1
=oby Idn1
UId2 [dual] If x ∧ i = x for all x, then i = 1
Idm1x ∨ x = x
Proof:x ∨ x
=(x ∨ x) ∧ 1by Idn2
=(x ∨ x) ∧ (x ∨ ¬x)by Cpl1
=x ∨ (x ∧ ¬x)by Dst1
=x ∨ 0by Cpl2
=xby Idn1
Idm2 [dual] x ∧ x = x
Bnd1x ∨ 1 = 1
Proof:x ∨ 1
=(x ∨ 1) ∧ 1by Idn2
=1 ∧ (x ∨ 1)by Cmm2
=(x ∨ ¬x) ∧ (x ∨ 1)by Cpl1
=x ∨ (¬x ∧ 1)by Dst1
=x ∨ ¬xby Idn2
=1by Cpl1
Bnd2 [dual] x ∧ 0 = 0
Abs1x ∨ (x ∧ y) = x
Proof:x ∨ (x ∧ y)
=(x ∧ 1) ∨ (x ∧ y)by Idn2
=x ∧ (1 ∨ y)by Dst2
=x ∧ (y ∨ 1)by Cmm1
=x ∧ 1by Bnd1
=xby Idn2
Abs2 [dual] x ∧ (x ∨ y) = x
UNgIf x ∨ xn = 1 and x ∧ xn = 0, then xn = ¬x
Proof:If x ∨ xn = 1 and x ∧ xn = 0, then
xn
=xn ∧ 1by Idn2
=xn ∧ (x ∨ ¬x)by Cpl1
=(xn ∧ x) ∨ (xn ∧ ¬x)by Dst2
=(x ∧ xn) ∨ (¬x ∧ xn)by Cmm2
=0 ∨ (¬x ∧ xn)by assumption
=(x ∧ ¬x) ∨ (¬x ∧ xn)by Cpl2
=(¬x ∧ x) ∨ (¬x ∧ xn)by Cmm2
=¬x ∧ (x ∨ xn)by Dst2
=¬x ∧ 1by assumption
=¬xby Idn2
DNg¬¬x = x
Proof:¬x ∨ x = x ∨ ¬x = 1by Cmm1, Cpl1
and¬x ∧ x = x ∧ ¬x = 0by Cmm2, Cpl2
hencex = ¬¬xby UNg
A1x ∨ (¬x ∨ y) = 1
Proof:x ∨ (¬x ∨ y)
=(x ∨ (¬x ∨ y)) ∧ 1by Idn2
=1 ∧ (x ∨ (¬x ∨ y))by Cmm2
=(x ∨ ¬x) ∧ (x ∨ (¬x ∨ y))by Cpl1
=x ∨ (¬x ∧ (¬x ∨ y))by Dst1
=x ∨ ¬xby Abs2
=1by Cpl1
A2 [dual] x ∧ (¬x ∧ y) = 0
B1(x ∨ y) ∨ (¬x ∧ ¬y) = 1
Proof:(x ∨ y) ∨ (¬x ∧ ¬y)
=((x ∨ y) ∨ ¬x) ∧ ((x ∨ y) ∨ ¬y)by Dst1
=(¬x ∨ (x ∨ y)) ∧ (¬y ∨ (y ∨ x))by Cmm1
=(¬x ∨ (¬¬x ∨ y)) ∧ (¬y ∨ (¬¬y ∨ x))by DNg
=1 ∧ 1by A1
=1by Idn2
B2 [dual] (x ∧ y) ∧ (¬x ∨ ¬y) = 0
C1(x ∨ y) ∧ (¬x ∧ ¬y) = 0
Proof:(x ∨ y) ∧ (¬x ∧ ¬y)
=(¬x ∧ ¬y) ∧ (x ∨ y)by Cmm2
=((¬x ∧ ¬y) ∧ x) ∨ ((¬x ∧ ¬y) ∧ y)by Dst2
=(x ∧ (¬x ∧ ¬y)) ∨ (y ∧ (¬y ∧ ¬x))by Cmm2
=0 ∨ 0by A2
=0by Idn1
C2 [dual] (x ∧ y) ∨ (¬x ∨ ¬y) = 1
DMg1¬(x ∨ y) = ¬x ∧ ¬y
Proof:by B1, C1, and UNg
DMg2 [dual] ¬(x ∧ y) = ¬x ∨ ¬y
D1(x∨(y∨z)) ∨ ¬x = 1
Proof:(x ∨ (y ∨ z)) ∨ ¬x
=¬x ∨ (x ∨ (y ∨ z))by Cmm1
=¬x ∨ (¬¬x ∨ (y ∨ z))by DNg
=1by A1
D2 [dual] (x∧(y∧z)) ∧ ¬x = 0
E1y ∧ (x∨(y∨z)) = y
Proof:y ∧ (x ∨ (y ∨ z))
=(y ∧ x) ∨ (y ∧ (y ∨ z))by Dst2
=(y ∧ x) ∨ yby Abs2
=y ∨ (y ∧ x)by Cmm1
=yby Abs1
E2 [dual] y ∨ (x∧(y∧z)) = y
F1(x∨(y∨z)) ∨ ¬y = 1
Proof:(x ∨ (y ∨ z)) ∨ ¬y
=¬y ∨ (x ∨ (y ∨ z))by Cmm1
=(¬y ∨ (x ∨ (y ∨ z))) ∧ 1by Idn2
=1 ∧ (¬y ∨ (x ∨ (y ∨ z)))by Cmm2
=(y ∨ ¬y) ∧ (¬y ∨ (x ∨ (y ∨ z)))by Cpl1
=(¬y ∨ y) ∧ (¬y ∨ (x ∨ (y ∨ z)))by Cmm1
=¬y ∨ (y ∧ (x ∨ (y ∨ z)))by Dst1
=¬y ∨ yby E1
=y ∨ ¬yby Cmm1
=1by Cpl1
F2 [dual] (x∧(y∧z)) ∧ ¬y = 0
G1(x∨(y∨z)) ∨ ¬z = 1
Proof:(x ∨ (y ∨ z)) ∨ ¬z
=(x ∨ (z ∨ y)) ∨ ¬zby Cmm1
=1by F1
G2 [dual] (x∧(y∧z)) ∧ ¬z = 0
H1¬((x∨y)∨z) ∧ x = 0
Proof:¬((x ∨ y) ∨ z) ∧ x
=(¬(x ∨ y) ∧ ¬z) ∧ xby DMg1
=((¬x ∧ ¬y) ∧ ¬z) ∧ xby DMg1
=x ∧ ((¬x ∧ ¬y) ∧ ¬z)by Cmm2
=(x ∧ ((¬x ∧ ¬y) ∧ ¬z)) ∨ 0by Idn1
=0 ∨ (x ∧ ((¬x ∧ ¬y) ∧ ¬z))by Cmm1
=(x ∧ ¬x) ∨ (x ∧ ((¬x ∧ ¬y) ∧ ¬z))by Cpl2
=x ∧ (¬x ∨ ((¬x ∧ ¬y) ∧ ¬z))by Dst2
=x ∧ (¬x ∨ (¬z ∧ (¬x ∧ ¬y)))by Cmm2
=x ∧ ¬xby E2
=0by Cpl2
H2 [dual] ¬((x∧y)∧z) ∨ x = 1
I1¬((x∨y)∨z) ∧ y = 0
Proof:¬((x ∨ y) ∨ z) ∧ y
=¬((y ∨ x) ∨ z) ∧ yby Cmm1
=0by H1
I2 [dual] ¬((x∧y)∧z) ∨ y = 1
J1¬((x∨y)∨z) ∧ z = 0
Proof:¬((x ∨ y) ∨ z) ∧ z
=(¬(x ∨ y) ∧ ¬z) ∧ zby DMg1
=z ∧ (¬(x ∨ y) ∧ ¬z)by Cmm2
=z ∧ ( ¬z ∧ ¬(x ∨ y))by Cmm2
=0by A2
J2 [dual] ¬((x∧y)∧z) ∨ z = 1
K1(x ∨ (y ∨ z)) ∨ ¬((x ∨ y) ∨ z) = 1
Proof:(x∨(y∨z)) ∨ ¬((x ∨ y) ∨ z)
=(x∨(y∨z)) ∨ (¬(x ∨ y) ∧ ¬z)by DMg1
=(x∨(y∨z)) ∨ ((¬x ∧ ¬y) ∧ ¬z)by DMg1
=((x∨(y∨z)) ∨ (¬x ∧ ¬y)) ∧ ((x∨(y∨z)) ∨ ¬z)by Dst1
=(((x∨(y∨z)) ∨ ¬x) ∧ ((x∨(y∨z)) ∨ ¬y)) ∧ ((x∨(y∨z)) ∨ ¬z)by Dst1
=(1 ∧ 1) ∧ 1by D1,F1,G1
=1by Idn2
K2 [dual] (x ∧ (y ∧ z)) ∧ ¬((x ∧ y) ∧ z) = 0
L1(x ∨ (y ∨ z)) ∧ ¬((x ∨ y) ∨ z) = 0
Proof:(x ∨ (y ∨ z)) ∧ ¬((x ∨ y) ∨ z)
=¬((x∨y)∨z) ∧ (x ∨ (y ∨ z))by Cmm2
=(¬((x∨y)∨z) ∧ x) ∨ (¬((x∨y)∨z) ∧ (y ∨ z))by Dst2
=(¬((x∨y)∨z) ∧ x) ∨ ((¬((x∨y)∨z) ∧ y) ∨ (¬((x∨y)∨z) ∧ z))by Dst2
=0 ∨ (0 ∨ 0)by H1,I1,J1
=0by Idn1
L2 [dual] (x ∧ (y ∧ z)) ∨ ¬((x ∧ y) ∧ z) = 1
Ass1x ∨ (y ∨ z) = (x ∨ y) ∨ z
Proof:by K1, L1, UNg, DNg
Ass2 [dual] x ∧ (y ∧ z) = (x ∧ y) ∧ z
Abbreviations
UIdUnique Identity
IdmIdempotence
BndBoundaries
AbsAbsorption law
UNgUnique Negation
DNgDouble negation
DMgDe Morgan's Law
AssAssociativity
Huntington 1904 Boolean algebra axioms
Idn1x ∨ 0 = x Idn2x ∧ 1 = x
Cmm1x ∨ y = y ∨ x Cmm2x ∧ y = y ∧ x
Dst1x ∨ (y∧z) = (x∨y) ∧ (x∨z) Dst2x ∧ (y∨z) = (x∧y) ∨ (x∧z)
Cpl1x ∨ ¬x = 1 Cpl2x ∧ ¬x = 0
Abbreviations
IdnIdentity
CmmCommutativity
DstDistributivity
CplComplements
The first axiomatization of Boolean lattices/algebras in general was given by the English philosopher and mathematician Alfred North Whitehead in 1898.[7][8] It included the above axioms and additionally x∨1=1 and x∧0=0. In 1904, the American mathematician Edward V. Huntington (1874–1952) gave probably the most parsimonious axiomatization based on ∧, ∨, ¬, even proving the associativity laws (see box).[9] He also proved that these axioms are independent of each other.[10] In 1933, Huntington set out the following elegant axiomatization for Boolean algebra.[11] It requires just one binary operation + and a unary functional symbol n, to be read as 'complement', which satisfy the following laws:
1. Commutativity: x + y = y + x.
2. Associativity: (x + y) + z = x + (y + z).
3. Huntington equation: n(n(x) + y) + n(n(x) + n(y)) = x.
Herbert Robbins immediately asked: If the Huntington equation is replaced with its dual, to wit:
4. Robbins Equation: n(n(x + y) + n(x + n(y))) = x,
do (1), (2), and (4) form a basis for Boolean algebra? Calling (1), (2), and (4) a Robbins algebra, the question then becomes: Is every Robbins algebra a Boolean algebra? This question (which came to be known as the Robbins conjecture) remained open for decades, and became a favorite question of Alfred Tarski and his students. In 1996, William McCune at Argonne National Laboratory, building on earlier work by Larry Wos, Steve Winker, and Bob Veroff, answered Robbins's question in the affirmative: Every Robbins algebra is a Boolean algebra. Crucial to McCune's proof was the computer program EQP he designed. For a simplification of McCune's proof, see Dahn (1998).
Further work has been done for reducing the number of axioms; see Minimal axioms for Boolean algebra.
Generalizations
Algebraic structures
Group-like
• Group
• Semigroup / Monoid
• Rack and quandle
• Quasigroup and loop
• Abelian group
• Magma
• Lie group
Group theory
Ring-like
• Ring
• Rng
• Semiring
• Near-ring
• Commutative ring
• Domain
• Integral domain
• Field
• Division ring
• Lie ring
Ring theory
Lattice-like
• Lattice
• Semilattice
• Complemented lattice
• Total order
• Heyting algebra
• Boolean algebra
• Map of lattices
• Lattice theory
Module-like
• Module
• Group with operators
• Vector space
• Linear algebra
Algebra-like
• Algebra
• Associative
• Non-associative
• Composition algebra
• Lie algebra
• Graded
• Bialgebra
• Hopf algebra
Removing the requirement of existence of a unit from the axioms of Boolean algebra yields "generalized Boolean algebras". Formally, a distributive lattice B is a generalized Boolean lattice, if it has a smallest element 0 and for any elements a and b in B such that a ≤ b, there exists an element x such that $a\land x=0$ and $a\lor x=b$. Defining a \ b as the unique x such that $(a\land b)\lor x=a$ and $(a\land b)\land x=0$, we say that the structure $(B,\land ,\lor ,\setminus ,0)$ is a generalized Boolean algebra, while $(B,\lor ,0)$ is a generalized Boolean semilattice. Generalized Boolean lattices are exactly the ideals of Boolean lattices.
A structure that satisfies all axioms for Boolean algebras except the two distributivity axioms is called an orthocomplemented lattice. Orthocomplemented lattices arise naturally in quantum logic as lattices of closed subspaces for separable Hilbert spaces.
See also
• List of Boolean algebra topics
• Boolean domain
• Boolean function
• Boolean logic
• Boolean ring
• Boolean-valued function
• Canonical form (Boolean algebra)
• Complete Boolean algebra
• De Morgan's laws
• Finitary boolean function
• Forcing (mathematics)
• Free Boolean algebra
• Heyting algebra
• Hypercube graph
• Karnaugh map
• Laws of Form
• Logic gate
• Logical graph
• Logical matrix
• Propositional logic
• Quine–McCluskey algorithm
• Two-element Boolean algebra
• Venn diagram
• Conditional event algebra
Notes
1. Strictly, electrical engineers tend to use additional states to represent other circuit conditions such as high impedance - see IEEE 1164 or IEEE 1364.
References
1. Givant & Halmos 2009, p. 20.
2. Davey & Priestley 1990, pp. 109, 131, 144.
3. Goodstein 2012, p. 21ff.
4. Stone 1936.
5. Hsiang 1985, p. 260.
6. Cohn 2003, p. 81.
7. Padmanabhan & Rudeanu 2008, p. 73.
8. Whitehead 1898, p. 37.
9. Huntington 1904, pp. 292–293.
10. Huntington 1904, p. 296.
11. Huntington 1933a.
Works cited
• Davey, B.A.; Priestley, H.A. (1990). Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press.
• Cohn, Paul M. (2003), Basic Algebra: Groups, Rings, and Fields, Springer, pp. 51, 70–81, ISBN 9781852335878
• Givant, Steven; Halmos, Paul (2009), Introduction to Boolean Algebras, Undergraduate Texts in Mathematics, Springer, ISBN 978-0-387-40293-2.
• Goodstein, R. L. (2012), "Chapter 2: The self-dual system of axioms", Boolean Algebra, Courier Dover Publications, pp. 21ff, ISBN 9780486154978
• Hsiang, Jieh (1985). "Refutational Theorem Proving Using Term Rewriting Systems". Artificial Intelligence. 25 (3): 255–300. doi:10.1016/0004-3702(85)90074-8.
• Huntington, Edward V. (1904). "Sets of Independent Postulates for the Algebra of Logic". Transactions of the American Mathematical Society. 5 (3): 288–309. doi:10.1090/s0002-9947-1904-1500675-4. JSTOR 1986459.
• Padmanabhan, Ranganathan; Rudeanu, Sergiu (2008), Axioms for lattices and boolean algebras, World Scientific, ISBN 978-981-283-454-6.
• Stone, Marshall H. (1936). "The Theory of Representations for Boolean Algebra". Transactions of the American Mathematical Society. 40: 37–111. doi:10.1090/s0002-9947-1936-1501865-8.
• Whitehead, A.N. (1898). A Treatise on Universal Algebra. Cambridge University Press. ISBN 978-1-4297-0032-0.
General references
• Brown, Stephen; Vranesic, Zvonko (2002), Fundamentals of Digital Logic with VHDL Design (2nd ed.), McGraw–Hill, ISBN 978-0-07-249938-4. See Section 2.5.
• Boudet, A.; Jouannaud, J.P.; Schmidt-Schauß, M. (1989). "Unification in Boolean Rings and Abelian Groups". Journal of Symbolic Computation. 8 (5): 449–477. doi:10.1016/s0747-7171(89)80054-9.
• Cori, Rene; Lascar, Daniel (2000), Mathematical Logic: A Course with Exercises, Oxford University Press, ISBN 978-0-19-850048-3. See Chapter 2.
• Dahn, B. I. (1998), "Robbins Algebras are Boolean: A Revision of McCune's Computer-Generated Solution of the Robbins Problem", Journal of Algebra, 208 (2): 526–532, doi:10.1006/jabr.1998.7467.
• Halmos, Paul (1963), Lectures on Boolean Algebras, Van Nostrand, ISBN 978-0-387-90094-0.
• Halmos, Paul; Givant, Steven (1998), Logic as Algebra, Dolciani Mathematical Expositions, vol. 21, Mathematical Association of America, ISBN 978-0-88385-327-6.
• Huntington, E. V. (1933a), "New sets of independent postulates for the algebra of logic" (PDF), Transactions of the American Mathematical Society, American Mathematical Society, 35 (1): 274–304, doi:10.2307/1989325, JSTOR 1989325.
• Huntington, E. V. (1933b), "Boolean algebra: A correction", Transactions of the American Mathematical Society, 35 (2): 557–558, doi:10.2307/1989783, JSTOR 1989783
• Mendelson, Elliott (1970), Boolean Algebra and Switching Circuits, Schaum's Outline Series in Mathematics, McGraw–Hill, ISBN 978-0-07-041460-0
• Monk, J. Donald; Bonnet, R., eds. (1989), Handbook of Boolean Algebras, North-Holland, ISBN 978-0-444-87291-3. In 3 volumes. (Vol.1:ISBN 978-0-444-70261-6, Vol.2:ISBN 978-0-444-87152-7, Vol.3:ISBN 978-0-444-87153-4)
• Sikorski, Roman (1966), Boolean Algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer Verlag.
• Stoll, R. R. (1963), Set Theory and Logic, W. H. Freeman, ISBN 978-0-486-63829-4. Reprinted by Dover Publications, 1979.
External links
• "Boolean algebra", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Stanford Encyclopedia of Philosophy: "The Mathematics of Boolean Algebra," by J. Donald Monk.
• McCune W., 1997. Robbins Algebras Are Boolean JAR 19(3), 263—276
• "Boolean Algebra" by Eric W. Weisstein, Wolfram Demonstrations Project, 2007.
• Burris, Stanley N.; Sankappanavar, H. P., 1981. A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2.
• Weisstein, Eric W. "Boolean Algebra". MathWorld.
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Dr Rudolf Winter
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Ffiseg Defnyddiau
Prifysgol Aberystwyth
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Paramagnetism in metals
Curie's law states that the susceptibility of a paramagnetic material goes with inverse temperature. This is true for many materials, but in most metals the susceptibility doesn't depend very strongly on temperature. To understand this, we need to see how the band structure influences the magnetisation of a metal sample.
In a metal, the Fermi energy, $E_F$, (the energy of the highest occupied state at absolute zero temperature) lies within an energy band, i.e. the conduction band. The detailed distribution of states within the band is governed by the density of states; here the bands are just shown as rectangular areas in which states are available to be occupied. Different bands are separated by band gaps in which there are no states available. The Fermi distribution (green curve) determines which of the available states will be occupied at a given temperature. At T=0K, the Fermi distribution is a step function with a sharp cut-off at the Fermi energy. As the temperature rises, the curve becomes more inclined, maintaining $E_F$ as a fixpoint - electrons are excited from the green triangle below $E_F$ into the green triangle above $E_F$. The range over which electrons can be excited thermally is of the order of $k_BT$.
In an external magnetic field, the potential energy of those states whose magnetic moments are aligned with the field is reduced by the magnetic interaction energy, and those aligned against the field have a correspondingly higher potential energy. Therefore, the parallel aligned states are nearer the bottom edge of the band. This means that only magnetic moments within a range of $k_BT$ either side of $E_F$ can flip their orientation according to the magnetic field. The 'flippable' fraction is $\frac{k_BT}{E_F}$, and the resulting magnetisation is the usual paramagnetic magnetisation term multiplied by this fraction: $$M_{\textrm{metal}}=\frac{Np_m^2B}{k_BT}\cdot\frac{k_BT}{E_F}=\frac{Np_m^2B}{E_F}\qquad,$$ which is indeed independent of temperature.
The discussion above makes no assumptions about the detail of the density of states within the conduction band. Despite this simplification, it predicts the magnetisation reasonably accurately and demonstrates the fact that the magnetisation of a metal is independent of temperature. We can derive a more accurate formula if we use the density of states according to the free electron gas model of a metal, i.e. treat the delocalised electrons of the atom as independent from the atom cores (nuclei and localised inner-shell electrons) and allow them to move freely like a gas. This model predicts the number $N$ of states per volume element at a particular energy $E$ as $$N(E)=\frac{1}{3\pi^2}\left(\frac{2m_eE}{\hbar^2}\right)^{\frac{3}{2}}\qquad,$$ and thus the density of states $D(E)$ (density here refers to how densely packed the states are on an energy scale) as $$D(E)=\frac{{\rm d}N}{{\rm d}E}=\frac{1}{2\pi^2}\left(\frac{2m_e}{\hbar^2}\right)^{\frac{3}{2}}\sqrt{E}\qquad,$$ represented by the two segments of parabola in the Figure.
When placed in a magnetic field, the potential energy of the parallel and anti-parallel aligned moments is shifted down and up, respectively, by the magnetic interaction $p_mB$, resulting in a relative shift of $2p_mB$ between the parabola segments for the two different orientations. The Fermi energy still applies across the whole electron gas, resulting in a larger population of electrons with parallel aligned moments (in the lowered parabola segment) than anti-parallel ones (in the raised segment). The (anti-)parallel population is $$N_{\uparrow\uparrow,\color{Red}{\uparrow\downarrow}}=\frac{1}{2}\int_{\overset{\color{Red}{+}}{-}p_mB}^{E_F}f(E)D(E\overset{+}{\color{Red}{-}}p_mB){\rm d}E\qquad,$$ where the density of states $D$ has been shifted according to the strength of the magnetic interaction, but the Fermi distribution $f$ remains unchanged since it applies to the electron gas as a whole. The integral consists of a part corresponding to the number of states in the absence of an external field and an almost trapezoidal area (shown in green) near the Fermi energy which is affected by the shift: $$N_{\uparrow\uparrow,\color{Red}{\uparrow\downarrow}}\approx\frac{1}{2}\int_0^{E_F}f(E)D(E){\rm d}E\overset{\color{Red}{+}}{-}\frac{1}{2}p_mBD(E_F)\qquad.$$ The magnetisation arises from the population difference: $$M=p_m(N_{\uparrow\uparrow}-N_{\uparrow\downarrow})=p_m^2D(E_F)B\qquad.$$ The density of states at the Fermi energy $D(E_F)$ in the free electron gas model is $D(E_F)=\frac{3N}{2E_F}$, so the magnetisation becomes $$M_{\textrm{metal}}=\frac{3Np_m^2B}{2E_F}\qquad,$$ confirming the result from the less sophisticated model apart from a factor $\frac{3}{2}$.
Next, we'll see how individual (para-)magnetic moments interact to give rise to much higher susceptibilities in ferromagnets.
Department of Physics, Aberystwyth University, Penglais, Aberystwyth SY23 3BZ, Wales
Adran Ffiseg, Prifysgol Aberystwyth, Penglais, Aberystwyth SY23 3BZ, Cymru
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Selecting survey parameters
1 Problem 8.15a
2 Problem 8.15b
3 Problem 8.15c
4 Problem 8.15d
5 Problem 8.15e
Problem 8.15a
Assume that you wish to survey a 70 × 70 {\displaystyle 70\times 70} km area where Eocene and Cretaceous anticlinal structures with their long axes north-south are expected, the minimum size of an economically viable structure being 2 km across. The maximum dip expected is 15 ∘ {\displaystyle 15^{\circ }} and reflectors are listed in Table 8.15a "The Recent" reflection will be useful in making static corrections. A noise test gave a prestack S/N = 0.5 {\displaystyle {\mbox{S/N}}=0.5} . Propose the line spacing and orientation of a reconnaissance survey.
Static corrections are corrections that are independent of traveltime; these include corrections for variations in the surface elevation and the weathered layer (see problem 8.18).
A noise test is discussed in problem 8.11.
Table 8.15a. Reflection data.
Depth (m)
V ¯ ( m / s ) {\displaystyle {\bar {V}}(m/s)}
t 0 ( s ) {\displaystyle t_{0}(s)}
V i ( m / s ) {\displaystyle V_{i}(m/s)}
Recent 300 2000 0.300 2000
Eocene 3000 3000 2.000 3180
Cretaceous 5000 3370 2.970 4140
A reconnaissance will mainly use east-west lines plus a few north-south lines to tie the survey together. Any prior knowledge of the area will help in locating the lines. This includes examination of the land surface to see if surface features may relate to deeper structure. We do not expect to locate all possible structures on the first reconnaissance so east-west lines will be spaced 10 km or more apart and the north-south lines about double this. We shall plan on about seven east-west and three to four north-south lines and we should run the east-west lines first so that we can use their interpretation to locate the north-south lines. We will then select a couple of portions of the area for more detailed surveying where we can infill with east-west lines 2 to 3 km apart plus additional north-south lines to tie the area together. We must keep an open mind about the prior knowledge that the anticlines are oriented north-south and we may alter line orientations as interpretation unfolds. We may wish to shoot additional lines more-or-less perpendicular to the strike of faults that may affect structures.
Problem 8.15b
What multiplicity is required to give S/N = 3.0 $ {\displaystyle {\mbox{S/N}}=3.0\$}
From problem 8.14a we know that S/N varies as n 1 / 2 {\displaystyle n^{1/2}} for random noise; so to increase S/N from 0.5 to 3.0 requires a multiplicity of ( 3.0 / 0.5 ) 2 = 36 {\displaystyle (3.0/0.5)^{2}=36} .
Problem 8.15c
What spread geometry should be used, that is, what are the required near- and far-offsets, group spacing to avoid aliasing for 15 to 40 Hz, and minimum number of channels?
Because the objectives are Eocene to Cretaceous, we will want maximum offsets of 5000 m and, since we expect to use the Recent reflection to make static corrections, we will also need short offset data. Hence, end-on spreads should extend from near the source to 5000 m. We should use 96 channels and 50-m group intervals although 48 channels and 100-m group intervals might suffice. We probably should use geophone spacing within a group no larger than 5 m and have the same group length and group interval.
Problem 8.15d
How long will the survey require, assuming production of 270 km/month can be achieved for 24-fold multiplicity?
The reconnaissance of seven east-west and three to four north-south lines, each being 75 km long, amounts to about 800 km and therefore will take about 3 months.
Problem 8.15e
Answer part (d) assuming 210 km/month production for 48-fold multiplicity.
Increasing the multiplicity from 24 to 48 will increase the survey cost but will be worthwhile considering the signal/noise improvement expected. The reduction in production rate will increase the time by about 3 weeks. Shortening the group interval might achieve the same improvement in S/N.
Determining vibroseis parameters Effect of signal/noise ratio on event picking
Seismic equipment Data processing
Effect of too many groups connected to the cable
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Determining vibroseis parameters
Effect of signal/noise ratio on event picking
Interpreting uphole surveys
Weathering and elevation (near-surface) corrections
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Blondeau weathering corrections
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Reflection field methods | CommonCrawl |
\begin{document}
\begin{abstract} By using the infinitesimally marking point to break the loop in the localization calculation as Kim and Lho, and Zinger's explicit formulas for double $J$-functions, we obtain a formula for genus one stable quasimaps invariants when the target is a complete intersection Calabi-Yau in projective space, which gives a new proof of Kim and Lho's mirror theorem for elliptic quasimap invariants. \end{abstract}
\maketitle
\section{Introduction} The moduli space of stable quotients was first constructed and studied by Marian, Oprea and Pandharipande \cite{MOP}. Cooper and Zinger \cite{CZ} calculated the $J$-function of stable quotients for projective complete intersections, and proved that it is related to the genus zero Gromov-Witten invariants by mirror map. Using the moduli space of stable quasimaps, which is a generalization of stable quotient by Ciocan-Fontanine, Kim and Maulik~\cite{FKM}, Ciocan-Fontanine, Kim proved that the genus zero stable quasimap invariants (including twisted cases) are related to stable map by mirror map in \cite{FK}. Later Kim and Lho \cite{KB} obtained the formula for genus one stable quasimap invariants without markings for projective complete intersections. Combining with the result in \cite{FK2} it recovered the results of Popa \cite{Popa} and Zinger \cite{Zin1} on genus one Gromov Witten invariants of projective complete intersections by mirror map. In this paper using the double $J$-functions of complete intersections in projective space proved by Zinger \cite{Zin2}, we give another proof of that formula.
Let $\ell$ be a nonnegative integer. $E:=\oplus_{i=1}^{\ell}L_i$ be the direct sum of line bundles over $\mathbb{P}^{n-1}$ with degree $a_i=\deg L_i>0$.
If $|\mathbf a|:=\sum_{k=1}^{\ell}a_k=n$, a transversal section $s$ of $E$ gives a Calabi-Yau manifold $X$, which is a complete intersection in $\mathbb{P}^{n-1}$.
Let $\overline{Q}_{1,0}(X,d)$ be the moduli space of genus one and degree $d$ stable quasimaps to $X$. It is a proper Deligne-Mumford stack and has perfect obstruction theory. Thus it has virtual cycle $[\overline{Q}_{1,0}(X,d)]^{\text{vir}}$ with zero virtual dimension. Let $$ G_{1,0}:=\sum_{d=1}^{\infty}q^d\deg[\overline{Q}_{1,0}(X,d)]^{\text{vir}}. $$
Let $\overline{Q}_{1,0}(\mathbb{P}^{n-1},d)$ be the moduli space of genus one and degree $d$ stable quasimaps to $\mathbb{P}^{n-1}$. It is a smooth Deligne-Mumford stack. Let $$\tilde\pi:\tilde{\cal C}\to \overline{Q}_{1,0}(\mathbb{P}^{n-1},d) $$ be the universal family, $\tilde{\cal S}$ be the universal bundle over $\tilde{\cal C}$. Let $\tilde\iota$ be the closed immersion $\tilde\iota: \overline{Q}_{1,0}(X,d)\to \overline{Q}_{1,0}(\mathbb{P}^{n-1},d)$. By the functoriality in \cite{KKP} we have $$ \tilde\iota_{*}[\overline{Q}_{1,0}(X,d)]^{\text{vir}}=\mathbf{e}(\tilde{\cal V}_1)\cap[\overline{Q}_{1,0}(\mathbb{P}^{n-1},d)], $$ where $\tilde{\cal V}_1:=\oplus_{i=1}^{\ell} \tilde{\pi}_{*}\tilde{\cal S}^{\vee\otimes a_i}$ is locally free. Thus \begin{equation}\label{Cal} \deg[\overline{Q}_{1,0}(X,d)]^{\text{vir}}=\deg(\mathbf{e}(\tilde{\cal V}_1)\cap[\overline{Q}_{1,0}(\mathbb{P}^{n-1},d)]). \end{equation}
The standard torus $\mathbb{T}=(\mathbb{C}^*)^n$ action on $\mathbb{P}^{n-1}$ induces an action on the moduli space $\overline{Q}_{1,0}(\mathbb{P}^{n-1},d)$. There are two different types of the fixed loci when applying localization method to calculate the degree on the right hand side of (\ref{Cal}). One is the loop type and the other is the vertex type. To calculate the degree of the loop type, we use an infinitesimally marking to break the loop as Kim and Lho \cite{KB}, see Section 2.2. But in this paper we work out the loop contribution by the double $J$-functions given by Zinger \cite{Zin2}, which are directly related to the hypergeometric series used in calculation.
In the forthcoming papers, we will extend the calculation to the genus one stable quasimap invariants with markings, and genus two stable quasimap invariants for projective complete intersections.
Let ${\cal A}_i(q)$ be as in Proposition \ref{equivred_prp_A}, ${\cal F}^{(0,0)}(\alpha_i,q)$ and $\dot\Phi^{(0)}(\alpha_i,q)$ be as in Corollary \ref{expan}. By the localization method, we prove the following result. \begin{theo}\label{main_2}For projective complete intersection Calabi-Yau $X$,
\begin{equation}\nonumber q\frac{d}{d q}G_{1,0}=\bigg\{\sum_{i\in\mathbf{n}}\bigg({\cal A}_{i}(q)+\frac{1}{24}q\frac{d}{d q}\bigg(c_{i}(\alpha){\cal F}^{(0,0)}(\alpha_i,q)-\log \dot\Phi^{(0)}(\alpha_i,q)\bigg)\bigg)\bigg\}\bigg|_{\alpha=0}, \end{equation} where $c_{i}(\alpha)=\sum_{k\neq i}\frac{1}{\alpha_k-\alpha_i}+\sum_{k=1}^{\ell}\frac{1}{a_k\alpha_i}$. \end{theo}
Let $L(q)=(1-\mathbf a^{\mathbf a}q)^{-\frac{1}{n}}$ and $\mu(q)=\int_{0}^q\frac{L(u)-1}{u}du$, where $\mathbf a^{\mathbf a}=\prod_{k=1}^{\ell} a_k^{a_k}$. For $p\in\mathbb{Z}^{\ge0}$, let $\dot{I}_p(q)\in\mathbb{Q}[[q]]$ be defined as in Section 3. We can recover the following result of Kim and Lho from Theorem \ref{main_2}. \begin{theo}\cite[Theorem 1.1]{KB} For projective complete intersection Calabi-Yau $X$,
$$ G_{1,0}=\frac{1}{2}\mathbb A(q)+\frac{1}{24}\bigg(\sum_{i=1}^{\ell}(\frac{n}{a_i}-\binom{n}{2})\mu(q)-\frac{n(\ell+1)}{2}\log L(q)\bigg), $$ where \begin{eqnarray*} \mathbb A(q)=\frac{n}{24}(n-1-2\sum_{r=1}^{\ell}\frac{1}{a_r})\mu(q)-\frac{3(n-1-\ell)^2+(n-2)}{24}\log(1-\mathbf a^{\mathbf a}q)\\ -\sum_{p=0}^{n-2-\ell}\binom{n-p-\ell}{2}\log \dot I_p(q). \end{eqnarray*} \end{theo}
{\bf Acknowledgement}: The author thanks Ionut Ciocan-Fontanine, Bumsig Kim and Hyenho Lho for useful discussion when the author visited KIAS. The author also thanks Huai-Liang Chang, Jun Li and Wanmin Liu for helpful discussions. This work was supported by Start-up Fund of Hunan University.
\section{Localization}
$\mathbb{P}^{n-1}$ has a torus action induced by the standard torus $\mathbb{T}=(\mathbb{C}^*)^n$ action on $\mathbb{C}^n$. The $\mathbb{T}$ fixed points of $\mathbb{P}^{n-1}$ are the 1-planes spanned by $e_i$, where $\{e_i\}_{i=1}^n$ is the standard basis of $\mathbb{C}^n$. We label it by $p_{i}$. Let $\mathbf{n}$ be the set $1\le i\le n$.
Denote by $H^*(B\mathbb{T},\mathbb{Q}):=\mathbb{Q}[\alpha_1,\cdots,\alpha_n]$, where $\alpha_i\!=\!\pi_i^*c_1(\gamma)$. Here $$\pi_i\!: (\mathbb{P}^{\infty})^n\longrightarrow\mathbb{P}^{\infty} \qquad\hbox{and}\qquad \gamma^{\vee}\longrightarrow\mathbb{P}^{\infty}$$ are the projection onto the $i$-th component and the tautological line bundle respectively. Denote by $\alpha=(\alpha_1,\cdots,\alpha_n)$. Let $\mathbb{Q}_{\alpha}$ be the fractional field of $\mathbb{Q}[\alpha_1,\cdots,\alpha_n]$.
Let $\mathbf{x}$ be the equivariant Chern class of the dual universal bundle. Then $$ H^*_{\mathbb{T}}(\mathbb{P}^{n-1})\cong \mathbb{Q}[\alpha_1,\cdots,\alpha_n][\mathbf{x}]/<\prod_{i=1}^n(\mathbf{x}-\alpha_i)>. $$
Let $$ \phi_{i}=\prod_{k\neq i}(\mathbf{x}-\alpha_k). $$ Then $\phi_{i}$ is the equivariant Poincar$\acute{e}$ dual of the points $p_{i}\in H_{\mathbb{T}}^*(\mathbb{P}^{n-1})$. The Euler class
$$\mathbf{e}(T_{\mathbb{P}^{n-1}})|_{p_{i}}=\prod_{k\neq i}(\alpha_i-\alpha_k)=\phi_{i}|_{p_{i}}$$
The Artiyah-Bott localization theorem states that \begin{equation}
\int_{\mathbb{P}^{n-1}}\eta=\sum\int_{p_{i}}\frac{\eta|_{p_{i}}}{\mathbf{e}(N_{p_{i}/\mathbb{P}^{n-1}})},\quad\text{for all}\,\eta\in H^*_{\mathbb{T}}(\mathbb{P}^{n-1}). \end{equation} Therefore \begin{equation}\label{AB}
\eta|_{p_{i}}=\int_{\mathbb{P}^{n-1}}\eta \phi_{i}. \end{equation}
So for $\eta\in H^*_{\mathbb{T}}(\mathbb{P}^{n-1})$ \begin{equation}\label{zero}
\eta|_{p_{i}}=0 \quad\text{for all}\quad i\in\mathbf{n}\quad\Longleftrightarrow\quad \eta=0. \end{equation}
Denote by
$$\overline{Q}_{g,k|m}(\mathbb{P}^{n-1},d)$$
the moduli space of genus $g$ degree $d$ quasimaps to $\mathbb{P}^{n-1}$ with ordinary $k$ pointed markings and infinitesimally weighted $m$ pionted markings (see \cite[Section 2]{FK1}). When $g=1$ and $k=0,\,m=1$, the genus one moduli space $\overline{Q}_{1,0|1}(\mathbb{P}^{n-1},d)$ is a smooth Deligne-Mumford stack since the obstruction vanishes. Let $$\pi:{\cal C}\to \overline{Q}_{1,0|1}(\mathbb{P}^{n-1},d) $$ be the universal family, ${\cal S}$ be the universal bundle over ${\cal C}$. Let ${\cal S}^{\vee}$ be the dual bundle of ${\cal S}$.
Let $\iota$ be the closed immersion $\iota: \overline{Q}_{1,0|1}(X,d)\to \overline{Q}_{1,0|1}(\mathbb{P}^{n-1},d)$, by the functoriality in \cite{KKP} we have $$
\iota_{*}[\overline{Q}_{1,0|1}(X,d)]^{\text{vir}}=\mathbf{e}({\cal V}_1)\cap[\overline{Q}_{1,0|1}(\mathbb{P}^{n-1},d)], $$ where ${\cal V}_1:=\oplus_{i=1}^{\ell} \pi_{*}{\cal S}^{\vee\otimes a_i}$ is locally free.
\subsection{Localization}
First we recall some facts about residues, which are from \cite[Section 1.2]{Zin1}
If $f$ is a rational function in $z$ and possibly other variables and $z_0\!\in\!S^2$, let ${\mathfrak R}_{z=z_0}f(z)$ denote the residue of the one-form $f(z)dz$ at $z\!=\!z_0$; thus, $${\mathfrak R}_{z=\infty}f(z)\equiv-{\mathfrak R}_{w=0}\big\{w^{-2}f(w^{-1})\big\}.$$ If $f$ involves variables other than $z$, ${\mathfrak R}_{z=z_0}f(z)$ is a function of the other variables. If $f$ is a power series in $q$ with coefficients that are rational functions in~$z$ and possibly other variables, let ${\mathfrak R}_{z=z_0}f(z)$ denote the power series in~$q$ obtained by replacing each of the coefficients by its residue at $z\!=\!z_0$. If $z_1,\ldots,z_k$ is a collection of points in~$S^2$, not necessarily distinct, we define $${\mathfrak R}_{z=z_1,\ldots,z_k}f(z) \equiv \sum_{y\in\{z_1,\ldots,z_k\}}\!\!\!\!\!\! {\mathfrak R}_{z=y}f(z).$$
Let $\hat\pi:\hat{\cal C}\to\overline{Q}_{0,2}(\mathbb{P}^{n-1},d)$ be the universal family, $\hat{\cal S}$ be the universal bundle over $\hat{\cal C}$. Let \begin{equation}\label{prod_e0} {\cal V}_{n;\mathbf a}^{(d)}:=\bigoplus_{k}R^0\hat\pi_*\big(\hat{\cal S}^{\vee\otimes a_k}\big)
\longrightarrow \overline{Q}_{0,2}(\mathbb{P}^{n-1},d),\end{equation} \begin{equation}\label{prod_e} \dot{\cal V}_{n;\mathbf a}^{(d)}:=\bigoplus_{k}R^0\hat\pi_*\big(\hat{\cal S}^{\vee\otimes a_k}(-\sigma_1)\big)
\longrightarrow \overline{Q}_{0,2}(\mathbb{P}^{n-1},d)\end{equation}
and
\begin{equation}\ddot{\cal V}_{n;\mathbf a}^{(d)}:=\bigoplus_{k}R^0\hat\pi_*\big(\hat{\cal S}^{\vee\otimes a_k}(-\sigma_2)\big)
\longrightarrow \overline{Q}_{0,2}(\mathbb{P}^{n-1},d).
\end{equation} where $\sigma_i$ is the section correspondent to the marking. Thus ${\cal V}_{n;\mathbf a}^{(d)}$, $\dot{\cal V}_{n;\mathbf a}^{(d)}$ and $\ddot{\cal V}_{n;\mathbf a}^{(d)}$ are $\mathbb{T}$-equivariant vector bundles. Let \begin{equation}\label{cZ2} \dot{{\cal Z}}(\mathbf{x},\hbar,q) := 1+\sum_{d=1}^{\infty}q^d \text{ev}_{1*}\!\!\left[\frac{\mathbf{e}(\dot{\cal V}_{n;\mathbf a}^{(d)})}{\hbar\!-\!\psi_1}\right] \in H_{\mathbb{T}}^*(\mathbb{P}^{n-1})[\hbar^{-1}]\big[\big[q\big]\big],\end{equation}
\begin{equation}\label{ZZdfn_e2} \ddot{{\cal Z}}(\mathbf{x},\hbar,q) := 1+\sum_{d=1}^{\infty}q^d \text{ev}_{1*}\!\!\left[\frac{\mathbf{e}(\ddot{\cal V}_{n;\mathbf a}^{(d)})}{\hbar\!-\!\psi_1}\right] \in H_{\mathbb{T}}^*(\mathbb{P}^{n-1})[\hbar^{-1}]\big[\big[q\big]\big].\end{equation}
For $s\in\mathbb{Z}^{\ge0}$, Let
\begin{equation} \dot{{\cal Z}}^{(s)}(\mathbf{x},\hbar,q) := \mathbf{x}^s+\sum_{d=1}^{\infty}q^d \text{ev}_{1*}\!\!\left[\frac{\mathbf{e}(\dot{\cal V}_{n;\mathbf a}^{(d)})\text{ev}_2^*\mathbf{x}^s}{\hbar\!-\!\psi_1}\right] \in H_{\mathbb{T}}^*(\mathbb{P}^{n-1})[\hbar^{-1}]\big[\big[q\big]\big],\end{equation}
\begin{equation} \ddot{{\cal Z}}^{(s)}(\mathbf{x},\hbar,q) := \mathbf{x}^s+\sum_{d=1}^{\infty}q^d \text{ev}_{1*}\!\!\left[\frac{\mathbf{e}(\ddot{\cal V}_{n;\mathbf a}^{(d)})\text{ev}_2^*\mathbf{x}^s}{\hbar\!-\!\psi_1}\right] \in H_{\mathbb{T}}^*(\mathbb{P}^{n-1})[\hbar^{-1}]\big[\big[q\big]\big].\end{equation}
Let \begin{equation}\label{ZZdfn2} \dot{{\cal Z}}(\hbar_1,\hbar_2,q) := \frac{[\overline{\Delta}]}{\hbar_1+\hbar_2}+\sum_{d=1}^{\infty}q^d\big\{\text{ev}_1\!\times\!\text{ev}_2\}_*\!\!\left[\frac{\mathbf{e}(\dot{\cal V}_{n;\mathbf a}^{(d)})} {(\hbar_1\!-\!\psi_1)(\hbar_2\!-\!\psi_2)}\right] \end{equation} $$ \in H_{\mathbb{T}}^*(\mathbb{P}^{n-1})[\hbar_1^{-1},\hbar_2^{-1}]\big[\big[q\big]\big],$$ where $[\overline{\Delta}]$ is the equivariant Poincar\'{e} dual of the diagonal class. Denote by \begin{eqnarray}\label{Z2ptdfn_e} &&\widetilde{{\cal Z}}_{ij}^*(\hbar_1,\hbar_2,q) \\
&&=\frac{1}{2\hbar_1\hbar_2}(\dot{{\cal Z}}(\hbar_1,\hbar_2,q)-\frac{[\overline{\Delta}]}{\hbar_1+\hbar_2})|_{p_{i}\times p_{j}}\nonumber\\ &&=\frac{1}{2\hbar_1\hbar_2}\sum_{d=1}^{\infty}q^d \int_{\overline{Q}_{0,2}(\mathbb{P}^{n-1},d)} \frac{\mathbf{e}(\dot{\cal V}_{n;\mathbf a}^{(d)})}{(\hbar_1\!-\!\psi_1)(\hbar_2\!-\!\psi_2)} \text{ev}_1^*\phi_{i}\text{ev}_2^*\phi_{j}\nonumber. \end{eqnarray}
Let \begin{eqnarray*} {\cal Y}(\mathbf{x},\hbar,q)=\sum_{d\ge0}q^d {\cal Y}_d(\hbar),\\ \dot{{\cal Y}}(\mathbf{x},\hbar,q)=\sum_{d\ge0}q^d \dot{{\cal Y}}_d(\hbar),\\ \ddot{{\cal Y}}(\mathbf{x},\hbar,q)=\sum_{d\ge0}q^d \ddot{{\cal Y}}_d(\hbar), \end{eqnarray*} where \begin{eqnarray*} {\cal Y}_d(\hbar)=\sum_{d}\frac{\prod_{k=1}^{\ell}\prod_{l=1}^{a_kd}(a_k\mathbf{x}+l\hbar)}{\prod_{l=1}^{d} \prod_{k=1}^{n}(\mathbf{x}-\alpha_k+l\hbar)},\\ \dot{{\cal Y}}_d(\hbar)=\sum_{d}\frac{\prod_{k=1}^{\ell}\prod_{l=1}^{a_kd}(a_k\mathbf{x}+l\hbar)}{\prod_{l=1}^{d}\bigg( \prod_{k=1}^{n}(\mathbf{x}-\alpha_k+l\hbar)-\prod_{k=1}^{n}(\mathbf{x}-\alpha_k)\bigg)},\\ \ddot{{\cal Y}}_d(\hbar)=\sum_{d}\frac{\prod_{k=1}^{\ell}\prod_{l=0}^{a_kd-1}a_k(\mathbf{x}+l\hbar)}{\prod_{l=1}^{d} \bigg(\prod_{k=1}^{n}(\mathbf{x}-\alpha_k+l\hbar)-\prod_{k=1}^{n}(\mathbf{x}-\alpha_k)\bigg)}. \end{eqnarray*}
Denote $\dot{I}_0(q)=\dot{{\cal Y}}(\mathbf{x},\hbar,q)|_{\mathbf{x}=\alpha=0\atop\hbar=1}$ and $\ddot{I}_0(q)=\ddot{{\cal Y}}(\mathbf{x},\hbar,q)|_{\mathbf{x}=\alpha=0\atop\hbar=1}$. \begin{theo}\cite[Theorem 3]{Zin2}\label{thm1'} If $\ell\!\in\!\mathbb{Z}^{\ge0}$, $n\!\in\!\mathbb{Z}^+$, and $\mathbf a\!\in\!(\mathbb{Z}^{>0})^{\ell}$ are such that
$|\mathbf a|:=\sum_{i=1}^{\ell}a_i=n$, then \begin{equation} \dot{{\cal Z}}(\mathbf{x},\hbar,q)=\frac{\dot{{\cal Y}}(\mathbf{x},\hbar,q)}{\dot{I}_0(q)} \in H_{\mathbb{T}}^*(\mathbb{P}^{n-1})\big[\big[\hbar^{-1},q\big]\big],\end{equation} and \begin{equation} \ddot{{\cal Z}}(\mathbf{x},\hbar,q)=\frac{\ddot{{\cal Y}}(\mathbf{x},\hbar,q)}{\ddot{I}_0(q)} \in H_{\mathbb{T}}^*(\mathbb{P}^{n-1})\big[\big[\hbar^{-1},q\big]\big].\end{equation} \end{theo}
\subsection{Insertion of $0+$ weighted marking}
Let $$\hat{ev}_1:\overline{Q}_{1,0|1}(\mathbb{P}^{n-1},d)\to [\mathbb{C}^{n}/\mathbb{C}^*]$$ be the evaluations map at the infinitesimally marking, see \cite[Section 2.3]{FK1}. Let $$\tilde\gamma\in H^2_{\mathbb{T}}([\mathbb{C}^{n}/\mathbb{C}^*])$$ be the lift of $\gamma\in H^2_{\mathbb{T}}(\mathbb{P}^{n-1})$, where $\gamma$ is the hyperplane class .
Let $$
<\tilde\gamma>_{1,0|1,d}:=\int_{\overline{Q}_{1,0|1}(\mathbb{P}^{n-1},d)}\mathbf{e}({\cal V}_1)\hat{ev}^*_1\tilde\gamma, $$ then $$
\sum_{d=1}^{\infty}q^d<\tilde\gamma>_{1,0|1,d}=q\frac{d}{d q}G_{1,0}. $$
In the rest of this section we use localization method to calculate the left hand side of the above equation. As described in \cite[Section~7]{MOP}, the fixed loci of the $\mathbb{T}$-action on $\overline{Q}_{1,0|1}(\mathbb{P}^{n-1},d)$ are indexed by connected decorated graphs. Such a graph can be described by set $(\Ver,\Edg)$ of vertices, A decorated graph is a tuple \begin{equation}\Gamma = \big(\Ver,\Edg;\mu,\mathfrak{d},\eta\big),\end{equation} where $(\Ver,\Edg)$ is a graph as above and $$\mu\!:\Ver\longrightarrow \mathbf{n}, \qquad \mathfrak{d}\!: \Ver\!\sqcup\!\Edg\longrightarrow\mathbb{Z}^{\ge0}\qquad\mbox{and}\qquad\eta:[1]\longrightarrow \Ver$$ are maps such that \begin{equation} \mu(v_1)\neq\mu(v_2) ~~~\hbox{if}~~ \{v_1,v_2\}\in\Edg, \qquad \mathfrak{d}(e)\neq0~~\forall\,e\!\in\!\Edg.\end{equation}
\noindent Let
$$|\Gamma|\equiv\sum_{v\in\Ver}\!\!\mathfrak{d}(v)+\sum_{e\in\Edg}\!\!\mathfrak{d}(e)$$ be the degree of~$\Gamma$. Denoted by
$$\mbox{val}(v)\equiv (\big|\{e\!\in\!\Edg\!: v\!\in\!e\}\big|
, \big|\{i\!\in\![1]\!: \eta(1)\!=\!v\}\big|) .$$ for the vertices $v\!\in\!\Ver$.
Let ${\cal Z}_{\Gamma}$ be the fixed locus of $\overline{Q}_{1,0|1}(\mathbb{P}^{n-1},d)$ corresponds to a decorated graph $\Gamma$. Thus
$${\cal Z}_{\Gamma}\approx \prod_{v\in\Ver}\!\!\overline{{\cal M}}_{g_v,\val(v)|\mathfrak{d}(v)}$$
up to a finite group qoutient, where $\overline{{\cal M}}_{g_v,(k,m)|\mathfrak{d}(v)}$ denotes the moduli space weighted pointed stable curves with $k$ ordinary markings and $m$ infinitesimally markings. When $m=0$, we abbreviated by $\overline{{\cal M}}_{g_v,k|\mathfrak{d}(v)}$. Let $$\pi_1:{\cal C}_{\overline{{\cal M}}_{g_v,\val(v)|\mathfrak{d}(v)}}\to \overline{{\cal M}}_{g_v,\val(v)|\mathfrak{d}(v)}$$ be the restriction of the universal family.
With $b_1,b_2,r\!\in\!\mathbb{Z}^{\ge0}$, let \begin{eqnarray*} &&{\cal F}_{n}^{(b_1,b_2)}(\alpha_i,q)=\sum_{d=1}^{\infty}\frac{q^d}{d!}
\int_{\overline{{\cal M}}_{0,2|d}} \mathcal{G}_{n,d}^{(b_1,b_2)}(\alpha_i,q), \end{eqnarray*} where \begin{eqnarray*} \mathcal{G}_{n,d}^{(b_1,b_2)}(\alpha_i,q)&:=&\frac{\prod_{k\neq i}(\alpha_i-\alpha_k)\mathbf{e}(\dot{\cal V}_{n;\mathbf a}^{(d)}(\alpha_i))\psi_1^{b_1}\psi_2^{b_2}}{\prod_{k\neq i} \mathbf{e}(R^0\pi_{1*}{\cal S}^{\vee}(\alpha_i-\alpha_k))}. \end{eqnarray*}
By the proof of \cite[Theorem 4]{CZ}, we have \begin{equation}\label{Fred_e}q\frac{d}{dq}{\cal F}_{n}^{(b_1,b_2)}(\alpha_i,q)=\frac{1}{b_1!}{\cal F}_{n}^{(0,0)}(\alpha_i,q)^{b_1} \frac{1}{b_2!}{\cal F}_{n}^{(0,0)}(\alpha_i,q)^{b_2}q\frac{d}{dq}{\cal F}_{n}^{(0,0)}(\alpha_i,q).\end{equation} By inductions on $b_2$, this gives $$ {\cal F}_{n}^{(0,b_2)}(\alpha_i,q)=\frac{1}{(b_2+1)!}{\cal F}_{n}^{(0,0)}(\alpha_i,q)^{b_2+1}. $$
Thus the $r=0$ case of \cite[Proposition 8.3]{CZ} is equivariant to $$ {\mathfrak R}_{\hbar=0}\big\{\mbox{e}^{-\frac{{\cal F}_{n}^{(0,0)}(\alpha_i,q)}{\hbar}}\dot{\cal Y}(\alpha_i,\hbar,q)\big\}=0. $$ Thus there is an expansion \begin{coro}\cite[(4-9)]{Zin2}\label{expan}$$\mbox{e}^{-\frac{{\cal F}_{n}^{(0,0)}(\alpha_i,q)}{\hbar}}\dot{\cal Y}(\alpha_i,\hbar,q)=\sum_{m=0}^{\infty}\dot{\Phi}^{(m)}(\alpha_i,q)\hbar^m.$$ \end{coro}
In the case of moduli space $\overline{Q}_{1,0|1}(\mathbb{P}^{n-1},d)$, they are two types of graph, either one distinguished vertex
or one loop, depending on whether the stable qusimaps they describe are constant or not. The graphs with one loop are called $A_{i}$-graphs. In a graph of the $A_{i}$-type, the marked point~$1$ is attached to some vertex $v_0\!\in\!\Ver$ that lies inside of the loop and is labeled~$i$.
Denote ${\cal A}_{i}(q)$ by the total contribution from type $A_{i}$ graphs, then \begin{prop}\label{equivred_prp_A}For every $i\in\mathbf{n}$, \begin{eqnarray*}
{\cal A}_{i}(q)&=&(\phi_{i}|_{p_{i}})^{-1}\bigg(q\frac{d}{dq}{\cal F}_{n}^{(0,0)}(\alpha_i,q)+\alpha_i\bigg)\\ &&{\mathfrak R}_{\hbar_1=0}\left\{{\mathfrak R}_{\hbar_2=0} \left\{\mbox{e}^{-{\cal F}_{n}^{(0,0)}(\alpha_i,q)(\frac{1}{\hbar_1}+\frac{1}{\hbar_2})}\widetilde{{\cal Z}}_{ii}^*(\hbar_1,\hbar_2,q)
\right\}\right\}. \end{eqnarray*}
\end{prop}
\begin{proof}
Let $\Gamma$ be a decorated graph of $A_{i}$ type, then we can break $\Gamma$ into a strand $\Gamma_{\pm}$ at $v_0=\eta(1)$, where $\pm$ are points attached to $v_0$. Let $\mu(v_0)=i$. Thus ${\cal Z}_{\Gamma}=\overline{{\cal M}}_{0,(2,1)|\mathfrak{d}(v_0)}\times{\cal Z}_{\Gamma_{\pm}}$, let \begin{equation}
\pi_p:{\cal Z}_{\Gamma}\to\overline{{\cal M}}_{0,(2,1)|\mathfrak{d}(v_0)},\quad \mbox{and}\quad \pi_{e}:{\cal Z}_{\Gamma}\to{\cal Z}_{\Gamma_{\pm}} \end{equation} be the projections, then \begin{eqnarray*} \qquad \mathbf{e}({\cal V}_1)&=&\pi_p^*\mathbf{e}(\dot{{\cal V}}^{(\mathfrak{d}(v_0))}_{n;\mathbf a})\pi_{e}^*\mathbf{e}(\dot{{\cal V}}^{(d-\mathfrak{d}(v_0))}_{n;\mathbf a})\\ \frac{\mathbf{e}(T_{\mu(v_0)}\mathbb{P}^{n-1})}{\mathbf{e}({\cal N}{\cal Z}_{\Gamma})}&=&\frac{\prod_{k\neq i}(\alpha_i-\alpha_k)}{\prod_{k\neq i} \mathbf{e}(R^0\pi_{1*}{\cal S}^{\vee}(\alpha_i-\alpha_k))}\\ &&\frac{\mathbf{e}(T_{\mu(v_0)}\mathbb{P}^{n-1})\mathbf{e}(T_{\mu(v_0)}\mathbb{P}^{n-1})} {\mathbf{e}({\cal N}{\cal Z}_{\Gamma_{\pm}})\,(\hbar_{+}\!-\!\pi_e^*\psi_{+})\,(\hbar_{-}\!-\!\pi_e^*\psi_{-})}, \end{eqnarray*}
where $\hbar_{\pm}\!\equiv\!c_1(L_{\pm}')\!\in\!H^*(\overline{{\cal M}}_{0,(2,1)|\mathfrak{d}(v_0)})$, $L_{\pm}'$ be the universal tangent line bundle at the marked point corresponding to $\pm$.
Let $e_{1}$ be the line $\{v_{1},v_0\}$, where $v_{1}$ is the nearest point to $+$. Let $e_{2}$ be the line $\{v_{2},v_0\}$, where $v_{2}$ is the nearest point to $-$. Let $\Gamma_e$ be the strand obtained by $\Gamma_{\pm}$ eliminating $e_{1},\,e_{2}$. Therefore we have \begin{eqnarray}\label{A-type} &&\sum_{d=1}^{\infty}q^d\int_{{\cal Z}_{\Gamma}}\frac{\mathbf{e}({\cal V}_1)\hat{ev}_{1}^{*}\tilde\gamma}{\mathbf{e}({\cal N}{\cal Z}_{\Gamma})}\\
&=&\sum_{d=1}^{\infty}\sum_{\mathfrak{d}(v_0)\ge0}\frac{q^{\mathfrak{d}(v_0)}}{\mathfrak{d}(v_0)!}\int_{\overline{{\cal M}}_{0,(2,1)|\mathfrak{d}(v_0)}}\frac{\hat{ev}_{1}^*\tilde\gamma \,\mathbf{e}(\dot{{\cal V}}^{(\mathfrak{d}(v_0))}_{n;\mathbf a}(\alpha_i))}{\prod_{k\neq i} \mathbf{e}(R^0\pi_{1*}{\cal S}^{\vee}(\alpha_i-\alpha_k))}\nonumber\\ && q^{d-\mathfrak{d}(v_0)}\int_{{\cal Z}_{\Gamma_{\pm}}}\frac{\mathbf{e}(\dot{{\cal V}}^{(d-\mathfrak{d}(v_0))}_{n;\mathbf a})(\alpha_i)\text{ev}_1^*\phi_{i} \text{ev}_2^*\phi_{i}}{\mathbf{e}({\cal N}{\cal Z}_{\Gamma_{\pm}})\,(\hbar_{+}\!-\!\pi_e^*\psi_{+})\,(\hbar_{-}\!-\!\pi_e^*\psi_{-})}\nonumber\\
&=&\sum_{d=1}^{\infty}\sum_{\mathfrak{d}(v_0)\ge0}\frac{q^{\mathfrak{d}(v_0)}}{\mathfrak{d}(v_0)!}\int_{\overline{{\cal M}}_{0,(2,1)|\mathfrak{d}(v_0)}}\frac{\hat{ev}_{1}^*\tilde\gamma \,\mathbf{e}(\dot{{\cal V}}^{(\mathfrak{d}(v_0))}_{n;\mathbf a}(\alpha_i))\hbar_{+}^{b_{1}}\hbar_{-}^{b_{2}}}{\prod_{k\neq i} \mathbf{e}(R^0\pi_{1*}{\cal S}^{\vee}(\alpha_i-\alpha_k))}\nonumber\\ &&q^{d-\mathfrak{d}(v_0)}\int_{{\cal Z}_{\Gamma_{\pm}}}\frac{\mathbf{e}(\dot{{\cal V}}^{(d-\mathfrak{d}(v_0))}_{n;\mathbf a}(\alpha_i))\text{ev}_1^*\phi_{i} \text{ev}_2^*\phi_{i}\pi_e^*\psi_{+}^{-(b_1+1)}\pi_e^*\psi_{-}^{-(b_2+1)}}{\mathbf{e}({\cal N}{\cal Z}_{\Gamma_{\pm}})}\nonumber\\
&=&\sum_{d=1}^{\infty}\sum_{\mathfrak{d}(v_0)\ge0}\frac{q^{\mathfrak{d}(v_0)}}{\mathfrak{d}(v_0)!}\int_{\overline{{\cal M}}_{0,(2,1)|\mathfrak{d}(v_0)}}\frac{\hat{ev}_{1}^*\tilde\gamma \,\mathbf{e}(\dot{{\cal V}}^{(\mathfrak{d}(v_0))}_{n;\mathbf a}(\alpha_i))\hbar_{+}^{b_{1}}\hbar_{-}^{b_{2}}}{\prod_{k\neq i} \mathbf{e}(R^0\pi_{1*}{\cal S}^{\vee}(\alpha_i-\alpha_k))}q^{d-\mathfrak{d}(v_0)}\nonumber\\ &&\dot{\mathfrak{C}}_{i}^{\mu(v_{1})}(\mathfrak{d}(e_{1}))\bigg(-\frac{\alpha_{\mu(v_{1})}-\alpha_i}{\mathfrak{d}(e_{1})}\bigg)^{-(b_1+1)}\ddot{\mathfrak{C}}_{i}^{\mu(v_{2})} (\mathfrak{d}(e_{2}))\bigg(-\frac{\alpha_{\mu(v_{2})}-\alpha_{i}}{\mathfrak{d}(e_{2})}\bigg)^{-(b_{2}+1)}\nonumber\\
&&\int_{{\cal Z}_{\Gamma_{e}}}\frac{\mathbf{e}(\dot{{\cal V}}^{(d-\mathfrak{d}(v_0))-\mathfrak{d}(e_1)-\mathfrak{d}(e_2)}_{n;\mathbf a}(\alpha_i))\text{ev}_1^*\phi_{\mu(v_{1})} \text{ev}_2^*\phi_{\mu(v_{2})}}{\mathbf{e}({\cal N}{\cal Z}_{\Gamma_{e}})(\hbar_1-\psi_1)(\hbar_2-\psi_2)}|_{\hbar_1=\frac{\alpha_{\mu(v_{1})}-\alpha_i}{\mathfrak{d}(e_{1})}, \hbar_2=\frac{\alpha_{\mu(v_{2})}-\alpha_i}{\mathfrak{d}(e_{2})}},\nonumber \end{eqnarray} where $$ \dot{\mathfrak{C}}_{i}^{\mu(v_1)}(d)=\frac{\prod_{r=1}^{\ell}\prod_{l=1}^{a_{r}d}(a_{r}\alpha_i+\frac{l}{d}(\alpha_{\mu(v_1)}-\alpha_i))} {d{\prod_{l=1}^{d}\prod_{m=1}^n\atop(l,m)\neq(d,k)}(\alpha_i-\alpha_m+\frac{l}{d}(\alpha_{\mu(v_1)}-\alpha_i))}, $$
$$ \ddot{\mathfrak{C}}_{i}^{\mu(v_2)}(d)=\frac{\prod_{r=1}^{\ell}\prod_{l=0}^{a_{r}d-1}(a_{r}\alpha_i+\frac{l}{d}(\alpha_{\mu(v_2)}-\alpha_i))} {d{\prod_{l=1}^{d}\prod_{m=1}^n\atop(l,m)\neq(d,k)}(\alpha_{i}-\alpha_m+\frac{l}{d}(\alpha_{\mu(v_2)}-\alpha_i))}. $$
Denote by \begin{equation}
\overline{{\cal F}}^{(b_1,b_2)}=\sum_{\mathfrak{d}(v_0)=0}^{\infty}\frac{q^{\mathfrak{d}(v_0)}}{\mathfrak{d}(v_0)!}\int_{\overline{{\cal M}}_{0,(2,1)|\mathfrak{d}(v_0)}}\frac{\phi_{i}|_{p_i}\hat{ev}_{1}^*\tilde\gamma \,\mathbf{e}(\dot{{\cal V}}^{(\mathfrak{d}(v_0))}_{n;\mathbf a}(\alpha_i))\psi_{1}^{b_{1}}\psi_{2}^{b_{2}}}{\prod_{k\neq i} \mathbf{e}(R^0\pi_{1*}{\cal S}^{\vee}(\alpha_i-\alpha_k))}. \end{equation}
Let $D_{1\hat{1},2}\subset \overline{{\cal M}}_{0,(2,1)|\mathfrak{d}(v_0)}$ be the divisor whose general element is a two-component rational curve, with one component carrying the marked point 1 and $\hat{1}$ and the other carrying the marked point 2, where $\hat{1}$ means the infinitesimally marking. Then $\psi_2=D_{1\hat{1},2}$ on $\overline{{\cal M}}_{0,(2,1)|\mathfrak{d}(v_0)}$, and \begin{eqnarray*} \overline{{\cal F}}^{(b_1,b_2)}&=&\overline{{\cal F}}^{(b_1,0)}{\cal F}_{n}^{(0,b_2-1)}(\alpha_i,q)\\ &=&\overline{{\cal F}}^{(0,0)}{\cal F}_{n}^{(0,b_1-1)}(\alpha_i,q){\cal F}_{n}^{(0,b_2-1)}(\alpha_i,q). \end{eqnarray*}
Because $\overline{{\cal M}}_{0,(2,1)|\mathfrak{d}(v_0)}$ canonical isomorphic to the universal curve $${\cal C}_{\overline{{\cal M}}_{0,2|\mathfrak{d}(v_0)}}\to \overline{{\cal M}}_{0,2|\mathfrak{d}(v_0)},$$ we have $$ \overline{{\cal F}}^{(0,0)}=q\frac{d}{dq}{\cal F}_{n}^{(0,0)}(\alpha_i,q)+\alpha_i. $$
By the recursion formula \cite[(7-12)]{Zin2} and the formulas in page 484 of \cite[Section 7]{Zin2} \begin{eqnarray*} &&\frac{1}{2}\sum_{\Gamma_{\pm}}q^{d-\mathfrak{d}(v_0)}\dot{\mathfrak{C}}_{i}^{\mu(v_1)}(\mathfrak{d}(e_{1}))\bigg(-\frac{\alpha_{\mu(v_{1})}-\alpha_i}{\mathfrak{d}(e_{1})}\bigg)^{-(b_1+1)} \ddot{\mathfrak{C}}_{i}^{\mu(v_2)}(\mathfrak{d}(e_{2}))\bigg(-\frac{\alpha_{\mu(v_{2})}-\alpha_{i}}{\mathfrak{d}(e_{2})}\bigg)^{-(b_{2}+1)}\\ &&
\int_{{\cal Z}_{\Gamma_{e}}}\frac{\mathbf{e}(\dot{{\cal V}}^{(d-\mathfrak{d}(v_0))-\mathfrak{d}(e_1)-\mathfrak{d}(e_2)}_{n;\mathbf a}(\alpha_i))\text{ev}_1^*\phi_{\mu(v_{1})} \text{ev}_2^*\phi_{\mu(v_{2})}}{\mathbf{e}({\cal N}{\cal Z}_{\Gamma_{e}})(\hbar_1-\psi_1)(\hbar_2-\psi_2)}|_{\hbar_1=\frac{\alpha_{\mu(v_{1})}-\alpha_i}{\mathfrak{d}(e_{1})}, \hbar_2=\frac{\alpha_{\mu(v_{2})}-\alpha_i}{\mathfrak{d}(e_{2})}} \end{eqnarray*}
$$ ={\mathfrak R}_{\hbar_1=\frac{\alpha_{\mu(v_{1})}-\alpha_i}{\mathfrak{d}(e_{1})}}\left\{{\mathfrak R}_{\hbar_2=\frac{\alpha_{\mu(v_{2})}-\alpha_{i}}{\mathfrak{d}(e_{2})}} \left\{(-\hbar_1)^{-b_1}(-\hbar_2)^{-b_2}\widetilde{{\cal Z}}_{ii}^*(\hbar_1,\hbar_2,q)
\right\}\right\}. $$ Then by the residue theorem on $S^2$, and the vanishing of the residue at $\infty$, which can be directly obtained from the expression of $\widetilde{{\cal Z}}_{ii}^*(\hbar_1,\hbar_2,q)$. We have \begin{equation}\label{A-type1} \frac{1}{2}\sum_{\Gamma_{\pm}}q^{d-\mathfrak{d}(v_0)}\int_{{\cal Z}_{\Gamma_{\pm}}}\frac{\mathbf{e}(\dot{{\cal V}}^{(d-\mathfrak{d}(v_0))}_{n;\mathbf a}(\alpha_i))\text{ev}_1^*\phi_{i} \text{ev}_2^*\phi_{i}\pi_e^*\psi_{+}^{-(b_1+1)}\pi_e^*\psi_{-}^{-(b_2+1)}}{\mathbf{e}({\cal N}{\cal Z}_{\Gamma_{\pm}})} \end{equation} $$ ={\mathfrak R}_{\hbar_1=0}\left\{{\mathfrak R}_{\hbar_2=0} \left\{(-\hbar_1)^{-b_1}(-\hbar_2)^{-b_2}\widetilde{{\cal Z}}_{ii}^*(\hbar_1,\hbar_2,q)
\right\}\right\}. $$ Combining formula (\ref{A-type}) and (\ref{A-type1}), \begin{eqnarray*} &&\frac{1}{2}\sum_{d=1}^{\infty}q^d\int_{{\cal Z}_{\Gamma}}\frac{\mathbf{e}({\cal V}_1)\hat{ev}_{1}^{*}\tilde\gamma}{\mathbf{e}({\cal N}{\cal Z}_{\Gamma})}\\
&=&(\phi_{i}|_{p_{i}})^{-1}\sum_{b_1\ge0,b_2\ge0}\bigg((q\frac{d}{dq}{\cal F}_{n}^{(0,0)}(\alpha_i,q)+\alpha_i) \frac{{\cal F}_{n}^{(0,0)}(\alpha_i,q)^{b_1}}{b_1!}\frac{{\cal F}_{n}^{(0,0)}(\alpha_i,q)^{b_2}} {b_2!}\bigg)\\ &&{\mathfrak R}_{\hbar_1=0}\left\{{\mathfrak R}_{\hbar_2=0} \left\{(-\hbar_1)^{-b_1}(-\hbar_2)^{-b_2}\widetilde{{\cal Z}}_{ii}^*(\hbar_1,\hbar_2,q)
\right\}\right\}. \end{eqnarray*}
Thus \begin{eqnarray*} {\cal A}_{i}&=&\frac{1}{2}\sum_{d=1}^{\infty}q^d\int_{{\cal Z}_{\Gamma}}\frac{\mathbf{e}({\cal V}_1)\hat{ev}_{1}^{*}\tilde\gamma}{\mathbf{e}({\cal N}{\cal Z}_{\Gamma})}\\
&=&(\phi_{i}|_{p_{i}})^{-1}\bigg(q\frac{d}{dq}{\cal F}_{n}^{(0,0)}(\alpha_i,q)+\alpha_i\bigg)\\ &&{\mathfrak R}_{\hbar_1=0}\left\{{\mathfrak R}_{\hbar_2=0} \left\{\mbox{e}^{-{\cal F}_{n}^{(0,0)}(\alpha_i,q)(\frac{1}{\hbar_1}+\frac{1}{\hbar_2})}\widetilde{{\cal Z}}_{ii}^*(\hbar_1,\hbar_2,q)
\right\}\right\}. \end{eqnarray*}
\end{proof}
When the domain curve is mapped to the fixed point $p_{i}$, then the correpondent decorated graph $\Gamma$ is a vertex. They will be called $B_{i}$-types. In a graph of the $B_{i}$-type, the infinitesimally marked point~$1$ is attached to a vertex labeled~$i$.
Let ${\cal B}_{i}(q)$ be the total contribution from type $B_{i}$ graphs. Then
\begin{prop}\label{equivred_prp_B} For every $i\in\mathbf{n}$, \begin{eqnarray*} {\cal B}_{i}(q)&=&\frac{1}{24}q\frac{d}{d q}\bigg(c_{i}(\alpha){\cal F}^{(0,0)}(\alpha_i,q)-\log \dot\Phi^{(0)}(\alpha_i,q)\bigg), \end{eqnarray*} where $c_{i}(\alpha)=\sum_{k\neq i}\frac{1}{\alpha_k-\alpha_i}+\sum_{k=1}^{\ell}\frac{1}{a_k\alpha_i}$. \end{prop} \begin{proof}
The proof is exact the same as the calculation of the vertex contribution in \cite{KB}. We sketch it as following. Let $\Gamma$ be a type $B_{i}$ decorated graph, it is just vertex $v$ over $p_{i}$. Thus ${\cal Z}_{\Gamma}=\overline{{\cal M}}_{1,(0,1)|d}$. The contribution
\begin{equation} {\cal B}_{i}(q)=\sum_{d=1}^{\infty}\frac{q^d}{d!}
\int_{\overline{{\cal M}}_{1,(0,1)|d}}\frac{\hat{ev}_{1}^{*}\tilde\gamma\,\mathbf{e}(\mathbb{E} ^\vee \otimes T_{p_{i}}\mathbb{P}^{n-1})}{\mathbf{e} (T_{p_{i}}\mathbb{P}^{n-1})} \mathbf{e}({\cal V}_1(\alpha_i))\mathrm{Q}_{v} \end{equation} $$ =q\frac{d}{d q}\bigg(\sum_{d=1}^{\infty}\frac{q^d}{d!}
\int_{\overline{{\cal M}}_{1,0|d}}\frac{\mathbf{e}(\mathbb{E} ^\vee \otimes T_{p_{i}}\mathbb{P}^{n-1})}{\mathbf{e} (T_{p_{i}}\mathbb{P}^{n-1})} \mathbf{e}({\cal V}_1(\alpha_i))\mathrm{Q}_{v}\bigg) $$ where $\mathbb{E}$ is the Hodge bundle, \begin{equation}\label{Q-def}
\mathrm{Q}_{v}=\frac{1}{\prod\limits_{k\neq i}\!\!\mathbf{e}(H^0(C_{v},{\mathscr O}_{C_{v}}(D_1)|_{D_1})(\alpha_i\!-\!\alpha_k))},\end{equation} ${\mathscr O}_{C_{v}}(D_1)\cong{\cal S}^{\vee}|_{C_{v}}$.
$$
\mathbf{e}({\cal V}_1(\alpha_i))=\frac{\mathbf{e}(E|_{p_{i}})}{\mathbf{e}(\mathbb{E}^{\vee}\otimes E|_{p_{i}} )}\widetilde{Q}, $$ where
\begin{equation}\label{Q-def}
\widetilde{Q}=\frac{1}{\prod_{i=1}^{\ell}\mathbf{e}(H^0(C_{v},{\mathscr O}_{C_{v}}(\widetilde{D}_i)|_{\widetilde{D}_i})(\alpha_i))},\end{equation}
${\mathscr O}_{C_{v}}(\widetilde{D}_i)\cong{\cal S}^{\vee\otimes a_i}|_{C_{v}}$.
Let $c_{i}(\alpha)$ determined by $$
1+c_{i}(\alpha)\mathbf{e}(\mathbb{E})=\frac{\mathbf{e}(\mathbb{E} ^\vee \otimes T_{p_{i}}\mathbb{P}^{n-1})}{\mathbf{e} (T_{p_{i}}\mathbb{P}^{n-1})}\frac{\mathbf{e}(E|_{p_{i}})}{\mathbf{e}(\mathbb{E}^{\vee}\otimes E|_{p_{i}} )}. $$ Thus $$c_{i}(\alpha)=\sum_{k\neq i}\frac{1}{\alpha_k-\alpha_i}+\sum_{k=1}^{\ell}\frac{1}{a_k\alpha_i}. $$ Denote by $F^{(1,0)}_{d}=Q_v\,\widetilde{Q}$, then \begin{eqnarray*}
&&\int_{\overline{{\cal M}}_{1,0|d}}\frac{\mathbf{e}(\mathbb{E} ^\vee \otimes T_{p_{i}}\mathbb{P}^{n-1})}{\mathbf{e} (T_{p_{i}}\mathbb{P}^{n-1})} \mathbf{e}({\cal V}_1(\alpha_i))\mathrm{Q}_{v})\\
&&=\int_{\overline{{\cal M}}_{1,0|d}}(1+c_{i}(\alpha)\mathbf{e}(\mathbb{E}))F^{(1,0)}_{d}. \end{eqnarray*}
For nonnegative integers $g$ and $m$, the above expression for $F^{(1,0)}_d$ also defined as an element in $H^*(\overline{M}_{g,m|d},\mathbb{Q}_{\alpha})$, which can be written as a polynomial of diagonal classes and the psi classes. By the proof in \cite[Theorem 2.6]{KB}, we have $$
\int_{\overline{{\cal M}}_{1,0|d}}\mathbf{e}(\mathbb{E})\,F^{(1,0)}_{d}=\frac{1}{24}\int_{\overline{{\cal M}}_{0,2|d}}F^{(0,2)}_{d}, $$ $$
\sum_{d=1}^{\infty}\frac{q^d}{d!}\int_{\overline{{\cal M}}_{1,0|d}}F^{(1,0)}_{d}
=\frac{1}{24}\log(\sum_{d=0}^{\infty}\frac{q^d}{d!}\int_{\overline{{\cal M}}_{0,3|d}}F^{(0,3)}_{d}). $$ Therefore \begin{eqnarray}\label{part1} && \sum_{d=1}^{\infty}\frac{q^d}{d!}
\int_{\overline{{\cal M}}_{1,0|d}}\mathbf{e}(\mathbb{E})\,F^{(1,0)}_{d}\\ &=&\frac{1}{24}\sum_{d=1}^{\infty}\frac{q^d}{d!}
\int_{\overline{{\cal M}}_{0,(2,0)|d}}F^{(0,2)}_{d}\nonumber\\ &=&\frac{1}{24}\sum_{d=1}^{\infty}\frac{q^d}{d!}
\int_{\overline{{\cal M}}_{0,(2,0)|d}}\frac{\prod_{k\neq i}(\alpha_i-\alpha_k)\mathbf{e}(\dot{\cal V}_{n;\mathbf a}^{(d)}(\alpha_i))}{\prod_{k\neq i} \mathbf{e}(R^0\pi_{1*}{\cal S}^{\vee}(\alpha_i-\alpha_k))}\nonumber\\ &=&\frac{1}{24}{\cal F}^{(0,0)}(\alpha_i,q).\nonumber \end{eqnarray} By \cite[Proposition 4.1]{Zin2}, we have \begin{equation}\label{part2}
\sum_{d=0}^{\infty}\frac{q^d}{d!}\int_{\overline{{\cal M}}_{0,3|d}}F^{(0,3)}_{d}=\frac{1}{\dot\Phi^{(0)}(\alpha_i,q)}. \end{equation} Thus combining (\ref{part1}) and (\ref{part2}), we have
\begin{eqnarray*} {\cal B}_{i}(q)&=&\frac{1}{24}q\frac{d}{d q}\bigg(c_{i}(\alpha){\cal F}^{(0,0)}(\alpha_i,q)-\log \dot\Phi^{(0)}(\alpha_i,q)\bigg). \end{eqnarray*} \end{proof} Combining Proposition \ref{equivred_prp_A} and Proposition \ref{equivred_prp_B}, we have \begin{theo} \label{The}For Calabi-Yau manifold $X\subset \mathbb{P}^{n-1}$ which is complete intersection,
\begin{equation}\label{form}
q\frac{d}{d q}G_{1,0}=\bigg\{\sum_{i\in\mathbf{n}}\bigg({\cal A}_{i}(q)+\frac{1}{24}q\frac{d}{d q}\bigg(c_{i}(\alpha){\cal F}^{(0,0)}(\alpha_i,q)-\log \dot\Phi^{(0)}(\alpha_i,q)\bigg)\bigg)\bigg\}\bigg|_{\alpha=0}, \end{equation} where $c_{i}(\alpha)=\sum_{k\neq i}\frac{1}{\alpha_k-\alpha_i}+\sum_{k=1}^{\ell}\frac{1}{a_k\alpha_i}$. \end{theo}
\section{Calculation} In this section we work out the explicit expression of (\ref{form}) by using the hypergeometric series and their properties established in \cite{Popa}, \cite{Zin1} and \cite{Zin2}.
We define power series $L_{n},\xi_{n}\in\mathbb{Q}_{\alpha}[\mathbf{x}][[q]]$ by \begin{alignat}{2} L_{n}&\in \mathbf{x}+q\mathbb{Q}_{\alpha}[\mathbf{x}][[q]], &\qquad
\tilde{\mathbf{s}}_n\big(L_{n}(\mathbf{x},q)\big)-q\mathbf a^{\mathbf a}L_{n}(\mathbf{x},q)^{|\mathbf a|}&=\tilde{\mathbf{s}}_n(\mathbf{x}), \\ \xi_{n}&\in q\mathbb{Q}_{\alpha}[\mathbf{x}][[q]],&\qquad \mathbf{x}+q\frac{d}{d q}\xi_{n}(\mathbf{x},q)&=L_{n}(\mathbf{x},q),\notag \end{alignat} where $\tilde{\mathbf{s}}_r(y)$ is the r-th elementary symmetric polynomial in $\{y-\alpha_k\}$. By \cite[(4-9)]{Zin2}, $\xi_{n}(\alpha_i,q)={\cal F}_{n}^{(0,0)}(\alpha_i,q)$. Let $$L(q)=(1-\mathbf a^{\mathbf a}q)^{-\frac{1}{n}},\quad\quad\mu(q)=\int_{0}^q\frac{L(u)-1}{u}du,$$ then
\begin{equation}\label{L-for}
L_{n}(\mathbf{x},q)=L(q)\mathbf{x}+\sum_{d=0}^{\infty}f_d(\mathbf{x},\alpha)q^d,
\end{equation}
where $f_d(\mathbf{x},\alpha)\in \mathbb{Q}_{\alpha}[\mathbf{x}]$ with $\mathbf{x}|f_d(\mathbf{x},\alpha)$, and has no $\mathbf{x}$ term with constant coefficient, $f_d(\mathbf{x},0)=0$.
\begin{equation}\label{xi-for}
\xi_{n}(\mathbf{x},q)=\mu(q)\mathbf{x}+\sum_{d=1}^{\infty}g_d(\mathbf{x},\alpha)q^d,
\end{equation}
where $g_d(\mathbf{x},\alpha)\in \mathbb{Q}_{\alpha}[\mathbf{x}]$ with $\mathbf{x}|g_d(\mathbf{x},\alpha)$, and has no $\mathbf{x}$ term with constant coefficient, $g_d(\mathbf{x},0)=0$.
By the residue theorem on $S^2$, \begin{eqnarray*} \sum_{i=1}^n\sum_{k\neq i}\frac{\xi_{n}(\alpha_i,q)}{\alpha_k-\alpha_i}&=&-\sum_{i}{\mathfrak R}_{z=\alpha_i}\sum_{k\neq j}\frac{\mu(q)z}{(z-\alpha_j)(z-\alpha_k)}\\ &&-\sum_{d=0}^{\infty}q^d\sum_{i}{\mathfrak R}_{z=\alpha_i}\sum_{k\neq j}\frac{g_d(z,\alpha)}{(z-\alpha_j)(z-\alpha_k)}\\ &=&{\mathfrak R}_{z=\infty}\sum_{k\neq j}\frac{\mu(q)z}{(z-\alpha_j)(z-\alpha_k)}\\ &&+\sum_{d=0}^{\infty}q^d{\mathfrak R}_{z=\infty}\sum_{k\neq j}\frac{g_d(z,\alpha)}{(z-\alpha_j)(z-\alpha_k)}\\ &=&-\binom{n}{2}\mu(q)+\sum_{d=0}^{\infty}q^d{\mathfrak R}_{z=\infty}\sum_{k\neq j}\frac{g_d(z,\alpha)}{(z-\alpha_j)(z-\alpha_k)}. \end{eqnarray*} By (\ref{xi-for}), $$
\bigg({\mathfrak R}_{z=\infty}\sum_{k\neq j}\frac{g_d(z,\alpha)}{(z-\alpha_j)(z-\alpha_k)}\bigg)\bigg|_{\alpha=0}=0. $$ Thus $$
\bigg(\sum_{i=1}^n\sum_{k\neq i}\frac{\xi_{n}(\alpha_i,q)}{\alpha_k-\alpha_i}\bigg)\bigg|_{\alpha=0}=-\binom{n}{2}\mu(q). $$ \begin{eqnarray*} \sum_{i=1}^{n}\frac{\xi_{n}(\alpha_i,q)}{a_k\alpha_i}&=&\frac{1}{a_k}\sum_{i=1}^n{\mathfrak R}_{z=\alpha_i} \sum_{j=1}^{n}\frac{\xi_{n}(z,q)}{z(z-\alpha_j)}\\ &=&\frac{1}{a_k}\sum_{i=1}^n{\mathfrak R}_{z=\alpha_i} \sum_{j=1}^{n}\frac{\mu(q)}{(z-\alpha_j)}\\ &&+\frac{1}{a_k}\sum_{d=0}^{\infty}q^d\sum_{i=1}^n{\mathfrak R}_{z=\alpha_i} \sum_{j=1}^{n}\frac{g_d(z,\alpha)}{z(z-\alpha_j)}\\ &=&-\frac{1}{a_k}{\mathfrak R}_{z=\infty} \sum_{j=1}^{n}\frac{\mu(q)}{(z-\alpha_j)}\\ &&-\frac{1}{a_k}\sum_{d=0}^{\infty}q^d\sum_{i=1}^n{\mathfrak R}_{z=\infty} \sum_{j=1}^{n}\frac{g_d(z,\alpha)}{z(z-\alpha_j)}\\ &&-\frac{1}{a_k}\sum_{d=0}^{\infty}q^d\sum_{i=1}^n{\mathfrak R}_{z=0} \sum_{j=1}^{n}\frac{g_d(z,\alpha)}{z(z-\alpha_j)}, \end{eqnarray*} by (\ref{xi-for}) \begin{eqnarray*} \frac{1}{a_k}\sum_{d=0}^{\infty}q^d\bigg(\sum_{i=1}^n{\mathfrak R}_{z=0,\infty}
\sum_{j=1}^{n}\frac{g_d(z,\alpha)}{z(z-\alpha_j)}\bigg)\bigg|_{\alpha=0}=0. \end{eqnarray*} Thus $$
\bigg(\sum_{k=1}^{\ell}\sum_{i=1}^{n}\frac{\xi_{n}(\alpha_i,q)}{a_k\alpha_i}\bigg)\bigg|_{\alpha=0} =\sum_{k=1}^{\ell}\frac{n}{a_k}. $$
By \cite[(4-10)]{Zin2}
$$
\dot\Phi^{(0)}(\alpha_i,q)|_{\alpha=0}=L(q)^{\frac{\ell+1}{2}}. $$ Therefore \begin{eqnarray}\label{B-va}
&&\bigg(\sum_{i=1}^n{\cal B}_{i}(q)\bigg)\bigg|_{\alpha=0}\\
&=&\frac{1}{24}\bigg(\sum_{i=1}^n\bigg(c_{i}(\alpha){\cal F}^{(0,0)}(\alpha_i,q)-\log \dot\Phi^{(0)}(\alpha_i,q)\bigg)\bigg)\bigg|_{\alpha=0}\nonumber\\ &=&\frac{1}{24}\bigg(\sum_{i=1}^{\ell}(\frac{n}{a_i}-\binom{n}{2})\mu(q)-\frac{n(\ell+1)}{2}\log L(q)\bigg).\nonumber \end{eqnarray}
For each $p\in\mathbf{n}$, let $\sigma_p$ be the $p$-th elementary symmetric polynomial in $\alpha_1,\ldots,\alpha_n$. Denote by $$\mathbb{Q}[\alpha]^{S_n}\equiv\mathbb{Q}[\alpha_1,\ldots,\alpha_n]^{S_n}\subset\mathbb{Q} [\alpha_1,\ldots,\alpha_n]$$ the subspace of symmetric polynomials, by $\mathcal{J}\subset\mathbb{Q}[\alpha]^{S_n}$ the ideal generated by $\sigma_1,\ldots,\sigma_{n-1}$, and~by $$\tilde\mathbb{Q}[\alpha]^{S_n} \equiv
\mathbb{Q}[\alpha_1,\ldots,\alpha_n]_{<(\alpha_j-\alpha_k)|j\neq k>}^{S_n} \subset \mathbb{Q}_{\alpha}$$ the subalgebra of symmetric rational functions in $\alpha_1,\ldots,\alpha_n$ whose denominators are products of $(\alpha_j\!-\!\alpha_k)$ with $j\!\neq\!k$. For each $i\!=\!1,\ldots,n$, let $$\tilde\mathbb{Q}_i[\alpha]^{S_{n-1}} \equiv
\mathbb{Q}[\alpha_1,\ldots,\alpha_n]_{<(\alpha_i-\alpha_k)|k\neq i>}^{S_{n-1}} \subset \mathbb{Q}_{\alpha}$$ be the subalgebra consisting of rational functions symmetric in $\{\alpha_k\!:k\!\neq\!i\}$ and with denominators that are products of $(\alpha_i\!-\!\alpha_k)$ with $k\!\neq\!i$.
\begin{lemm}\label{good0} Let $f(z,\alpha)\in \mathcal{J}\mathbb{Q}[\alpha][z]$. Then for $\frac{f(\alpha_j,\alpha)}{\prod_{k\neq j}(\alpha_j-\alpha_k)^m}$ with $m\ge0$, we have \begin{eqnarray}
&&\bigg(\sum_{j=1}^n\frac{f(\alpha_j, \alpha)}{\prod_{k\neq j}(\alpha_j-\alpha_k)^{m+1}}\bigg)\bigg|_{\alpha=0}=0. \end{eqnarray} \end{lemm} \begin{proof}
By the residue theorem on $S^2$, when $m=0$, \begin{eqnarray*}
&&\bigg(\sum_{j=1}^n\frac{f(\alpha_j,\alpha)}{\prod_{k\neq j}(\alpha_j-\alpha_k)}\bigg)\bigg|_{\alpha=0}\\ &=&\bigg(\sum_{j=1}^n{\mathfrak R}_{z=\alpha_j}
\bigg\{\frac{ f(z,\alpha)}{\prod_{k=1}^n(z-\alpha_k)}\bigg\}\bigg)\bigg|_{\alpha=0}\nonumber\\ &=&-\bigg({\mathfrak R}_{\infty}
\bigg\{\frac{ f(z,\alpha)}{\prod_{k=1}^n(z-\alpha_k)}\bigg\}\bigg)\bigg|_{\alpha=0}\nonumber\\ &=&0.\nonumber \end{eqnarray*}
When $m\ge1$, \begin{eqnarray*}
&&\bigg(\sum_{j=1}^n\frac{f(\alpha_j,\alpha)}{\prod_{k\neq j}(\alpha_j-\alpha_k)^{m+1}}\bigg)\bigg|_{\alpha=0}\\ &=&\bigg(\sum_{j=1}^n{\mathfrak R}_{z=\alpha_j}
\bigg\{\sum_{i=1}^n\frac{ (z-\alpha_i)^m f(z,\alpha)}{\prod_{k=1}^n(z-\alpha_k)^{m+1}}\bigg\}\bigg)\bigg|_{\alpha=0}\nonumber\\ &=&-\bigg({\mathfrak R}_{\infty}
\bigg\{\sum_{i=1}^n\frac{ (z-\alpha_i)^mf(z,\alpha)}{\prod_{k=1}^n(z-\alpha_k)^{m+1}}\bigg\}\bigg)\bigg|_{\alpha=0}\nonumber\\ &=&0.\nonumber \end{eqnarray*} \end{proof}
\begin{lemm}\label{good'0} Let $d,m\in\mathbb{Z}^{\ge0}$ satisfy, $ m\neq (n-1)(d+1)$. Then
\begin{eqnarray}
&&\bigg(\sum_{j=1}^n\frac{\alpha_j^{m}}{\prod_{k\neq j}(\alpha_j-\alpha_k)^{d+1}}\bigg)\bigg|_{\alpha=0}=0. \end{eqnarray} \end{lemm} \begin{proof}
By the residue theorem on $S^2$, when $d=0$ \begin{eqnarray*}
&&\bigg(\sum_{j=1}^n\frac{\alpha_j^{m}}{\prod_{k\neq j}(\alpha_j-\alpha_k)}\bigg)\bigg|_{\alpha=0}\\ &=&\bigg(\sum_{j=1}^n{\mathfrak R}_{z=\alpha_j}
\bigg\{\frac{ z^{m}}{\prod_{k=1}^n(z-\alpha_k)}\bigg\}\bigg)\bigg|_{\alpha=0}\nonumber\\ &=&-\bigg({\mathfrak R}_{\infty}
\bigg\{\frac{ z^{m}}{\prod_{k=1}^n(z-\alpha_k)}\bigg\}\bigg)\bigg|_{\alpha=0}\nonumber\\ &=&0.\nonumber \end{eqnarray*} When $d\ge 1$, \begin{eqnarray*}
&&\bigg(\sum_{j=1}^n\frac{\alpha_j^{m}}{\prod_{k\neq j}(\alpha_j-\alpha_k)^{d+1}}\bigg)\bigg|_{\alpha=0}\\ &=&\bigg(\sum_{j=1}^n{\mathfrak R}_{z=\alpha_j}
\bigg\{\sum_{i=1}^n\frac{ (z-\alpha_i)^dz^m}{\prod_{k=1}^n(z-\alpha_k)^{d+1}}\bigg\}\bigg)\bigg|_{\alpha=0}\nonumber\\ &=&-\bigg({\mathfrak R}_{\infty}
\bigg\{\sum_{i=1}^n\frac{ (z-\alpha_i)^dz^m}{\prod_{k=1}^n(z-\alpha_k)^{d+1}}\bigg\}\bigg)\bigg|_{\alpha=0}\nonumber\\ &=&0.\nonumber \end{eqnarray*} \end{proof}
Let \begin{equation} F(w,q)\equiv\sum_{d=0}^{\infty}q^d \, \frac{\prod_{k=1}^{\ell}\prod\limits_{r=1}^{a_kd}(a_kw\!+\!r)} {\prod\limits_{r=1}^{d}(w\!+\!r)^n} \in \mathbb{Q}(w)\big[\big[q\big]\big],\end{equation}
\begin{equation}\dot{F}(w,q)\equiv\sum_{d=0}^{\infty}q^d \, \frac{\prod_{k=1}^{\ell}\prod\limits_{r=1}^{a_kd}(a_kw\!+\!r)} {\prod\limits_{r=1}^{d}\big((w\!+\!r)^n-w^n\big)} \in \mathbb{Q}(w)\big[\big[q\big]\big],\end{equation}
and \begin{equation}\ddot{F}(w,q)\equiv\sum_{d=0}^{\infty}q^d \, \frac{\prod_{k=1}^{\ell}\prod\limits_{r=0}^{a_kd-1}(a_kw\!+\!r)} {\prod\limits_{r=1}^{d}\big((w\!+\!r)^n-w^n\big)} \in \mathbb{Q}(w)\big[\big[q\big]\big].\end{equation}
These are power series in $q$ with constant term 1 whose coefficients are rational functions in $w$ which are regular at~$w=0$. We denote the subgroup of all such power series by~${\cal P}$ and define \begin{equation} \begin{aligned} &{\mathbf D}\!:\mathbb{Q}(w)\big[\big[q\big]\big]\longrightarrow \mathbb{Q}(w)\big[\big[q\big]\big], &\quad& \mathbf{M}:{\cal P}\longrightarrow{\cal P} \qquad\hbox{by}\\ &{\mathbf D} H(w,q)\equiv \left\{1+\frac{q}{w}\frac{d}{d q}\right\}H(w,q), &\quad& \mathbf{M} H(w,q)\equiv{\mathbf D}\bigg(\frac{H(w,q)}{H(0,q)}\bigg). \end{aligned}\end{equation} For $s\in\mathbb{Z}^{\ge0}$, let \begin{equation} \dot{I}_s(q)\equiv \mathbf{M}^s\dot{F}(0,q), \qquad \ddot{I}_s(q)\equiv \mathbf{M}^s\ddot{F}(0,q).\end{equation} By \cite[(4.14)]{Popa}, for $p\ge0$, $$ \mathbf{M}^{p}\dot{F}(w,q)=\mathbf{M}^{p+\ell}\ddot{F}(w,q). $$
Let $$\mathfrak{D}^0\check{{\cal Y}}(\mathbf{x},\hbar,q)=\frac{\check{{\cal Y}}(\mathbf{x},\hbar,q)}{\check{I}_0(q)}, \quad \mathfrak{D}^s\check{{\cal Y}}(\mathbf{x},\hbar,q)= \frac{1}{\check{I}_s(q)} \left\{\mathbf{x}+\hbar\, q\frac{d}{d q}\right\}\mathfrak{D}^{s-1}\check{{\cal Y}}(\mathbf{x},\hbar,q)$$ for all $s\!\in\!\mathbb{Z}^+$ and $(\check{{\cal Y}},\check{I})\!=\!({\cal Y},I), (\dot{\cal Y},\dot{I}), (\ddot{\cal Y},\ddot{I})$. For $r,s,s'\ge 0$, there exists ${\cal C}_{s,s'}^{(r)}\in\mathbb{Q}[\alpha_1,\cdots,\alpha_n][[q]]$, such that \begin{equation}\label{good3} \hbar^s\sum_{s'=0}^{\infty}\sum_{r=0}^{s'}{\cal C}_{s,s'}^{(r)}(q)\mathbf{x}^{s'-r}\hbar^{-s'} =\mathfrak{D}^s{\cal Y}(\mathbf{x},\hbar,q).\end{equation} \begin{gather}\label{good4}
\mathfrak{D}^s\check{{\cal Y}}(\mathbf{x},\hbar,q)\big|_{\alpha=0}= \mathbf{x}^s\mathfrak{D}^s\check{F}(\mathbf{x}/\hbar,q),\qquad\hbox{where}\\ \mathfrak{D}^0\check{F}(w,q)=\frac{\check{F}(w,q)}{\check{I}_0(q)}\,,\quad \mathfrak{D}^s\check{F}(w,q)=\frac{1}{\check{I}_s(q)}\left\{1+\frac{q}{w}\frac{d}{d q}\right\} \mathfrak{D}^{s-1}\check{F}(w,q) \quad\forall\,s\!\in\!\mathbb{Z}^+, \end{gather} with $(\check{{\cal Y}},\check{F},\check{I})\!=\!({\cal Y}, F,I), (\dot{\cal Y},\dot F,\dot{I}),(\ddot{\cal Y},\ddot F,\ddot{I})$. \begin{theo}\cite[Theorem 4]{Zin2}\label{main1}If $\ell\!\in\!\mathbb{Z}^{\ge0}$, $n\!\in\!\mathbb{Z}^+$, and $\mathbf a\!\in\!(\mathbb{Z}^{>0})^{\ell}$ are such that
$|\mathbf a|=\sum_{i=1}^{\ell}\!=\!n$, then \begin{equation}\label{main_1} \dot{{\cal Z}}(\hbar_1,\hbar_2,q) =\frac{1}{\hbar_1+\hbar_2}\sum_{s_1,s_2,r\ge0\atop s_1+s_2+r=n-1}(-1)^r\sigma_r\dot{{\cal Z}}^{(s_1)}(\mathbf{x},\hbar_1,q)\ddot{{\cal Z}}^{(s_2)}(\mathbf{x},\hbar_2,q), \end{equation} where $\sigma_r\in\mathbb{Q}_{\alpha}$ is the $r$-th elementary symmetric polynomial in $\alpha_1,\cdots,\alpha_n$. There exists $\widetilde{{\cal C}}_{s_1,s_2}^{(r)}\in\mathbb{Q}[\alpha][[q]]$ such that $$ \check{{\cal Z}}^{(s)}(\mathbf{x},\hbar,q)=\check{{\cal Y}}^{(s)}(\mathbf{x},\hbar,q):=\sum_{r=0}^s\sum_{s'=0}^{s-r} \widetilde{{\cal C}}_{s-\ell^*(\mathbf a),s'-\ell^*(\mathbf a)}^{(r)}(q)\,\hbar^{s-r-s'}\mathfrak{D}^{s'}\check{{\cal Y}}(\mathbf{x},\hbar,q), $$ where $(\check{{\cal Z}},\ell^*)=(\dot{\cal Z},0),(\ddot{\cal Z},\ell)$, $\check{{\cal Y}}=\dot{\cal Z},\ddot{\cal Z}$. \end{theo}
\begin{lemm}\label{good}
\begin{eqnarray}
&&\bigg(\sum_{i=1}^n(\phi_{i}|_{p_{i}})^{-1}\bigg(q\frac{d}{dq}{\cal F}_{n}^{(0,0)}(\alpha_i,q)+\alpha_i\bigg)\nonumber\\ &&{\mathfrak R}_{\hbar_1=0}\left\{{\mathfrak R}_{\hbar_2=0} \left\{\mbox{e}^{-{\cal F}_{n}^{(0,0)}(\alpha_i,q)(\frac{1}{\hbar_1}+\frac{1}{\hbar_2})}\widetilde{{\cal Z}}_{ii}^*(\hbar_1,\hbar_2,q)
\right\}\right\}\bigg)\bigg|_{\alpha=0}\nonumber\\
&=&\bigg(\sum_{i=1}^n(\phi_{i}|_{p_{i}})^{-1}\bigg(q\frac{d}{dq}{\cal F}_{n}^{(0,0)}(\alpha_i,q)+\alpha_i\bigg)\alpha_i^{n-1}
{\mathfrak R}_{\hbar_1=0}\bigg\{{\mathfrak R}_{\hbar_2=0}\nonumber\\ &&\frac{1}{2(\hbar_1+\hbar_2)\hbar_1\hbar_2} \bigg\{\mbox{e}^{-{\cal F}_{n}^{(0,0)}(\alpha_i,q)(\frac{1}{\hbar_1}+\frac{1}{\hbar_2})}
\mathbb{F}(\alpha_i/\hbar_1,\alpha_i/\hbar_2,q)\bigg\}\bigg\}\bigg)\bigg|_{\alpha=0},\nonumber \end{eqnarray}
where \begin{equation*}\begin{split} \mathbb{F}(w_1,w_2,q) &=\sum_{p=0}^{n-1-\ell}\frac{\mathbf{M}^p\dot{F}(w_1,q)}{\dot I_p(q)} \frac{\mathbf{M}^{n-1-p}\ddot{F}(w_2,q)}{\ddot{I}_{n-1-p}(q)}\\ &\qquad +\sum_{p=1}^{\ell}\frac{\mathbf{M}^{n-1+p}\ddot{F}(w_1,q)}{\ddot{I}_{n-1+p}(q)} \frac{\mathbf{M}^{n-p}\dot{F}(w_2,q)}{\dot{I}_{n-p}(q)}. \end{split}\end{equation*} \end{lemm} \begin{proof} By Theorem \ref{main1}, \begin{eqnarray*} &&\sum_{s_1,s_2,r\ge0\atop s_1+s_2+r=n-1}(-1)^r\sigma_r\alpha^{s_1+s_2}_i+2(\hbar_1+\hbar_2)\hbar_1\hbar_2\widetilde{{\cal Z}}^*_{ii} \\ &&=\sum_{s_1,s_2\ge0\atop s_1+s_2=n-1}\dot{{\cal Y}}^{(s_1)}(\alpha_i,\hbar_1,q)\ddot{{\cal Y}}^{(s_2)}(\alpha_i,\hbar_2,q)\\ &&+\sum_{s_1,s_2\ge0,r>0\atop s_1+s_2+r=n-1}(-1)^r\sigma_r\dot{{\cal Y}}^{(s_1)}(\alpha_i,\hbar_1,q)\ddot{{\cal Y}}^{(s_2)}(\alpha_i,\hbar_2,q). \end{eqnarray*} By the definition of $\dot{{\cal Y}}^{(s_1)}(\mathbf{x},\hbar_1,q)$ and $\ddot{{\cal Y}}^{(s_2)}(\mathbf{x},\hbar_1,q)$, and (\ref{xi-for}), the $q$ coefficient of \begin{eqnarray*} {\mathfrak R}_{\hbar_1=0}\bigg\{{\mathfrak R}_{\hbar_2=0}\frac{1}{2(\hbar_1+\hbar_2)\hbar_1\hbar_2} \bigg\{\mbox{e}^{-{\cal F}_{n}^{(0,0)}(\alpha_i,q)(\frac{1}{\hbar_1}+\frac{1}{\hbar_2})}&&\\ \sigma_r\dot{{\cal Y}}^{(s_1)}(\alpha_i,\hbar_1,q)\ddot{{\cal Y}}^{(s_2)}(\alpha_i,\hbar_2,q)\bigg\}\bigg\}&& \end{eqnarray*} satisfies the condition in Lemma \ref{good0}, for $r>0$. Thus
\begin{eqnarray}
&&\bigg(\sum_{i=1}^n(\phi_{i}|_{p_{i}})^{-1}\bigg(q\frac{d}{dq}{\cal F}_{n}^{(0,0)}(\alpha_i,q)+\alpha_i\bigg)\nonumber\\ &&{\mathfrak R}_{\hbar_1=0}\left\{{\mathfrak R}_{\hbar_2=0} \left\{\mbox{e}^{-{\cal F}_{n}^{(0,0)}(\alpha_i,q)(\frac{1}{\hbar_1}+\frac{1}{\hbar_2})}\widetilde{{\cal Z}}_{ii}^*(\hbar_1,\hbar_2,q)
\right\}\right\}\bigg)\bigg|_{\alpha=0}\nonumber\\
&=&\bigg(\sum_{i=1}^n(\phi_{i}|_{p_{i}})^{-1}\bigg(q\frac{d}{dq}{\cal F}_{n}^{(0,0)}(\alpha_i,q)+\alpha_i\bigg)\nonumber\\
&&{\mathfrak R}_{\hbar_1=0}\bigg\{{\mathfrak R}_{\hbar_2=0}\frac{1}{2(\hbar_1+\hbar_2)\hbar_1\hbar_2} \bigg\{\mbox{e}^{-{\cal F}_{n}^{(0,0)}(\alpha_i,q)(\frac{1}{\hbar_1}+\frac{1}{\hbar_2})}\nonumber\\ &&
\sum_{s_1,s_2\ge0\atop s_1+s_2=n-1}\dot{{\cal Y}}^{(s_1)}(\alpha_i,\hbar_1,q)\ddot{{\cal Y}}^{(s_2)}(\alpha_i,\hbar_2,q)\bigg\}\bigg\}\bigg)\bigg|_{\alpha=0}.\nonumber \end{eqnarray} By (\ref{good4}) and (\ref{xi-for}), the $q$ coefficient of \begin{eqnarray*} &&{\mathfrak R}_{\hbar_1=0}\bigg\{{\mathfrak R}_{\hbar_2=0}\frac{1}{2(\hbar_1+\hbar_2)\hbar_1\hbar_2} \bigg\{\mbox{e}^{-{\cal F}_{n}^{(0,0)}(\alpha_i,q)(\frac{1}{\hbar_1}+\frac{1}{\hbar_2})}\nonumber\\ && \bigg(\sum_{s_1,s_2\ge0\atop s_1+s_2=n-1}\dot{{\cal Y}}^{(s_1)}(\alpha_i,\hbar_1,q)\ddot{{\cal Y}}^{(s_2)}(\alpha_i,\hbar_2,q)- \alpha_i^{n-1}\mathbb{F}(\alpha_i/\hbar_1,\alpha_i/\hbar_2,q)\bigg)\bigg\}\bigg\} \end{eqnarray*}
is of the form as in Lemma \ref{good0} and Lemma \ref{good'0}. Thus \begin{eqnarray}
&&\bigg(\sum_{i=1}^n(\phi_{i}|_{p_{i}})^{-1}\bigg(q\frac{d}{dq}{\cal F}_{n}^{(0,0)}(\alpha_i,q)+\alpha_i\bigg)\nonumber\\ &&{\mathfrak R}_{\hbar_1=0}\left\{{\mathfrak R}_{\hbar_2=0} \left\{\mbox{e}^{-{\cal F}_{n}^{(0,0)}(\alpha_i,q)(\frac{1}{\hbar_1}+\frac{1}{\hbar_2})}\widetilde{{\cal Z}}_{ii}^*(\hbar_1,\hbar_2,q)
\right\}\right\}\bigg)\bigg|_{\alpha=0}\nonumber\\
&=&\bigg(\sum_{i=1}^n(\phi_{i}|_{p_{i}})^{-1}\bigg(q\frac{d}{dq}{\cal F}_{n}^{(0,0)}(\alpha_i,q)+\alpha_i\bigg)\alpha_i^{n-1}
{\mathfrak R}_{\hbar_1=0}\bigg\{{\mathfrak R}_{\hbar_2=0}\nonumber\\ && \frac{1}{2(\hbar_1+\hbar_2)\hbar_1\hbar_2}\bigg\{\mbox{e}^{-{\cal F}_{n}^{(0,0)}(\alpha_i,q)(\frac{1}{\hbar_1}+\frac{1}{\hbar_2})}
\mathbb{F}(\alpha_i/\hbar_1,\alpha_i/\hbar_2,q)\bigg\}\bigg\}\bigg)\bigg|_{\alpha=0}.\nonumber \end{eqnarray}
\end{proof} \begin{lemm}\cite[Lemma 5.4]{Popa}\label{good2} \begin{eqnarray*} {\mathfrak R}_{h_1=0}{\mathfrak R}_{h_2=0}\left\{ \frac{\mbox{e}^{-\mu(q)\alpha_i(\hbar_1^{-1}+\hbar_2^{-1})}}{\hbar_1\hbar_2(\hbar_1\!+\!\hbar_2)} \mathbb{F}(\alpha_i/\hbar_1,\alpha_i/\hbar_2,q)\right\}&= \alpha_i^{-1}L(q)^{-1}q\frac{d}{dq}\mathbb A(q), \end{eqnarray*} \end{lemm} where \begin{eqnarray*} \mathbb A(q)=\frac{n}{24}(n-1-2\sum_{r=1}^{\ell}\frac{1}{a_r})\mu(q)-\frac{3(n-1-\ell)^2+(n-2)}{24}\log(1-\mathbf a^{\mathbf a}q)\\ -\sum_{p=0}^{n-2-\ell}\binom{n-p-\ell}{2}\log \dot I_p(q). \end{eqnarray*} By Lemma \ref{good}, Lemma \ref{good2} and (\ref{good0}) \begin{eqnarray}\label{A-va}
&&\bigg(\sum_{i=1}^n{\cal A}_i\bigg)\bigg|_{\alpha=0}\\
&=&\bigg(\sum_{i=1}^n(\phi_{i}|_{p_{i}})^{-1}\bigg(q\frac{d}{dq}{\cal F}_{n}^{(0,0)}(\alpha_i,q)+\alpha_i\bigg)\nonumber\\ &&{\mathfrak R}_{\hbar_1=0}\left\{{\mathfrak R}_{\hbar_2=0} \left\{\mbox{e}^{-{\cal F}_{n}^{(0,0)}(\alpha_i,q)(\frac{1}{\hbar_1}+\frac{1}{\hbar_2})}\widetilde{{\cal Z}}_{ii}^*(\hbar_1,\hbar_2,q)
\right\}\right\}\bigg)\bigg|_{\alpha=0}\nonumber\\
&=&\frac{1}{2}q\frac{d}{dq}\mathbb A(q).\nonumber \end{eqnarray}
\begin{theo}\label{main_3} For projective complete intersection Calabi-Yau $X$,
$$ G_{1,0}=\frac{1}{2}\mathbb A(q)+\frac{1}{24}\bigg(\sum_{i=1}^{\ell}(\frac{n}{a_i}-\binom{n}{2})\mu(q)-\frac{n(\ell+1)}{2}\log L(q)\bigg), $$ where \begin{eqnarray*} \mathbb A(q)=\frac{n}{24}(n-1-2\sum_{r=1}^{\ell}\frac{1}{a_r})\mu(q)-\frac{3(n-1-\ell)^2+(n-2)}{24}\log(1-\mathbf a^{\mathbf a}q)\\ -\sum_{p=0}^{n-2-\ell}\binom{n-p-\ell}{2}\log \dot I_p(q). \end{eqnarray*} \end{theo} \begin{proof} The proof is just combining Theorem \ref{The}, (\ref{A-va}) and (\ref{B-va}). \end{proof}
\end{document} | arXiv |
\begin{document}
\title{Rank Preserving Maps on CSL Algebras}
\author{Jaedeok Kim}
\address{Department of Mathematical Computing and Information Sciences, Jacksonville State University, Jacksonville, AL 36265}
\email{[email protected]}
\author{Robert L. Moore} \address{Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487 }\email{[email protected]}
\subjclass{Primary 47L35, ; Secondary 47A67} \date{January 1, 1994 and, in revised form, June 22, 1994.}
\keywords{Completely Distributive CSL Algebras, Rank Preserving Maps}
\begin{abstract} We give a description of a weakly continuous rank preserving map on a reflexive algebra on complex Hilbert space with commutative completely distributive subspace lattice. We show that the implementation of a rank preserving map can be described by the combination of two different types of maps. We also show that a rank preserving map can be implemented by only one type if the corresponding lattice is irreducible. We present some examples of both types of rank preserving map. \end{abstract}
\maketitle
\section{PRELIMINARIES}
It has been of special interest to study the linear maps between two
nonself\-adjoint operator algebras over the last several decades.
These maps include isomorphisms, isometries and rank preserving
maps etc. In \cite{MR33:4703}, Ringrose proved that an isomorphism between two
nest algebras must be spatially implemented by making effective
use of rank one operators. In \cite{MR87k:47103}, Gilfeather and Moore showed
that an isomorphism acting between two CSL algebras with
completely \mbox{distributive} lattices need not be spatially or
quasi-spatially implemented. But they also showed that if an
isomorphism preserves the rank of all finite rank operators, then
it must be quasi-spatially implemented. In \cite{MR17:09095}, Panaia has shown that
if $\mathfrak{A}_i=\operatorname{Alg}{\mathfrak{L}}_i$ where the $\mathfrak{L}_i$ are finite distributive
subspaces lattices, then every rank preserving algebraic isomorphism of $\mathfrak{A}_1$ onto
$\mathfrak{A}_2$ is quasi-spatial. The characterization of
the rank preserving maps on nest algebras was done by the Chinese
mathematicians, Shu-Yun Wei and Sheng-Zhao Hou \cite{MR2000m:47099}. Part of their results showed that some rank preserving maps $\Phi$ on a nest algebra can be described as $\Phi(T)=ATB^*$ for some $A,B \in B(\mathcal{H})$. Use of rank one operators was an integral part of their proof. In this paper, we shall describe the rank preserving maps on completely distributive commutative subspace lattice algebras (CDCSL algebras).
In this paper we show that a rank preserving map $\Phi$ on a completely distributive algebra, CDC algebra, with certain condition can be described as $\Phi(A)=UAV^*$ for some densely defined linear transformations $U$ and $V$.
Let $\mathcal{H}$ be a complex separable Hilbert space. A \emph{subspace lattice} $\mathfrak{L}$ is a strongly closed lattice of orthogonal projections on $\mathcal{H}$, containing $0$ and $I$. If $\mathfrak{L}$ is a subspace lattice, $\operatorname{Alg}{\mathfrak{L}}$ denotes the algebra of all bounded operators on $\mathcal{H}$ that leave invariant every projection in $\mathfrak{L}$. $\operatorname{Alg}{\mathfrak{L}}$ is a weakly closed subalgebra of $B(\mathcal{H})$, the algebra of all bounded operators on $\mathcal{H}$. Dually, if $\mathfrak{A}$ is a subalgebra of $B(\mathcal{H})$, then $\operatorname{Lat}{\mathfrak{A}}$ is the lattice of all projections invariant for each operator in $\mathfrak{A}$. An algebra $\mathfrak{A}$ is reflexive if $\mathfrak{A}=\operatorname{Alg}\operatorname{Lat}{\mathfrak{A}}$ and a lattice $\mathfrak{L}$ is reflexive if $\mathfrak{L}=\operatorname{Lat}\operatorname{Alg}{\mathfrak{L}}$. A lattice is a \emph{commutative subspace lattice}, or CSL, if each pair of projections in $\mathfrak{L}$ commute; $\operatorname{Alg}{\mathfrak{L}}$ is then called a CSL algebra. All lattices in this paper will be commutative.
In \cite{MR51:1420}, Arveson showed that every commutative subspace lattice is reflexive. A totally ordered (and hence commutative) subspace lattice is a nest and the associated algebra is a nest algebra. We use the convention that a subspace is identified with the orthogonal projection onto the subspace. Thus $E \subseteq F$ is the same as $E \leq F$.
We have two operations between subspaces: meet($\wedge$) and join($\vee$).
Given any family $\{E_{\alpha}\}_{\alpha \in I}$ of subspaces of a Hilbert space $\mathcal{H}$, $\wedge_{\alpha\in I}E_{\alpha}$ denotes the greatest subspace contained in each $E_{\alpha}$ and $\vee_{\alpha\in I}E_{\alpha}$ denotes the smallest subspace containing each $E_{\alpha}$.
We now restrict our attention to a special type of lattice: completely distributive lattice. One of the advantages of completely distributive lattices over other lattices is the abundance of the rank one operators in the associated algebra; these operators will be building blocks in the subsequent discussion. A \emph{completely distributive lattice} is a complete lattice which satisfies distributive laws expressed for families of arbitrary cardinality. To be more precise, let $\mathfrak{L}$ be a complete lattice and $I$ be an arbitrary index set. For each $\alpha \in I$, let $J_{\alpha}$ also be an arbitrary index set, and for each $\beta \in J_{\alpha}$, let $E_{\beta \alpha}$ denote an element of $\mathfrak{L}$. $\Pi$ will denote the Cartesian product of all the $J_{\alpha}$, i.e. the collection of all choice functions $\varphi:I \rightarrow\cup_{\alpha \in I}J_{\alpha}$ satisfying $\varphi(\alpha) \in J_{\alpha}$, for all $\alpha$. $\mathfrak{L}$ is completely distributive if it satisfies the following two identities.
\begin{enumerate} \item $\wedge_{\alpha \in I}(\vee_{\beta \in J_{\alpha}}E_{\beta\alpha})=\vee_{\varphi \in \Pi}(\wedge_{\alpha \in I}E_{\varphi(\alpha)\alpha})$
\item $\vee_{\alpha \in I}(\wedge_{\beta \in J_{\alpha}}E_{\beta\alpha})=\wedge_{\varphi \in \Pi}(\vee_{\alpha \in I}E_{\varphi(\alpha)\alpha})$ \end{enumerate}
If $\mathfrak{L}$ is completely distributive and commutative, we will call $\operatorname{Alg}{\mathfrak{L}}$ a CDC algebra. For further discussion about the characteristics of CDC algebra, we need to introduce the notion of rank one operators. We will let $x\otimes y^{\ast}$ denote the rank one operator defined on $\mathcal{H}$ by $(x\otimes y^{\ast})(f)=\langle f,y \rangle x$.
There are plenty of rank one operators and finite rank operators in a nest algebra. Moreover, these operators have played a central role in the theory of nest algebra. However, there are many examples of commutative subspace lattices whose corresponding algebras do not contain any rank one operators \cite{MR84h:47048}. But as we discussed earlier, the complete distributivity of a lattice guarantees a reasonable supply of rank one operators. Furthermore, one of the surprising characteristics of the CDC algebra is stated in the following lemma. This lemma is due to Laurie and Longstaff \cite{MR85b:47052}.
\begin{lemma}\label{wk} Let $\mathfrak{L}$ be a commutative completely distributive subspace lattice on a separable Hilbert space $\mathcal{H}$. Let $R_{\mathfrak{L}}$ denote the linear span of the rank one operators in $\operatorname{Alg} \mathfrak{L}$. Then $R_{\mathfrak{L}}$ is dense in $\operatorname{Alg}\mathfrak{L}$ in any of the weak, strong, ultraweak or ultrastrong topologies. \end{lemma}
It is an interesting question to ask when $x \otimes y^{\ast}$ belongs to $\operatorname{Alg}\mathfrak{L}$. The following lemma, due to Longstaff \cite{MR53:1294}, is the answer to this question, and the use of this lemma is essential.
\begin{lemma}\label{lsl1} The operator $x \otimes f^{\ast} $ belongs to $\operatorname{Alg} \mathfrak{L}$ if and only if there is a projection $E \in \mathfrak{L}$ such that $x \in E $ and $f \in E_{-}^{\perp}$. \end{lemma}
Let $\mathfrak{L}$ be a subspace lattice. For $M \in
\mathfrak{L}$, define $M_{-}$ by $M_{-} = \vee \{N|M \neq N, N \in \mathfrak{L}\}$. We end this section with a couple of lemmas that will be used repeatedly.
\begin{lemma}\label{cdl1} Let $\mathfrak{L}$ be a commutative completely distributive lattice. Then
\[\vee \{N | N \in \mathfrak{L}, N_{-} \neq I \}=I \]
and
\[\vee \{N_{-}^{\perp}| N \in \mathfrak{L}, N \neq 0 \}=I \] \end{lemma}
\begin{lemma}\label{ul} Let $\mathfrak{L}$ be a subspace lattice and $\{E_k\}_{k=1}^n$ be a finite subset of $\mathfrak{L}$. Then $(\vee_{k=1}^n E_k)_-=\vee_{k=1}^n (E_k)_- $ and $(\wedge_{k=1}^n E_k)_- \leq \wedge_{k=1}^n (E_k)_- $ \end{lemma} \begin{proof} It is sufficient to prove that the statements are true in case $n=2$. Let $E$ and $F$ denote two elements in $\mathfrak{L}$. Since $E \leq E \vee F$ and $F \leq E \vee F$, we have $E_- \leq (E \vee F)_-$ and $F_- \leq (E \vee F)_-$. These imply that \[ E_- \vee F_- \leq (E \vee F)_-. \]
On the other hand, if $K \in \{G
\in \mathfrak{L} \, | \, G \ngeq E \vee F \}$, then $K \in \{G \in
\mathfrak{L} \, | \, G \ngeq E\}$ or $K \in \{G \in \mathfrak{L}
\, | \, G \ngeq F\}$, so $K \leq E_-$ or $K \leq F_-$. These imply that $K \leq E_- \vee F_-$. Hence $ (E \vee F)_- \leq E_- \vee F_-.$ This proves $(1)$.
Let $E$ and $F$ be two elements in $\mathfrak{L}$. Since $E \wedge F \leq E$ and $E \wedge F \leq F$, $(E \wedge F)_- \leq E_-$ and $(E \wedge F)_- \leq F_-$. Therefore $(E \wedge F)_- \leq E_- \wedge F_-$. The results follow. \end{proof}
Now let's discuss two important types of linear maps on $\operatorname{Alg} \mathfrak{L}$.
\begin{definition} Let $\mathfrak{L}$ be a commutative subspace lattice. \begin{enumerate} \item A linear map $\Phi:\operatorname{Alg}\mathfrak{L}\rightarrow \operatorname{Alg}\mathfrak{L}$ is called an \emph{isomorphism} if $\Phi$ is a bijection and a multiplication preserving map. \item A linear map $\Phi:\operatorname{Alg} \mathfrak{L}\rightarrow \operatorname{Alg}\mathfrak{L}$ is called a \emph{rank preserving map} if $rank(\Phi(A))=rank(A)$ for each finite rank operator $A$ in $\operatorname{Alg}\mathfrak{L}$. \end{enumerate} \end{definition}
The automatic norm continuity of an isomorphism is proved by Gilfeather and Moore in \cite{MR87k:47103}. In order to discuss more results about isomorphisms, we need the following terminology. An isomorphism $\Phi$ is said to be \emph{spatially implemented} if there is a bounded invertible operator $T$ on $\mathcal{H}$ so that $\Phi(A)=TAT^{-1}$ for all $A \in \operatorname{Alg}(\mathfrak{L})$. An isomorphism is said to be \emph{quasi}-\emph{spatially implemented} if there is a one-to-one operator with dense domain $\mathcal{D}$ so that $\Phi(A)Tf=TAf$ for all $A \in \operatorname{Alg}\mathcal{L}$ and for all $f \in \mathcal{D}$.
The following theorem characterizes isomorphisms of CSL algebras. The theorem is due to Gilfeather and Moore \cite{MR87k:47103}.
\begin{theorem} Let $\mathfrak{L}_1$ and $\mathfrak{L}_2$ be commutative subspace lattices on Hilbert spaces $\mathcal{H}_1$ and $\mathcal{H}_2$, respectively, and let $\mathfrak{L}_1$ be completely distributive. Let $\rho:\operatorname{Alg}\mathfrak{L_1}\rightarrow \operatorname{Alg} \mathfrak{L_2}$ be an algebraic isomorphism. The followings are equivalent. \begin{enumerate} \item $\rho$ is quasi-spatially implemented by a closed, injective linear transformation $T:H_1 \rightarrow H_2$ whose range and domain are dense. \item $\rho$ is a rank-preserving map. \end{enumerate} \end{theorem}
The weak continuity of a rank preserving map $\Phi$ will be used in the proof of the main Theorem. However, we set the assumption that \emph{each rank preserving map $\Phi$ is weakly continuous} from this early stage to avoid any misunderstanding.
\section{RANK PRESERVING LINEAR MAPS} We begin with a lemma which will play a key role in our result. For notational convenience, we will let $a \sim b$ represent the linear dependency of two nonzero vectors $a,b$ in $\mathcal{H}$. It can be easily verified that $a \sim b$ is an equivalence relation.
Suppose $\mathfrak{L}$ is a commutative subspace lattice and $\Phi$ is a rank preserving map from $\operatorname{Alg} \mathfrak{L}$ to $\operatorname{Alg} \mathfrak{L}$. Let $N$ be an element in $\mathfrak{L}$ with $\dim N \geq 2, \, \dim N_-^{\perp} \geq 2$. Let $x,y$ be any two linearly independent vectors in $N$ and $f,g$ be any two linearly independent vectors in $N_-^{\perp}$. Then we can form four different rank one operators in $\operatorname{Alg}\mathfrak{L}$ as follows: $x \otimes f^{\ast},y \otimes f^{\ast},x \otimes g^{\ast},y \otimes g^{\ast}$.
Since $\Phi$ is a rank preserving map on $\operatorname{Alg}\mathfrak{L}$, we can consider another four rank one operators in $\operatorname{Alg} \mathfrak{L}$ which are images of each of the four rank one operators and write them as follows. \begin{align*}
\Phi(x \otimes f^{\ast})&= u \otimes v^{\ast} &&\cdots \cdots (A) \\
\Phi(y \otimes f^{\ast})&= p \otimes q^{\ast} &&\cdots \cdots (B) \\
\Phi(x \otimes g^{\ast})&= w \otimes z^{\ast} &&\cdots \cdots (C) \\
\Phi(y \otimes g^{\ast})&= r \otimes s^{\ast} &&\cdots \cdots (D) \end{align*}
\begin{lemma} (Four Vectors Lemma)\label{fvl}
Let $\mathfrak{L}$ be a commutative subspace lattice and $\Phi$ be a rank preserving map on $\operatorname{Alg}\mathfrak{L}$. Let $N$ be a subspace in $\mathfrak{L}$ with $ \dim N \geq 2, \, \dim N_-^{\perp} \geq 2$. If we consider the eight rank one operators described above, then either $(1)$ or $(2)$ holds for all $x,y \in N$ and $f,g \in N_-^{\perp}$.
\begin{enumerate}
\item $u \sim w, v \sim q, p \sim r \, and \, z \sim s$
\item $v \sim z, u \sim p, w \sim r \, and \, q \sim r$ \end{enumerate} \end{lemma}
\begin{proof} Let's consider the following six equalities which are obtained by adding two of $ (A), (B),(C) \, and \, (D) $.
$(A)+(C):\Phi(x \otimes(f+g)^*)=u\otimes v^*+w\otimes z^*$
$(A)+(B):\Phi((x + y)\otimes f^*)=u\otimes v^*+p\otimes q^*$
$(B)+(D):\Phi(y \otimes(f+g)^*)=p\otimes q^*+r\otimes s^*$
$(C)+(D):\Phi((x+y) \otimes g^*)=w\otimes z^*+r\otimes s^*$
$(A)+(D):\Phi(x \otimes f^*+y \otimes g^*)=u\otimes v^*+r\otimes s^*$
$(B)+(C):\Phi(x \otimes g^*+y\otimes f^*)=p\otimes q^*+w\otimes z^*$
Since $x\otimes (f+g)^*$ is a rank one operator and $\Phi$ is a rank preserving map, $u\otimes v^*+w\otimes z^*$ is also a rank one operator. This fact implies either $u\sim w$ or $v\sim z$. Similarly we can argue that either $u\sim p$ or $v\sim q$ is true by the equality $(A)+(B)$, either $p\sim r$ or $q\sim s$ is true by the equality $(B)+(D)$ and $w\sim r$ or $z\sim s$ is true by the fourth equality $(C)+(D)$. Apart from the first four equalities, the map $\Phi$ maps a rank two operator into a rank two operator in the equality $(A)+(D)$ and$(B)+(C)$, which gives the following results \[u \nsim r, v\nsim s, p\nsim w \, \textrm{and} \, q\nsim z .\] \underline{Case 1:} Now let's suppose $u\sim w$ is true. Then it is easily deduced that $v\sim q$. Otherwise $u\sim p$ is true implying $w \sim p$, which contradicts the relation $w \nsim p$. Similarly, we can show that $p\sim r, z\sim s$ are true, so the result follows. \\ \underline{Case 2:} Suppose $v\sim z$. The proof of this case is almost identical to the proof of Case 1. \end{proof}
\begin{remark} If we assume that \emph{Case 1} holds, then $u \nsim p$. \end{remark}
If we consider the case $\dim N=1$ and $\dim N_{-}^{\perp}\geq 2$, we have the following two equalities. \begin{align*}
\Phi(x \otimes f^{\ast})&= u \otimes v^{\ast} &&\cdots \cdots (E)
\\
\Phi(x \otimes g^{\ast})&= p \otimes q^{\ast} &&\cdots \cdots (F) \end{align*} where $x \in N$ and $f,g$ are two linearly independent vectors in $N_{-}^{\perp}$.
The following lemma is analogous to Lemma \ref{fvl} for this case.
\begin{lemma}\label{tvl}
Let $\mathfrak{L}$ be a commutative subspace lattice and $\Phi$ be a rank preserving map on $\operatorname{Alg}\mathfrak{L}$. Let $N$ be a subspace in $\operatorname{Alg}\mathfrak{L}$ with $\dim N=1$ and $\dim N_{-}^{\perp}\geq 2$. If we consider the two equalities described above, then either $(1)$ or $(2)$ holds for all $x \in N$ and $f,g \in N_-^{\perp}$.
\begin{enumerate}
\item $ u \sim p , \, v \nsim q $
\item $ u \nsim p , \, v \sim q $
\end{enumerate} \end{lemma}
\begin{proof} By adding the two equalities, the following equality is obtained.
$(E)+(F): \Phi(x \otimes (f+g)^*)=u \otimes v^* + p \otimes
q^*$
From the fact that $\Phi$ maps a rank one operator into a rank one operator,
it can be easily deduced that either $ u \sim p$ or $ v \sim q$
is true. Suppose $u \sim p$ and $v \sim q$. Then there exist
$\lambda, \gamma \in \mathbb{C}$ such that $\lambda u \otimes v^* + \gamma p
\otimes q^*=0$. Therefore $ \Phi(x \otimes
(\overline{\lambda} f + \overline{\gamma} g)^*) = \lambda u \otimes v^* + \gamma
p\otimes q^*=0$ which is a contradiction that $\Phi$ maps a rank one operator into
a rank zero operator, so the result follows.
\end{proof}
The next lemma states a different version of Lemma \ref{tvl} for the case $\dim N \geq 2$ and
$\dim N_{-}^{\perp}=1$. The proof of this lemma is almost same
as the proof of Lemma \ref{tvl}.
\begin{lemma} \label{tvl2}
Let $\mathfrak{L}$ be a
commutative subspace lattice and $\Phi$ be a rank preserving map
on $\operatorname{Alg}\mathfrak{L}$. Let $N$ be a subspace in $\operatorname{Alg}\mathfrak{L}$
with $\dim N \geq 2$ and $\dim N_{-}^{\perp}=1$. If we consider the following two equalities,
\begin{align*}
\Phi(x \otimes f^{\ast})&= u \otimes v^{\ast} &&\cdots \cdots (G)
\\
\Phi(y \otimes f^{\ast})&= p \otimes q^{\ast} &&\cdots \cdots (H)
\end{align*}
where $ x,y $ are two linearly independent vectors in $N$ and
$f \in N_-^{\perp}$, then one of the following holds.
\begin{enumerate}
\item $ u \sim p , \, v \nsim q $
\item $ u \nsim q , \, v \sim q $
\end{enumerate} \end{lemma}
With these three lemmas in hand, we now can show the following important result. \begin{lemma}\label{ml} Let $\mathfrak{L}$ be a commutative subspace lattice and $\Phi$ be a rank preserving map on $\operatorname{Alg}\mathfrak{L}$. Let $N$ be a nonzero element in $\mathfrak{L}$ with $N_{-}\neq I$. Then at least one of the following holds. \begin{enumerate} \item There exist a linear map $U$ from $N$ to $\mathcal{H}$ and a linear map $V$ from $N_{-}^{\perp}$ to $\mathcal{H}$ such that $\Phi(x \otimes f^*)=U(x) \otimes V(f)^*$ for all $x \in N$ and $f \in N_-^{\perp}.$
\item There exist a conjugate linear map $U$ from $N_{-}^{\perp}$ to $\mathcal{H}$ and a conjugate linear map $V$ from $N$ to $\mathcal{H}$ such that $\Phi(x \otimes f^*)=U(f) \otimes V(x)^*$ for all $x \in N$ and $f \in N_-^{\perp}.$ \end{enumerate} \end{lemma}
\begin{proof} Suppose $(1)$ holds for all $x,y \in N$ and $f,g \in N_-^{\perp}$ in Lemma \ref{fvl}, Lemma \ref{tvl} and Lemma \ref{tvl2}.
\underline{Case 1:} Suppose that $\dim N=1$ and $\dim N_{-}^{\perp}=1$. Fix $x_1 \in N, f_1 \in
N_-^{\perp}$. Set $\Phi(x_1 \otimes f_1^*)=u_1 \otimes
v_1^*$. For any $x \in N, f \in N_-^{\perp}$, there exist
$\lambda, \gamma \in \mathbb{C}$ such that $x=\lambda x_1, f=\gamma
f_1$. If we define $U:N \rightarrow \mathcal{H} $ and $V:
N_-^{\perp} \rightarrow \mathcal{H}$ by
$U(x)=\lambda u_1$ and $V(f)=\gamma v_1$, then these maps are
clearly well defined linear maps. However, if we define $U:N \rightarrow \mathcal{H} $
and $V:N_-^{\perp} \rightarrow \mathcal{H}$ by $U(x)=\lambda v_1$ and $V(f)=\gamma
u_1$, these maps are also well defined linear maps. Therefore
both conclusions $(1)$ and $(2)$ hold in this special case.
\underline{Case 2:} Suppose that $\dim N=1$ and $\dim N_{-}^{\perp}\geq 2$. Fix $x_1 \in N$. It follows from Lemma \ref{tvl} (1) that it is possible to to define a map $V$ on $N_-^{\perp}$ by $\Phi(x_1 \otimes f^*)=u_1 \otimes V(f) $. For any $f, g \in N_{-}^{\perp}, t\in \mathbb{C}$, since $\Phi$ is linear,
\begin{align*}
\Phi(x_1\otimes (f+tg)^*)&=\Phi(x_1\otimes
f^*)+\Phi(x_1\otimes(tg)^*)\\
&=u_1\otimes V(f)^* +\overline{t}u_1\otimes V(g)^*.
\end{align*}
From the definition of $V$, $\Phi(x_1\otimes (f+tg)^*)=u_1\otimes V(f+tg)^*$. By comparing the two equalities, we see the map $V:N_-^{\perp} \rightarrow \mathcal{H}$
is a linear map on $N_{-}^{\perp}$.
For any $x \in N$, there exists $\lambda \in \mathbb{C}$ such that $x=\lambda x_1$. Now we can define a linear map $U:N \rightarrow
\mathcal{H}$ by $U(x)=\lambda u_1$. Therefore the conclusion $(1)$ holds with this assumption. Likewise, the conclusion $(2)$ follows if we assume $(2)$ in Lemma \ref{tvl} is true.
\underline{Case 3:} Suppose that $\dim N \geq 2$ and $\dim N_{-}^{\perp}= 1$. The proof of this case is almost identical to the proof of case 2.
\underline{Case 4:}
Let's fix $x_1\in N$. By Lemma \ref{fvl} (1), we can define a
map $V$ from $N_{-}^{\perp}$ to $\mathcal{H}$ by $\Phi(x_1 \otimes f^*)=u_1 \otimes
V(f)^*$. For any $f, g \in N_{-}^{\perp}, t\in \mathbb{C}$, since
$\Phi$ is linear,
\begin{align*}
\Phi(x_1\otimes (f+tg)^*)&=\Phi(x_1\otimes
f^*)+\Phi(x_1\otimes(tg)^*) \\
&=u_1\otimes V(f)^* +\overline{t}u_1\otimes V(g)^*.
\end{align*}
From the definition of $V$, $\Phi(x_1\otimes (f+tg)^*)=u_1\otimes
V(f+tg)^*$.
By comparing the two equalities, we see the map $V$ is a linear
map on $N_{-}^{\perp}$. Likewise, we can define a linear map $U$
from $N$ to $\mathcal{H}$ by $\Phi(x\otimes {f_1}^*)=U(x)\otimes
{v_1}^*$ for some fixed $f_1, v_1 \in N_{-}^{\perp}$. By
considering
\begin{align*}
\Phi(x_1\otimes {f_1}^*)&=U(x_1)\otimes{v_1}^*\\
&=u_1\otimes V(f_1)^*\\
&=u_1\otimes{v_1}^* ,
\end{align*}
we can make an observation that $U(x_1)=u_1$ and $V(f_1)=v_1$.
Now for any $x\in N, f\in N_{-}^{\perp}$, we may write
$\Phi(x\otimes f^*)$ as follows.
\[ \Phi(x\otimes f^*)=\alpha(x,f)U(x)\otimes V(f)^*\]
where $\alpha(x,f)$ is a complex valued function.
We claim that $\alpha(x,f)=1$ for all $x$ and $f$. Note that
$\Phi(x_1\otimes f^*)=\alpha(x_1,f)u_1\otimes
V(f)^*=u_1\otimes V(f)^*$, so $\alpha(x_1,f)=1$ for any $f\in
N_{-}^{\perp}$. Similarly, we observe that $\alpha(x,f_1)=1$ for any
$x\in N$. Consider
\begin{align*}
\Phi((x+sx')\otimes f^*)&=\alpha(x+sx',f)(U(x+sx')\otimes
V(f)^*)\\
&=\alpha(x+sx',f)[(U(x)+sU(x'))\otimes V(f)^*]\\
&=\alpha(x+sx',f)U(x)\otimes V(f)^*+ s\alpha(x+sx',f)U(x')\otimes V(f)^*
\end{align*}
where $x$ and $x'$ are linearly independent vectors in $N$.
On the other hand,
\begin{align*}
\phi((x+sx')\otimes f^*)&=\Phi(x\otimes f^*)+s\Phi(x'\otimes f^*)\\
&=\alpha(x,f)U(x)\otimes V(f)^* +s\alpha(x',f)U(x')\otimes
V(f)^*.
\end{align*}
We get the following equality by comparing the two equalities
above: \[\alpha(x+sx',f)U(x)+s\alpha(x+sx',f)U(x')=\alpha(x,f)U(x)+s\alpha(x',f)U(x')\] for all $s\in \mathbb{C}$. Since $U(x)$ and $U(x')$ are linearly independent, $\alpha(x,f)=\alpha(x+sx',f)=\alpha(x',f)$. Hence, $\alpha$ is independent of $x$. Likewise we can show that $\alpha$ is independent of $f$. Therefore $\alpha \equiv 1.$ The result follows. \end{proof}
What Lemma~\ref{ml} says is that whether a given subspace $N$ satisfies $(1)$ or $(2)$ in the lemma is a characteristic of the subspace which is associated with the rank preserving map $\Phi$.
\begin{definition}
Let $\mathfrak{L}$ be a commutative subspace lattice and $\Phi$ be a rank preserving map on $\operatorname{Alg}\mathfrak{L}$. Let $N$ be a nonzero subspace in $\operatorname{Alg} \mathfrak{L}$. The subspace $N$ is called \emph{consistent with respect to} $\Phi$ if $\Phi$ and $N$ satisfy $(1)$ in Lemma \ref{ml}. The subspace $N$ is called \emph{twisted with respect to} $\Phi$ if $\Phi$ and $N$ satisfy $(2)$ in Lemma \ref{ml}. \end{definition}
In a nest algebra, it is impossible that both $N$ and $N^{\perp}$ are in a nest $\mathcal{N}$ (a totally ordered lattice) if $N \neq 0$ and $N \neq I.$ But a commutative subspace lattice $\mathfrak{L}$ can have $N$ and $N^{\perp}$ both in it with the assumption $N \neq 0$ and $N \neq I.$ If there exists a subspace $N \in \mathfrak{L}$ with $N_- \neq I$ such that $\dim N=1$ and $N^{\perp} \in \mathfrak{L}$, then $N_-=\vee\{E \in
\mathfrak{L}\,|\, E \ngeq N\} \geq N^{\perp}$ since $N^{\perp} \in \mathfrak{L}$ and $N^{\perp} \ngeq N$. Hence $N_-^{\perp} \leq N$. Since $N_- \neq I$, $N_-^{\perp}=N$. \underline{Case 1} in the proof of Lemma~\ref{ml} shows that such $N$ is both consistent and twisted with respect to $\Phi$.
\begin{definition} A nonzero subspace $N$ in $\mathfrak{L}$ with $\dim N=1$ and $N^{\perp} \in \mathfrak{L}$ is called \emph{isolated}. \end{definition}
\begin{lemma} \label{isl} Let $\Phi$ be a rank preserving map on $\operatorname{Alg}\mathfrak{L}$ and $N$ be a subspace in $\mathfrak{L}$. Then $N$ is isolated if and only if $N$ is both consistent and twisted with respect to $\Phi$. \end{lemma} \begin{proof} It suffices to prove sufficiency. Suppose $N$ is both consistent and twisted with respect to $\Phi$. Then there exist linear maps $U_1,V_2$ defined on $N$, and $U_2,V_1$ defined on $N_-^{\perp}$ so that \[\Phi(x \otimes f^*)=U_1x \otimes (V_1f)^*=U_2f \otimes (V_2x)^*\] \[\Phi(y \otimes f^*)=U_1y \otimes (V_1f)^*=U_2f \otimes (V_2y)^*\] for arbitrary $x,y \in N$ and $f \in N_-^{\perp}$. It can be observed that $U_1x \sim U_2f$ and $U_2f \sim U_1y$, so $U_1x \sim U_1y$. Since the map $U_1$ can not have nonzero kernel, $x \sim y$. Using the same idea as this with $x \in N$ and $f,g \in N_-^{\perp}$, we can argue that $f \sim g$. These facts imply that $\dim N=\dim N_-^{\perp}=1$. Observe that if $x \in N$, $x \otimes x^*$ is the projection on $N$ since $\dim N=1$, so $x \otimes x^* \in \operatorname{Alg}\mathfrak{L}$. Therefore, $x \in N_-^{\perp}$. Then $N=N_-^{\perp}$. Thus $N$ is isolated. \end{proof}
\begin{lemma}\label{sl} Let $M$ and $N$ be two subspaces in $\mathfrak{L}$ with $N \leq M$. Let $\Phi$ be a rank preserving map on $\operatorname{Alg}\mathfrak{L}$. Then the followings are true. \begin{enumerate} \item If $M$ is consistent with respect to $\Phi$, then $N$ is also consistent with respect to $\Phi$. \item If $M$ is twisted with respect to $\Phi$, then $N$ is also twisted with respect to $\Phi$. \end{enumerate}
\end{lemma}
\begin{proof} First, let $M$ be a consistent element in $\mathfrak{L}$ with respect to $\Phi$. Without loss of generality, we assume $\dim N \geq 2$. Suppose that $N$ is twisted with respect to $\Phi$. Let $U_M,V_M$ and $U_N,V_N$ denote the two maps in lemma \ref{ml} defined on $M$ and $N$ ,respectively. By the assumption, we can choose two linearly independent vectors $x,y \in N$ and a vector $f \in M_-^{\perp}$. Note that $N_-^{\perp} \geq M_-^{\perp}$ since $N \leq M$. Since $M$ is consistent with respect to $\Phi$ and $N$ is twisted with respect to $\Phi$, we have \[\Phi(x \otimes f^*)= U_Mx \otimes (V_Mf)^*=U_Nf \otimes (V_Nx)^* \]
\[ \Phi(y \otimes f^*)= U_My \otimes (V_Mf)^*=U_Nf \otimes (V_Ny)^*. \] Then there exist two complex numbers $\lambda$ and $\gamma$ such that $U_Mx=\lambda U_Nf$ and $U_My=\gamma U_Nf$. But this contradicts the fact that $U_Mx$ and $U_My$ are linearly independent by Lemma \ref{fvl}, so the result $(1)$ follows. The proof of $(2)$ is essentially the same as the proof of $(1)$. \end{proof}
\begin{lemma}\label{vil} Let $\Phi$ be a rank preserving map on $\operatorname{Alg} \mathfrak{L}$. Let $M$ and $N$ be two non-zero, non-isolated subspaces in $\mathfrak{L}$ with $M_- \neq I$. If $M$ is consistent with respect to $\Phi$, and $N$ is twisted with respect to $\Phi$, then \[M \wedge N=0\] and \[M_-^{\perp} \wedge N_-^{\perp}=0.\] \end{lemma}
\begin{proof} Suppose $M \wedge N \neq 0$. Then $M \wedge N$ is a non-zero subspace of $M$ and $N$. By Lemma \ref{isl}, $M \wedge N$ is isolated, so $M \wedge N=(M \wedge N)_-^{\perp}=\langle e \rangle$ for some $e \in M \wedge N$. By Lemma \ref{ul}, we have $(M \wedge N)_- \leq M_- \wedge N_-$. It follows that $(M \wedge N)_-^{\perp} \geq M_-^{\perp} \vee N_-^{\perp}$. Thus $M_-^{\perp}=N_-^{\perp}=(M \wedge N)_-^{\perp}=\langle e \rangle$. On the other hand, we have $M_- \geq N$ since $N \ngeq M$, so $\langle e\rangle=M_-^{\perp} \leq N^{\perp}$. Therefore $N \wedge N^{\perp} \geq \langle e\rangle \neq 0$ which gives a contradiction. Thus \[M \wedge N=0.\]
Observe that $M \vee N$ is neither consistent nor twisted with respect to $\Phi$ by Lemma \ref{sl}. This implies $(M \vee N)_-=I$ by Lemma \ref{ml}. Therefore \[M_-^{\perp} \wedge N_-^{\perp}=(M_- \vee N_-)^{\perp}=(M \vee N)_-^{\perp}=0.\] \end{proof}
\begin{theorem}\label{dct} Let $\mathfrak{L}$ be a completely distributive commutative subspace lattice and $\Phi$ be a rank preserving map on $\operatorname{Alg}
\mathfrak{L}$. Let \[M_i=\vee\{E \in \mathfrak{L}\,|\,E \, \textrm{is isolated}\},\]
\[M_c=\vee\{E \in \mathfrak{L}\,|\, E \, \textrm{is consistent with respect to} \, \Phi \} \ominus M_i\] and
\[M_t=\vee\{E \in \mathfrak{L}\,|\, E \, \textrm{is twisted with respect
to}\, \Phi \ \} \ominus M_i.\]
Then \[\operatorname{Alg}\mathfrak{L}=\operatorname{Alg}(M_i\mathfrak{L}M_i) \oplus \operatorname{Alg}(M_c\mathfrak{L}M_c) \oplus \operatorname{Alg}(M_t\mathfrak{L}M_t) .\]
\end{theorem}
\begin{proof} By Lemma \ref{vil}, it is clear that $M_i \oplus M_c \oplus M_t=I$. Since $M_i^{\perp} \in \mathfrak{L}$, $M_c, M_t \in \mathfrak{L}.$ Note that $M_c^{\perp}=M_i \vee M_t.$ Hence $M_c^{\perp} \in \mathfrak{L}$. Likewise, $M_t^{\perp} \in \mathfrak{L}$. The result follows. \end{proof}
Remember that the operators $U$ and $V$ were constructed with
the subspace $N$ fixed. We now refer to them as $U_N$ and $V_N$ and try to fit together the $U_N$'s and $V_N$'s into operators $U$ and $V$ defined on the whole space $\mathcal{H}$ such that \[\Phi(x \otimes f^*)=Ux \otimes (Vf)^* \] whenever $x \otimes f^* \in \operatorname{Alg}\mathfrak{L}$. Note that if $M_-
\neq I$, if $M<N$, and if $x \in M$ and $f \in N_-^{\perp}$, then $x \in N$ and $\Phi(x \otimes f^*)=(U_N x) \otimes (V_N f)^*$. On the other hand, $f \in N_-^{\perp} <M_-^{\perp}$, so $\Phi(x \otimes f^*)=(U_M x) \otimes (V_M f)^*$. Thus there exists a complex number $\lambda$ such that $U_M x=\lambda U_N x$ and $V_N f=\overline{\lambda}V_M f$. Since $x$ and $f$ may vary independently, $\lambda$ does not depend on $x$ and $f$ but only on $M$ and $N$. We call it $\lambda_{MN}$ and have
\[ U_M=\lambda_{MN} U_N|M \,\,\, \textrm{and} \,\,\, V_N=\overline{\lambda_{MN}}V_M|N_-^{\perp} .\]
Let $\mathfrak{F}$ be the collection of all subspaces $N$ in $\mathfrak{L}$ such that $N \neq 0$ and $N_- \neq I$.
\begin{remark} In the remainder of this paper we assume that every subspace in $\mathfrak{F}$ is consistent with respect to $\Phi$. \end{remark}
Suppose that $M$ and $N$ lie in $\mathfrak{F}$. We will say that $M$ and $N$ are \emph{comparable} if either $M \leq N$ or $N \leq M$. Suppose $M$ and $N$ are comparable. If $M \leq N$, $\lambda_{MN}$ has already been defined. If $N \leq M$, define \[ \lambda_{MN}= \frac{1}{\lambda_{NM}}\,\, .\] Thus $\lambda_{MN}$ is defined whenever $M$ and $N$ are comparable, and it is easy to check that $\lambda_{LN}=\lambda_{LM}\lambda_{MN}$ whenever each pair from $\{L,M,N\}$ is comparable.
\begin{definition} Let $M$ and $N$ be two subspaces in $\mathfrak{F}$. We define a \emph{chain} from $M$ to $N$ to be a finite sequence of subspaces $\{M_0,M_1,...,M_n\}$, each $M_k$ in $\mathfrak{F}$, such that $M_0=M, M_n=N$, and such that $M_k$ is comparable to $M_{k+1}$ for each $k=0,1,...,n-1$.
If $M=N$, the chain $\{M_0,M_1,...,M_n\}$ is called a \emph{cycle of length} n. \end{definition}
Suppose $M,N \in \mathfrak{F}.$ If there is a chain $\{M_0,M_1,...,M_n\}$ from $M$ to $N$, we want to define $\lambda_{MN}$ to be $\lambda_{MM_1}\lambda_{M_1M_2} \cdots \lambda_{M_{n-2}M_{n-1}}\lambda_{M_{n-1}N}$. Since there may be more than one chain from $M$ to $N$, we need to show that such a product is well defined. The following lemma will prove this. An elaborate proof of the lemma can be found in \cite{MR87k:47103}.
\begin{lemma} \label{lprt} Let $\{M_0,M_1,...,M_n\}$ be a cycle in $\mathfrak{F}$. Then \[ \lambda_{M_0M_1}\lambda_{M_1M_2} \cdots \lambda_{M_{n-2}M_{n-1}}\lambda_{M_{n-1}M_n}=1 .\]
\end{lemma}
By making use of the Lemma \ref{lprt}, we can now show that \[\lambda_{MN}=\lambda_{MM_1}\lambda_{M_1M_2} \cdots \lambda_{M_{n-2}M_{n-1}}\lambda_{M_{n-1}N}\] is well defined if there is a chain $\{M_0,M_1,...,M_n\}$ from $M$ to $N$.
Suppose that there are two chains $\{M_0,M_1,M_2,...,M_{n-2},M_{n-1},M_n\}$ and \\ $\{M_0,M_1',M_2',...,M_{n-2}',M_{n-1}',M_n\}$ from $M$ to $N$. We now form a cycle \[\{M,M_1,...,M_{n-1},N,M_{n-1}',...,M_1',M\}.\] By applying Lemma \ref{lprt} to this cycle, it follows that \[\lambda_{MM_1} \cdots \lambda_{M_{n-1}N}\lambda_{NM_{n-1}'} \cdots \lambda_{M_1'M}=1.\] From this equation we get, \[\lambda_{MM_1} \cdots \lambda_{M_{n-1}N}= \frac{1}{\lambda_{NM_{n-1}'}\cdots \lambda_{M_1'M}}=\lambda_{MM_1'}\cdots \lambda_{M_{n-1}'N}. \] This shows that $\lambda_{MN}$ is well defined.\\
Recall that $\mathfrak{F}$ denotes the collection of all $N$ in $\mathfrak{L}$ such that $N \neq 0$ and $N_- \neq I$. Fix $N \in \mathfrak{L}$ and let
\[\mathfrak{G}_N^n=\{M \in \mathfrak{F}\,|\,M \, \textrm{can be connected to} \, N \, \textrm{by a chain of length} \, k \leq n\}.\] Let $\mathfrak{G}_N=\cup_{n}\mathfrak{G}_N^n$.
\begin{lemma}\label{du} Let $N \in \mathfrak{F}$. Then there exist linear transformations
$U$ with dense domain in $\vee \{M \, | \, M \in \mathfrak{G}_N\}$
and $V$ with dense domain in $\vee \{M_-^{\perp} \, | \, M \in \mathfrak{G}_N \}$ such that $\Phi(x \otimes f^*)=Ux \otimes (Vf)^*$ whenever there is $M \in \mathfrak{G}_N$ for which $x \in M$ and $f \in M_-^{\perp}$. \end{lemma}
\begin{proof} For $M \in \mathfrak{G}_N$, we have associated operators $U_M$ and $V_M$ such that \[\Phi(x \otimes f^*)=U_Mx \otimes (V_Mf)^*,\] whenever $x \in M$ and $f \in M_-^{\perp}$. Let $\widetilde{U}_M=\lambda_{NM}U_M$ and $\widetilde{V}_M=\overline{\lambda}_{MN} V_M$. Since $\lambda_{MN}\lambda_{NM}=1$, we have $\Phi(x \otimes f^*)= \widetilde{U}_Mx \otimes (\widetilde{V}_Mf)^*$ for $x \in M$ and $f \in M_-^{\perp}$. Note that for $L,M \in \mathfrak{G}_N$ with $L \leq M$, if $\{N,N_1,...,N_{n-1},M \}$ is a chain from $N$ to $M$, then $\{N,N_1,...,N_{n-1},M,L\}$ is a chain from $N$ to $L$. Therefore $\lambda_{NL}=\lambda_{NM}\lambda_{ML}$. Thus, if $x \in L$, we have $\widetilde{U}_{L}x=\lambda_{NL}U_{L}x=\lambda_{NM}\lambda_{ML}U_{L}x$. On the other hand, by definition of $\lambda_{LM}$ we have $U_Lx=\lambda_{LM}U_Mx$, so $\widetilde{U}_{L}x=\lambda_{NM}\lambda_{ML}\lambda_{LM}U_{M}x=\lambda_{NM}U_{M}x =\widetilde{U}_{M}x$. Thus, if $L \leq M$, $\widetilde{U}_L$ and $\widetilde{U}_M$ agree on $L$. Let $\mathcal{M} =
\{x_1+\cdots+x_n \, | \, \textrm{for some positive integer} \, n \, , \, x_i \in M \, \textrm{for some} \, M \in \mathfrak{G}_N \}$. Define a linear transformation $U$ on $\mathcal{M}$ by
$U|M=\widetilde{U}_M$. By the coherence of the $\widetilde{U}_M$, the map $U$ is well defined. Similarly, we can define
$V|M_-^{\perp}=\widetilde{V}_M$. If $x \in M, \, f \in M_-^{\perp}$ for $M \in \mathfrak{G}_N$, then we have $\Phi(x \otimes f^*)=Ux \otimes (Vf)^*$. \end{proof}
\begin{lemma}\label{orth}
For each $N \in \mathfrak{F}$, let $G_N=\vee\{M \, | \, M \in
\mathfrak{G}_N\}$ and $F_N=\vee\{M_-^{\perp} \, | \, M \in \mathfrak{G}_N \}$. Let $N,N' \in \mathfrak{F}$. \begin{enumerate} \item If $N' \in \mathfrak{G}_N$, then $G_N=G_{N'}$ and $F_N=F_{N'}$
\item If $N' \notin \mathfrak{G}_N$, then $G_N \perp G_{N'}$ and $F_N \perp F_{N'}$ \end{enumerate} \end{lemma}
\begin{proof} Suppose that $N' \in \mathfrak{G}_N$. Note that if there exists a chain connecting $N$ to $N'$ and one connecting $N'$ to $M$, then there is a chain connecting $N$ to $M$, whence $\mathfrak{G}_N=\mathfrak{G}_{N'}$. Thus $G_N=G_{N'}$ and $F_N=F_{N'}$. Suppose that $N' \notin \mathfrak{G}_N$; then, for any $M' \in \mathfrak{G}_{N'}$, $M' \notin \mathfrak{G}_N$. For such an $M'$, consider the projection $NM'$. If $NM' \neq 0$, then $NM' \in \mathfrak{G}_N \cap \mathfrak{G}_{M'}=\mathfrak{G}_{M'} \cap \mathfrak{G}_N$. Since this is impossible, it must be that $NM'=0$. Likewise, if $M \in \mathfrak{G}_N$, $MM'=0$ and hence $G_NG_{N'}=0$. \end{proof}
\begin{theorem} \label{mpn} Let $\mathfrak{L}$ be a completely distributive commutative subspace lattice and let $\Phi$ be a weakly continuous rank preserving map defined on $\operatorname{Alg}\mathfrak{L}$. If we assume that $\mathfrak{F}=\mathfrak{G}_N$ for some $N \in \mathfrak{F}$ (i.e. $\operatorname{Alg} \mathfrak{L}$ is irreducible.), then there exist two densely defined linear transformations $U,V$ such that \[\Phi(A)=UAV^*\] for all $A \in \operatorname{Alg}\mathfrak{L}$. \end{theorem}
\begin{proof} By Lemma \ref{du}, there exist densely defined maps $U$ and $V$ such that $\Phi(x \otimes f^*)=Ux \otimes (Vf)^*$ whenever $x \otimes f^* \in \operatorname{Alg}\mathfrak{L}$. Since $\mathfrak{F}=\mathfrak{G}_N$ for some $N \in \mathfrak{F}$, the domain of $U$ is the nonclosed linear span of $\mathfrak{F}=\{E
\in \mathfrak{L}\,|\, E_-\neq I \}$ and the domain of $V$ is the nonclosed linear span of $\{M_-^{\perp} \,|\, M \in \mathfrak{F} \}$. By Lemma \ref{cdl1}, these domains are dense in $\mathcal{H}$. Recall that $R_{\mathfrak{L}}$ denote the linear span of the rank one operators in $\operatorname{Alg}\mathfrak{L}$. Therefore, for any operator $A \in R_{\mathfrak{L}}$, $\Phi(A)=UAV^*$ by the linearity of $\Phi$. By Lemma \ref{wk}, in CDC algebra it is guaranteed that there exists a net of operators in $R_{\mathfrak{L}}$ weakly converging to any operator in $\operatorname{Alg}\mathfrak{L}$. Let $A$ be an operator in $\operatorname{Alg}\mathfrak{L}$. Then there exists a net $\{A_{\alpha}\}_{\alpha \in J}$ such that $A_{\alpha}$ converges weakly to $A$. For each $h \in \operatorname{dom}V^*$ and $k \in \operatorname{dom}U^*$, \begin{align*} \langle UA_{\alpha}V^*h,k \rangle &= \langle A_{\alpha}V^*h, U^*k \rangle \\ &\rightarrow \langle AV^*h,U^*k \rangle \,\, \textrm{since} \, A_{\alpha} \, \textrm{weakly converges to} \, A \\ &= \langle UAV^*h,k \rangle . \end{align*} Thus \[\Phi(A_{\alpha})=UA_{\alpha}V^* \overset{w}{\longrightarrow} UAV^*. \] On the other hand, $\Phi(A_{\alpha})$ converges weakly to $\Phi(A)$ since $\Phi$ is weakly continuous. Thus $\Phi(A)=UAV^*$ for $A \in \operatorname{Alg}\mathfrak{L}$.
\end{proof}
Now we are finally in a position to state and prove the main result of this paper. \begin{theorem} Let $\mathfrak{L}$ be a completely distributive commutative subspace lattice and let $\Phi$ be a weakly continuous rank preserving map defined on $\operatorname{Alg}\mathfrak{L}$. If we assume that every subspace is consistent with respect to $\Phi$, then there exist two densely defined linear transformations $U,V$ such that \[\Phi(A)=UAV^*\] for all $A \in \operatorname{Alg}\mathfrak{L}$. \end{theorem} \begin{proof} Let $N \in \mathfrak{F}$ and consider the collection $\mathfrak{G}_N$. If $\mathfrak{G}_N = \mathfrak{F}$, then this is nothing but the case of Proposition \ref{mpn}. Suppose there exists $M \in \mathfrak{F}$ with $M \notin \mathfrak{G}_N$ such that $G_N G_M=G_M G_N=0$. In this way we can form a sequence $\{G_{N_i}\}$ of mutually orthogonal projections in $\mathfrak{L}$. We shall suppress the $N$ and write simply $G_i$. The separability of $\mathcal{H}$ guarantees that there are no more than countably many $G_i$'s. We have $\vee G_i=I$ since $\vee \{N \,:\,N \in \mathfrak{F}\}=I$, whence $G_j^{\perp}=\vee_{i \neq j}G_i$ is also in $\mathfrak{L}$. Therefore, the algebra $\operatorname{Alg}\mathfrak{L}$ can be written as the direct sum $\sum_i \oplus \operatorname{Alg}(G_i\mathfrak{L}G_i)$. For any $A \in \operatorname{Alg}\mathfrak{L}$, we can write $A=\sum_i \oplus A_i$ where $A_i \in \operatorname{Alg}(G_i\mathfrak{L}G_i).$ By Proposition~\ref{mpn}, there exist two densely defined linear maps $U_i$ and $V_i$ such that $\Phi(A_i)=U_iA_iV_i^*$ for each $i$. If we define $U=\sum_i \oplus U_i$ and $V=\sum_i \oplus V_i$, then $U$ and $V$ are densely defined on $\mathcal{H}$. Then the result follows. \end{proof}
\begin{corollary} Let $\mathcal{H}$ be a finite dimensional Hilbert space and let $\mathfrak{L}$ be a commutative subspace lattice consisting of subspaces in $\mathcal{H}.$ Let $\Phi$ be a rank preserving map on $\operatorname{Alg} \mathfrak{L}$. If we assume that every element in $\mathfrak{L}$ is consistent with respect to $\Phi$, then there exists a map $\Psi$ defined by $\Psi(A)=A\Phi^{-1}(I)$ such that $\Psi \circ \Phi$ is an isomorphism on $\operatorname{Alg} \mathfrak{L}.$ \end{corollary}
\begin{proof} By Main Theorem, there are two linear maps $U$ and $V$ such that $\Phi(A)=UAV^*$. Since $\mathcal{H}$ is finite dimensional, it is true that $\Phi(I)=UV^*$ is invertible. Therefore $\Psi(A)=A\Phi^{-1}(I)=A(UV^*)^{-1}$ is a well defined map. Then \begin{align*} (\Psi \circ \Phi)(A)&=\Phi(A)(V^*)^{-1}U^{-1} \end{align*} \begin{align*}
&=UAV^*(V^*)^{-1}U^{-1} \\
&=UAU^{-1} \end{align*} Thus $\Psi \circ \Phi$ is an isomorphism implemented by $U.$ \end{proof}
\section{EXAMPLES} In this section we present some examples of rank preserving maps. Most of the examples we discuss in this section will be either $\mathcal{A}_{2n}$ or $\mathcal{A}_{\infty}$. The precise definition of these algebras is given in \cite{MR85g:47062}. But for our discussion it is enough to say that the algebras $\mathcal{A}_{2n}$ are tridiagonal matrices, of size $2n \times 2n$, of the form \[ \begin{bmatrix} * & * & & & & & & & & * \\
& * & & & & & & & & \\
& * & * & * & & & & & & \\
& & & * & & & & & & \\
& & & * & * & * & & & & \\
& & & & & \ddots & & & & \\
& & & & & & & * & & \\
& & & & & & & * & & \\
& & & & & & & * & * & * \\
& & & & & & & & & * \end{bmatrix} \] where all nonstarred entries are $0$. It can be observed that the associated lattice consists of certain diagonal projections, hence it is commutative and completely distributive. Thus the algebra $\mathcal{A}_{2n}$ is reflexive. The algebra $\mathcal{A}_{\infty}$ consists of infinite matrices of the form \[ \begin{bmatrix} * & * & & & \\
& * & & & \\
& * & * & * & \\
& & & * & \\
& & & * & \\
& & & & \ddots \end{bmatrix} .\]
Once again the associated lattice is commutative and completely distributive, so the algebra $\mathcal{A}_{\infty}$ is reflexive. \\ \begin{example} Consider $\mathcal{A}_4$. Define a map $\Phi_1:\mathcal{A}_4 \rightarrow \mathcal{A}_4$ by \[
\left[ \begin{matrix} a & b & 0 & h \\ 0 & c & 0 & 0 \\ 0 & d & e & f \\ 0 & 0 & 0 & g \end{matrix} \right] \mapsto \left[ \begin{matrix} e & f & 0 & d \\ 0 & g & 0 & 0 \\ 0 & h & a & b \\ 0 & 0 & 0 & c \end{matrix} \right] .\] Then it is easy to check that $\Phi_1$ is rank preserving. Moreover, the map $\Phi_1$ is implemented by \[ U = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix} .\] In other words, the map $\Phi_1$ is an isomorphism such that \[\Phi_1(A)=UAU^{-1}.\] If we define a map $\Phi_2:\mathcal{A}_4 \rightarrow \mathcal{A}_4$ by \[
\left[ \begin{matrix} a & b & 0 & h \\ 0 & c & 0 & 0 \\ 0 & d & e & f \\ 0 & 0 & 0 & g \end{matrix} \right] \mapsto \left[ \begin{matrix} g & f & 0 & h \\ 0 & e & 0 & 0 \\ 0 & d & c & b \\ 0 & 0 & 0 & a \end{matrix} \right] ,\] then $\Phi_2$ is also a rank preserving map. But the map $\Phi_2$ is not an isomorphism. Instead, we can write the map $\Phi_2$ as \[\Phi_2(A)=VA^{T}V^{-1}\] where \[ V = \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{bmatrix} .\]
\end{example}
\begin{example} Let $\mathcal{H}$ be a separable Hilbert space and $\mathcal{B}=\{e_k\}_{k=1}^{\infty}$ be an orthonormal basis for $\mathcal{H}$. Let $\mathcal{N}=\{0,[e_1],[e_1,e_2],[e_1,e_2,e_3],....\}$. Then $\mathcal{N}$ is a nest and the corresponding nest algebra is \[ \operatorname{Alg}\mathcal{N} = \begin{bmatrix} * & * & * & * & * & \cdots \\ 0 & * & * & * & * & \cdots \\ 0 & 0 & * & * & * & \cdots \\ 0 & 0 & 0 & * & * & \cdots \\ 0 & 0 & 0 & 0 & * & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix} .\]
Let $S$ denote the unilateral shift operator such that $Se_k=e_{k+1}$. If we define $\Phi:\operatorname{Alg}\mathcal{N} \longrightarrow \operatorname{Alg}\mathcal{N}$ by $\Phi(A)=SAS^*$, then \[ \Phi : \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & \cdots \\ 0 & a_{22} & a_{23} & a_{24} & a_{25} & \cdots \\ 0 & 0 & a_{33} & a_{34} & a_{35} & \cdots \\ 0 & 0 & 0 & a_{44} & a_{45} & \cdots \\ 0 & 0 & 0 & 0 & a_{55} & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix} \mapsto \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & \cdots \\ 0 & a_{11} & a_{12} & a_{13} & a_{14} & \cdots \\ 0 & 0 & a_{22} & a_{23} & a_{24} & \cdots \\ 0 & 0 & 0 & a_{33} & a_{34} & \cdots \\ 0 & 0 & 0 & 0 & a_{44} & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix} .\] From this fact we can observe that $\Phi$ is a rank preserving map on $\operatorname{Alg}\mathcal{N}$ but it is not an onto map. \end{example}
\begin{example} Consider the algebra $\mathcal{A}_{\infty}$ which is described at the beginning of Section 3. Let $\mathfrak{L}$ be the lattice associated with $\mathcal{A}_{\infty}$. We can easily argue that for any $M,N \in \mathfrak{L}$, $\mathfrak{G}_M=\mathfrak{G}_N$. Therefore $\mathcal{A}_{\infty}$ is irreducible. Let $\Phi$ be a map on $\mathcal{A}_{\infty}$ defined by \[ \Phi: \begin{bmatrix} a & b & 0 & 0 & 0 & 0 & \cdots\\ 0 & c & 0 & 0 & 0 & 0 & \\ 0 & d & e & f & 0 & 0 & \\ 0 & 0 & 0 & g & 0 & 0 & \\ 0 & 0 & 0 & h & i & j & \\
\vdots & & & & & & \ddots \end{bmatrix} \longrightarrow \begin{bmatrix} a & \frac{1}{2} b & 0 & 0 & 0 & 0 & \cdots\\ 0 & c & 0 & 0 & 0 & 0 & \\ 0 & \frac{3}{2} d & e & \frac{3}{4} f & 0 & 0 & \\ 0 & 0 & 0 & g & 0 & 0 & \\ 0 & 0 & 0 & \frac{5}{4} h & i & \frac{5}{6} j & \\
\vdots & & & & & & \ddots \end{bmatrix} .\] Then it is obvious that the map $\Phi:\mathcal{A}_{\infty} \rightarrow \mathcal{A}_{\infty}$ preserves rank and it can also be observed that the map $\Phi$ is implemented by an unbounded operator \[ U= \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & \cdots\\ 0 & 2 & 0 & 0 & 0 & \\ 0 & 0 & 3 & 0 & 0 & \\ 0 & 0 & 0 & 4 & 0 & \\ 0 & 0 & 0 & 0 & 5 & \\
\vdots & & & & & \ddots \end{bmatrix} .\] In other words, \[\Phi(A)=UAU^{-1}\] for all $A \in \operatorname{Alg} \mathfrak{L}.$
\end{example}
\end{document} | arXiv |
$\star$ Consider an open-address hash table with a load factor $\alpha$. Find the nonzero value $\alhpa$ for which the expected number of probes in an unsuccessful search equals twice the expected number of probes in a successful search. Use the upper bounds given by Theorems 11.6 and 11.8 for there expected number of probes. | CommonCrawl |
\begin{document}
\title{\bf Mean reflected stochastic differential equations with two constraints} \author{Adrian Falkowski and Leszek S\l omi\'nski \footnote{Corresponding author. E-mail address: [email protected]; Tel.: +48-566112954; fax: +48-566112987.}\\
\small Faculty of Mathematics and Computer Science, Nicolaus Copernicus University,\\ \small ul. Chopina 12/18, 87-100 Toru\'n, Poland}
\date{} \maketitle \begin{abstract} We study the problem of the existence, uniqueness and stability of solutions of reflected stochastic differential equations (SDEs) with a minimality condition depending on the law of the solution (and not on the paths). We require that some functionals depending on the law of the solution lie between two given c\`adl\`ag constraints. Applications to investment models with constraints are given. \end{abstract} {\em Key Words}: stochastic differential equations with constraints, the Skorokhod problem,
reflecting boundary condition.\\
{\em AMS 2000 Subject Classification}: Primary: 60H20; Secondary: 60G22.
\section{Introduction}
In this paper we consider reflected SDEs of the form \begin{equation}\label{eq1.1} X_t = X_0 +\int_0^t f(s,X_{s-})\,dM_s+\int_0^tg(s,X_{s-})\, dV_s + k_t,\quad t\in\ensuremath{\R^+}, \end{equation} In (\ref{eq1.1}), $f,g:\ensuremath{\R^+}\times\ensuremath{\mathbb R}\rightarrow\ensuremath{\mathbb R}$ are continuous functions, $M$ is a martingale, $V$ is an adapted process of bounded variation and $k$ is a deterministic function which for given two sided Lipschitz continuous function $h:\ensuremath{\R^+}\times\ensuremath{\mathbb R}\rightarrow\ensuremath{\mathbb R}$ compensates reflections of the functional of $X$ of the form $Eh(t,X_t)$, $t\in\ensuremath{\R^+}$, on c\`adl\`ag constraints $l$, $u$ such that $l_t\leq u_t$, $t\in\ensuremath{\R^+}$ (for a precise definition see Section 3).
Reflected SDEs were introduced by Skorokhod \cite{Sk} in the case where $l=0$, $u=+\infty$, $M$ is a standard Wiener process and $V_t=t$, $t\in\ensuremath{\R^+}$. The minimality condition in Skorokhod's equation was depending on the paths of the solution $X$. This implies that the compensating reflection part is a nondecreasing stochastic process and $X_t\geq 0$, $t\in\ensuremath{\R^+}$. Since the pioneering Skorokhod's work reflected SDEs have been intensively studied by many authors and his results were generalized to larger classes of constraints and larger classes of driving processes (see, e.g., \cite{BKR,ChL,DN,KLRS,rutkowski:1980,Sl1,sl-wo/10,sl-wo/13,ta}). In the all mentioned above papers the minimality condition, which characterizes the compensating reflections part, depends on the paths of the solution $X$.
In recent papers by Briand, Elie and Hu \cite{briand:2018} and Briand Ghannoum and Labart \cite{briand:2018a} new type of reflected SDEs was introduced. They were motivated by the mean field game theory. In this new type of equations the compensating reflection part depends not on the paths but on the law of the solution $X$. In this case the compensating reflection part is a nondecreasing function and $EX_t\geq0$, $t\in\ensuremath{\R^+}$, or more generally, $Eh(t,X_t)\geq0$, $t\in\ensuremath{\R^+}$, for given two sided Lipschitz continuous function $h:\ensuremath{\R^+}\times\ensuremath{\mathbb R}\rightarrow\ensuremath{\mathbb R}$. This in fact means that the mean minimality condition is considered. A similar problem was considered by Djehiche, Elie and Hamad\`ene \cite{ha}. In the present paper, we generalize this type of reflected SDEs to the case of equations of the form (\ref{eq1.1}) with two c\`adl\`ag constraints $l$, $u$ such that \begin{equation} \label{eq1.2}Eh(t,X_t)\in[l_t,u_t],\quad t\in\ensuremath{\R^+}, \end{equation} and the compensating reflection part $k$ is not necessarily of bounded variation.
The paper is organized as follows. In Section 2, for given c\`adl\`ag integrable process $Y$ we consider the Skorokhod problem for $Y$ with mean minimality condition. Its solution is a pair $(X,k)$ such that \begin{equation} \label{eq1.3} X_t=Y_t+k_t,\quad t\in\ensuremath{\R^+}, \end{equation} and moreover, (\ref{eq1.2}) is satisfied and $k$ is a deterministic function not necessarily of bounded variation. We observe that $(X,k)$ is strictly connected with the solution of appropriately defined classical deterministic Skorokhod problem. Using this we prove the existence and uniqueness of solutions of (\ref{eq1.3}) and provide an explicit formula for the function $k$. We also show Lipschitz continuity of the mapping $(Y,h,l,u)\mapsto(X,k)$. More precisely, we prove that if $(X^i,k^i)$ is a solution associated with an integrable process $Y^i$, function $h^i$ and barriers $l^i,u^i$, $i=1,2$, then there exists $C>0$ such that for every $q\in\ensuremath{\R^+}$,
\[\sup_{t\leq q}|k^1_t-k^2_t|\leq C\big(\parallel h^1-h^2\parallel_{[0,q]\times\ensuremath{\mathbb R}}
+ \sup_{t\leq q}E|Y^1_t-Y^2_t|+\sup_{t\leq q}
\max(|l^1_t-l^2_t|,|u^1_t-u^2_t|)\big)\] and
\[E\sup_{t\leq q}|X^1_t-X^2_t|
\leq (C+1)E\sup_{t\le q}|Y^1_t-Y^2_t|+ C\big(\parallel h^1-h^2\parallel_{[0,q]\times\ensuremath{\mathbb R}}+\sup_{t\leq q}\max(|l^1_t-l^2_t|,|u^1_t-u^2_t|)\big),\] where $\parallel h^1-h^2\parallel_{[0,q]\times\ensuremath{\mathbb R}}
=\sup_{(t,x)\in[0,q]\times\ensuremath{\mathbb R}}|h^1(t,x)-h^2(t,x)|$. We also give results on the Skorokhod problem with mean minimality condition and one barrier: lower (if $u=+\infty$) and upper (if $l=-\infty$).
In Section 3, we study the general problem of stability of solutions to (\ref{eq1.3}) with respect to the convergence of associated processes and barriers. We give conditions ensuring stability with respect to the convergence in law and in probability in the Skorokhod topology $J_{1}$. As an application, we propose a practical scheme of approximations of (\ref{eq1.3}) based on discrete approximations of the barriers $l,u$ and the process $Y$.
Section 4 is devoted to the study of weak and strong solutions of (\ref{eq1.1}). We prove the existence and uniqueness of a strong solution of (\ref{eq1.1}) provided that $f,g$ satisfy the linear growth condition and are Lipschitz continuous and show by example that equations of the form (\ref{eq1.1}) can be useful in the study of investment models with constraints. Moreover, we prove that the solution of (\ref{eq1.1}) can be approximated by a discrete scheme constructed with the analogy to the Euler scheme and we prove its convergence in probability in the Skorokhod topology $J_{1}$. We also show some stability results for solutions of the form \begin{equation}\label{eq4.9} X^n_t = X^n_0 +\int_0^t f^n(s,X^n_{s-})\,d M^n_s +\int_0^tg^n(s,X^n_{s-})\, dV^n_s + k^n_t,\quad t\in\ensuremath{\R^+}, \end{equation} under the assumption that the sequence of driving martingales $\{M^n\}$ and processes with locally bounded variations $\{V^n\}$ satisfy the condition corresponding to the so-called condition (UT) introduced by Stricker \cite{st} (see also \cite{jmp}). As a consequence, we prove existence of weak solution of the SDE (\ref{eq1.1}) provided that $f,g$ are continuous and satisfy the linear growth condition. In case (\ref{eq1.1}) has the weak uniqueness property, we formulate a theorem on convergence of solutions in law in the Skorokhod topology $J_{1}$.
In the paper, we will use the following notation. $\ensuremath{{D}(\Rp,\R^{d})}$ is the space of c\`adl\`ag mappings $x:\ensuremath{\R^+}\to\ensuremath{\mathbb R}^d$, i.e. mappings which are right continuous and admit left-hands limits
equipped with the Skorokhod topology $J_1$ (for the definition and many useful results on $J_1$ topology see, e.g., \cite{js}). Every process $X$ appearing in the sequel is assumed to have trajectories in the space $\ensuremath{{D}(\Rp,\R)}$. We denote by $\ensuremath{{\mathbb L}}^1$ the space of integrable random variables and by $\mathcal{{D}}$ the space of $({\cal F}_t)$-adapted process $X$ such that for every $q\in\ensuremath{\R^+}$ the family of random variables $\{X_t,t\leq q\}$ is uniformly integrable. Note that our class $\mathcal{{D}}$ is larger than usually considered Doobs class. For a semimartingale $X$, $[X]$ stands for the quadratic variation process of $X$ and $\langle X\rangle_t $ stands for the predictable compensator
of $[X]$. For a process with locally finite variation $K$ we denote by $|K|_t$ its total variation $[0,t]$. If additionally $|K|$ is locally integrable, then $\widetilde{K}$ stands for the predictable compensator of $K$.
For $x\in\ensuremath{{D}(\Rp,\R^{d})}$, $t>0$, we set $x_{t-}=\lim_{s\uparrow t}x_s$, $\Delta x_t=x_t-x_{t-}$ and $v_{p}(x)_{[a,b]} = \sup_\pi
\sum_{i=1}^m |x_{t_i}-x_{t_{i-1}}|^p <\infty$, where the supremum is taken over all subdivisions $\pi=\{a=t_0<\ldots<t_n=b\}$ of
$[a,b]$. $V_p(x)_{[a,b]}=(v_p(x)_{[a,b]})^{1/p}$ and $\bar V_p(x)_{[a,b]}=V_p(x)_{[a,b]}+|x_a|$ is the usual variation norm. For simplicity of notation, we write $v_p(x)_q=v_p(x)_{[0,q]}$,
$V_p(x)_q= V_p(x)_{[0,q]}$ and $\bar V_p(x)_q= \bar V_p(x)_{[0,q]}$. Note that $V_1(x)_q=v_1(x)_q=|x|_q$, $q\in\ensuremath{\R^+}$. Recall also that for $\eta>0$ and $q\in\ensuremath{\R^+}$ the number $N_{\eta}$ of $\eta$-oscillations of $x$ on $[0,q]$ is the largest integer $k$ such that one can find $0\leq t_1\leq t_2\leq\dots\leq t_{2k-1}\leq t_{2k}\leq q$ satisfying
$|x_{t_{2i-1}}-x_{t_{2i}}|>\eta$, $i=1,2,\dots,k$.
By ${\displaystyle \mathop{\rightarrow}_{\cal D}}$ and ${\displaystyle\mathop{\rightarrow}_{\cal P}}$ we denote the convergence in law and in probability, respectively.
\section{The Skorokhod problem with mean minimality condition}
Let $ l,u\in\ensuremath{{D}(\Rp,\R)}$ be such that $l\le u$, $(\Omega, {\cal F}, ({\cal F}_t), P)$ be a filtered probability space and $h:\ensuremath{\R^+}\times\ensuremath{\mathbb R}\rightarrow \ensuremath{\mathbb R}$ be a $\ensuremath{{\cal B}}(\ensuremath{\R^+})\times\ensuremath{{\cal B}}(\ensuremath{\mathbb R})$-measurable function for which there exists a constant $\mu_h>0$ such that \begin{equation}\label{eq2.1}
|h(t,x)|\le \mu_h(1+|x|),\quad t\in\ensuremath{\R^+},\, x\in\ensuremath{\mathbb R}.\end{equation} \begin{definition}\label{def1}Let $Y$ be an $({\cal F}_t)$-adapted process with trajectories in $\ensuremath{{D}(\Rp,\R)}$, $l,u\in\ensuremath{{D}(\Rp,\R)}$, $l\leq u$. Let $h:\times\ensuremath{\R^+}\times\ensuremath{\mathbb R}\rightarrow \ensuremath{\mathbb R}$ satisfy (\ref{eq2.1}) and $Eh(0,Y_0)\in[l_0,u_0]$. We say that a pair $(X,k)\in{\cal D}\times\ensuremath{{D}(\Rp,\R)}$ with $k_0=0$ is a solution of the Skorokhod problem) with mean minimality condition and two constraints)
associated with $h,Y,l,u$ ($(X,k)=\SP{Y}$ for short) if \begin{enumerate}[\bf(i)] \item $X_t=Y_t+k_t$, \item $Eh(t,X_t)\in[l_t,u_t],\quad t\in\ensuremath{\R^+}$, \item for every $0\leq t\leq q$, \begin{eqnarray*} k_q-k_t\ge 0,&&\quad\mbox{\rm if}\,\,Eh(s,X_s)<u_s\,\,\mbox{\rm for all}\,\,s\in(t,q],\\ k_q-k_t\le 0,&&\quad\mbox{\rm if}\,\,Eh(s,X_s)>l_s\,\,\mbox{\rm for all}\,\,s\in(t,q], \end{eqnarray*} and for every $t\in\ensuremath{\R^+}$, $ \Delta k_t\ge 0$, if $Eh(t,X_t)<u_t$ and $\Delta k_t\le 0$, if $Eh(t,X_t)>l_t$. \end{enumerate} \end{definition} We consider the following assumption on $h$.
\begin{enumerate} \item[(H)] $h$ satisfies (\ref{eq2.1}), $x\mapsto h(t,x)$ is strictly increasing for $t\in\ensuremath{\R^+}$
and there exist constants $\lambda_h,c_h^2,c_h^1>0$ such that
\begin{equation}\label{eq2.2}|h(t,x)-h(s,x)|\le \lambda_h|t-s|, \quad t,s\in\ensuremath{\R^+},\,x\in\ensuremath{\mathbb R}\end{equation} and
\begin{equation}c_h^1|x-y|\le |h(t,x)-h(t,y)|\le c_h^2|x-y|, \quad t\in\ensuremath{\R^+},\,x,y\in\ensuremath{\mathbb R}.\label{eq2.3} \end{equation} \end{enumerate}
For given $t\in\ensuremath{\R^+}$, $Y\in\ensuremath{{\mathbb L}}^1$ and $h$ satisfying {(H)} we define new map $H(t,\cdot,Y):\ensuremath{\mathbb R}\to\ensuremath{\mathbb R}$ by \begin{equation}\label{eq2.4}H(t,z,Y)=Eh(t,Y-EY+z),\quad z\in\ensuremath{\mathbb R}.\end{equation} By (\ref{eq2.1}), $H(t,\cdot,Y)$ is well defined. Moreover, by {(H)}, it is strictly increasing, continuous and $\lim_{z\to-\infty}H(t,z,Y)=-\infty$, $\lim_{z\to+\infty}H(t,z,Y)=+\infty$. Hence there exists strictly increasing and continuous inverse map $H^{-1}(t,\cdot,Y):\ensuremath{\mathbb R}\to\ensuremath{\mathbb R}$. Clearly, for every $z\in\ensuremath{\mathbb R}$, \begin{equation} \label{eq2.5}H^{-1}(t,z,Y)=\bar z\quad{\rm iff}\quad Eh(t,Y-EY+\bar z)=z. \end{equation} It is obvious that if $h(t,x)=x$, $x\in\ensuremath{\mathbb R}$, hen $H^{-1}(t,z,Y)=z$, $z\in\ensuremath{\mathbb R}$. \begin{lemma} Assume \mbox{\rm(H)} and let $Y=(Y_t)\in{\cal D}$. If $\bar z=(\bar z_t)\in\ensuremath{{D}(\Rp,\R)}$, then \[z=(z_t=H(t,\bar z_t,Y_t))\in\ensuremath{{D}(\Rp,\R)}. \] Similarly, if $z=(z_t)\in\ensuremath{{D}(\Rp,\R)}$, then \[\bar z=(\bar z_t=H^{-1}(t,z_t,Y_t))\in\ensuremath{{D}(\Rp,\R)}. \] \label{lem1} \end{lemma} \begin{proof} It is easy to observe that by the Lebesgue dominated convergence theorem, the function $t\mapsto z_t=Eh(t,Y_t-EY_t+\bar z_t)$ is c\`adl\`ag. To check the second conclusion assume that $z=(z_t)\in\ensuremath{{D}(\Rp,\R)}$. If $t_n\to t$, $t_n\geq t$, then from the right continuity of $z,$ and $Y$, $z_{t_n}\to z_t$ and $Y_{t_n}\to Y_t$
$P$-a.s., which implies that the sequence $\{\bar z_{t_n}\}$ is bounded (if not, there exists a subsequence $(n')\subset(n)$ such that $\lim_{t_{n'}}|Eh(t_{n'},Y_{t_{n'}}-EY_{t_{n'}}+\bar z_{t_{n'}})|=+\infty$). Consequently, we may and will assume that for some subsequence $(n')\subset(n)$, $\bar z_{t_{n'}}\to z'$. Then using once again the Lebesgue dominated convergence shows that $z'$ satisfies the equation \[ z_t=Eh(t,Y_t-EY_t+z'), \] which implies that $\bar z_t=z'$ and completes the proof of the right continuity of $\bar z=(\bar z_t)$. Similarly we show that if $t_n\to t$, $t_n< t$, then there exists a limit $\bar z_{t-}$ of $\{\bar z_{t_n}\}$ satisfying \[z_{t-}=Eh(t,Y_{t-}-EY_{t-}+\bar z_{t-}). \] \end{proof} \begin{theorem} Assume \mbox{\rm(H)}. If $Y=(Y_t)\in{\cal D}$, $l,u\in\ensuremath{{D}(\Rp,\R)}$ and $Eh(0,Y_0)\in[l_0,u_0]$, then there exists a unique solution of the Skorokhod problem $(X,k)=\SP{Y}$ . Moreover, \begin{equation}\label{eq2.6} k_t=-\max(0\wedge\inf_{0\leq u\leq t}(EY_u-\bar l_u), \sup_{0\leq s\leq t}[(EY_s-\bar u_s) \wedge\inf_{0\leq u\leq t}(EY_u-\bar l_u)]),\quad t\in\ensuremath{\R^+}, \end{equation} where $\bar l=H^{-1}\circ l$, $\bar u=H^{-1}\circ u$, i.e. $\bar l_t=H^{-1}(t,l_t,Y_t)$, $\bar u_t=H^{-1}(t,u_t,Y_t)$, $t\in\ensuremath{\R^+}$. \label{thm1} \end{theorem} \begin{proof} In the proof we use results from the theory of deterministic Skorokhod problem (see Appendix). First we show the existence of solutions of the Skorokhod problem with mean minimality condition. Set $y_t=EY_t$, $t\in\ensuremath{\R^+}$, and observe that $y=(y_t)\in\ensuremath{{D}(\Rp,\R)}$. By Lemma \ref{lem1}, $\bar l=H^{-1}\circ l\in\ensuremath{{D}(\Rp,\R)}$, $\bar u=H^{-1}\circ u\in\ensuremath{{D}(\Rp,\R)}$. Since $H^{-1}$ is strictly increasing, \[EY_0=H^{-1}(0,Eh(0,Y_0),Y_0)\in[\bar l_0,\bar u_0].\] By Theorem \ref{thm5.1}, there exists a unique solution of the Skorokhod problem $(x,k)=SP_{\bar l}^{\bar u}(y)$ such that for every $0\leq t\leq q$, \begin{eqnarray*} k_q-k_t\geq0&&\quad\mbox{\rm if }x_s<H^{-1}(s,u_s,Y_s)\,\,\mbox{\rm for all }s\in(t,q],\\ k_q-k_t\leq0&&\quad\mbox{\rm if }x_s>H^{-1}(s,l_s,Y_s)\,\,\mbox{\rm for all }s\in(t,q], \end{eqnarray*} and for every $t\in\ensuremath{\R^+}$, $ \Delta k_t\geq0$ if $x_t<H^{-1}(t,u_t,Y_t)$ and $\Delta k_t\leq0$ if $x_t>H^{-1}(t,l_t,Y_t)$.
Set \[ X_t=Y_t+k_t,\quad t\in\ensuremath{\R^+}, \] and note that $(X,k)=\SP{Y}$. Indeed, since $H$ is strictly increasing, \[ Eh(t,X_t)=Eh(t,Y_t+k_t)=H(t,x_t,Y_t) \in[H\circ\bar l_t,H\circ u_t]=[l_t,u_t],\quad t\in\ensuremath{\R^+}, \] which gives condition (ii) from Definition \ref{def1}. To check condition (iii) it is sufficient to observe that for every $s\in\ensuremath{\R^+}$, \[x_s<H^{-1}(s,u_s,Y_s)\quad {\rm iff}\quad Eh(s,X_s)<u_s\] and \[x_s>H^{-1}(s,l_s,Y_s)\quad {\rm iff}\quad Eh(s,X_s)>l_s.\]
It is easy to prove that every solution of the Skorokhod problem with mean minimality condition is of the form given above. Indeed, using the arguments used previously one can check that if $(X',k')=\SP{Y}$, then $(x',k')=SP_{\bar l}^{\bar u}(y)$, where $x'_t=H^{-1}(t,Eh(t,X'_t),Y_t)$, $t\in\ensuremath{\R^+}$. By the uniqueness of the deterministic Skorokhod problem (see Theorem \ref{thm5.1}), $(x',k')=(x,k)$, which implies that $X'_t=Y_t+k_t=X_t$, $t\in\ensuremath{\R^+}$. Finally, observe that the form of the compensating reflection part $k$ follows easily from (\ref{eq5.1}). \end{proof}
\begin{proposition} \label{prop1} Assume that $h^i$, $i=2$, satisfy \mbox{ \rm(H)} and $Y^i\in{\cal D}$, $i=1,2$, are processes defined on the same probability space If $l^i,u^i\in\ensuremath{{D}(\Rp,\R)}$ are such that $l^i\leq u^i$, $l^i_0\leq Eh^i(0,Y_0)\leq u^i_0$, $(X^i, k^i)=\mathbb{SP}^{u^i}_{l^i}(h^i,Y^i)$, $i=1,2$, and $C_h=\max((c_h^1)^{-1},2c_h^2(c^1_h)^{-1}+1)$, then for every $q\in\ensuremath{\R^+}$, \begin{equation}
\label{eqprop}\sup_{t\leq q}|k^1_t-k^2_t|\leq C_h\big(\parallel h^1-h^2\parallel_{[0,q]\times\ensuremath{\mathbb R}}
+ \sup_{t\leq q}E|Y^1_t-Y^2_t|+\sup_{t\leq q}
\max(|l^1_t-l^2_t|,|u^1_t-u^2_t|)\big) \end{equation} and
\begin{eqnarray}\label{eqprop1}E\sup_{t\leq q}|X^1_t-X^2_t|
&\leq& (C_h+1)E\sup_{t\le q}|Y^1_t-Y^2_t|+ C_h\big(\parallel h^1-h^2\parallel_{[0,q]\times\ensuremath{\mathbb R}}\nonumber\\
&&\qquad+\sup_{t\leq q}\max(|l^1_t-l^2_t|,|u^1_t-u^2_t|)\big), \end{eqnarray} where $\parallel h^1-h^2\parallel_{[0,q]\times\ensuremath{\mathbb R}}
=\sup_{(t,x)\in[0,q]\times\ensuremath{\mathbb R}}|h^1(t,x)-h^2(t,x)|$. \end{proposition} \begin{proof} From the proof of Theorem \ref{thm1} we know that $k^i$ are compensation reflection parts of solutions of the deterministic Skorokhod problems $SP_{\bar l^i}^{\bar u^i}(y^i)$ with $y^i_t=EY^i_t$, $\bar l^i_t=H_i^{-1}(t,l^i_t,Y^i_t)$, $\bar u^i_t=H_i^{-1}(t,u^i_t,Y^i_t)$, $t\in\ensuremath{\R^+}$, $i=1,2$. By (\ref{eq5.3}), \begin{equation}\label{eq2.7}
\sup_{t\leq q}|k^1_t-k^2_t|\leq\sup_{t\leq q}|EY^1_t-EY^2_t|
+\sup_{t\leq q}\max(|\bar l^1_t-\bar l^2_t|,|\bar u^1_t-\bar u^2_t|). \end{equation}
Fix $t\in[0,q]$. To estimate $|\bar l^1_t-\bar l^2_t|$ observe that \begin{eqnarray*} l^1_t-l^2_t&=&Eh^1(t,Y^1_t-EY^1_t+\bar l^1_t)-Eh^2(t,Y^2_t-EY^2_t+\bar l^2_t)\\ &=&Eh^1(t,Y^1_t-EY^1_t+\bar l^1_t)-Eh^2(t,Y^1_t-EY^1_t+\bar l^1_t)\\ &&\,+Eh^2(t,Y^1_t-EY^1_t+\bar l^1_t)-Eh^2(t,Y^2_t-EY^2_t+\bar l^1_t)\\ &&\,+Eh^2(t,Y^2_t-EY^2_t+\bar l^1_t)-Eh^2(t,Y^2_t-EY^2_t+\bar l^2_t)\\ &=&I^1_t+I^2_t+I^3_t. \end{eqnarray*} Without loss of generality we may assume that $\bar l^1_t>\bar l^2_t$. Then, by (\ref{eq2.3}), there exists $C\in[c_h^1,c_h^2]$ such that $I^3_t=C(\bar l^1_t-\bar l^2_t)$ , which implies that
\begin{eqnarray*}|\bar l^1_t-\bar l^2_t|&\leq& (1/c_h^1)(|I^1_t|+|I^2_t| +|l^1_t-l^2_t|)\\
&\leq&(1/c_h^1)(\sup_{x\in \ensuremath{\mathbb R}}|h^1(t,x)-h^2(t,x)|+
2c_h^2E|Y^1_t-Y^2_t| +|l^1_t-l^2_t|).\end{eqnarray*} Similarly we estimate $|\bar u^1_t-\bar u^2_t|$. From the above and (\ref{eq2.7}) we deduce (\ref{eqprop}). Estimete (\ref{eqprop1}) easily follows from (\ref{eqprop}). \end{proof}
\begin{corollary} \label{cor1}Assume \mbox{\rm ({H})}. If $Y\in{\cal D}$, $l,u\in\ensuremath{{D}(\Rp,\R)}$ are such that $l\leq u$, $l_0\leq Eh(0,Y_0)\leq u_0$, $(X, k)=\mathbb{SP}^u_l(h,Y)$, then for every $t,q\in\ensuremath{\R^+}$ such that $t\leq q$, \begin{equation}
\label{eqcor}\sup_{t\leq s\leq q}|k_s-k_t|\leq C_h\big( \sup_{t\leq s\leq q}E|Y_s-Y_t|+\lambda_h(q-t)+\sup_{t\leq s\leq q}\max(|l_s-l_t|,|u_s-u_t|)\big)\end{equation}
and\begin{eqnarray}\label{eqcor1}E\sup_{t\leq s\leq q}|X_s-X_t|&\leq&\nonumber
(C_h+1)E\sup_{t\leq s\leq q}|Y_s-Y_t|\\
&&\qquad+C_h\big(\lambda_h(q-t)+ \sup_{t\leq s\leq q}\max(|l_s-l_t|,|u_s-u_t|)\big).\end{eqnarray}\end{corollary} \begin{proof} Fix $t\leq q$ and in Proposition \ref{prop1} put $Y^1=Y$, $l^1=l$, $u^1=u$, $h^1=h$ and $Y^2=Y_{\cdot\wedge t}$, $l^2=l_{\cdot\wedge t}$, $u^2=u_{\cdot\wedge t}$, $h^2=h_{\cdot\wedge t}$. To complete the proof it is sufficient to observe that by (\ref{eq2.2}), \[
\sup_{(t,x)\in[0,q]\times\ensuremath{\mathbb R}}|h^1(t,x)-h^2(t,x)|
\leq \sup_{(s,x)\in[t,q]\times\ensuremath{\mathbb R}}|h(s,x)-h(t,x)|\leq \lambda_h(q-t). \] \end{proof}
\begin{remark}\label{rem_dwie} (a) In the above definition we can replace condition (iii) with the following one: \begin{enumerate} \item [{\bf (iii')}]for every $t\le q\in\ensuremath{\R^+}$ such that $\inf_{s\in[t,q]}(u_s-l_s)>0$, $k$ is a function of bounded variation on $[t,q]$ and \begin{equation}\label{eq_wrminidwie} \int_{[t,q]} [Eh(s,X_s)-l_s]\,dk_s\le 0 \text{ and }\int_{[t,q]} [Eh(s,X_s)-u_s]\,dk_s\le0. \end{equation} \end{enumerate}
Simple calculation shows that (i), (ii), (iii) and (i), (ii), (iii') are equivalent.
(b) Note that by \eqref{eq_wrminidwie}, if $\Delta k_t>0$, then $Eh(t,X_t)=l_t$ and hence $X_t=Y_t-EY_t+\bar l_t$. Similarly, if $\Delta k_t<0$, then $Eh(t,X_t)=u_t$ and $X_t=Y_t-EY_t+\bar u_t$. Consequently, \begin{equation}\label{eq_kwzor} k_t=\max(\min(k_{t-},\bar u_t-EY_t),\bar l_t-EY_t)),\quad t\in\ensuremath{\R^+}. \end{equation} \end{remark}
Under the additional assumption that the compensating reflection part $k$ has bounded variation the minimality condition {\bf (iii)} can be written in the following simpler form: for every
$t\in\ensuremath{\R^+}$, \[ \int_{0}^t [Eh(s,X_s)-l_s]\,dk^+_s=0 \text{ and }\int_{0}^t [u_s-Eh(s,X_s)]\,dk^-_s=0,\]
where $k^{(+)}$, $k^{(-)}$ are nondecreasing right continuous functions with $k_0=k^{(+)}_0=k^{(-)}_0=0$ such that $k^{(+)}$ increases only on $\{t;Eh(t,X_t)={l}_t\}$ and $k^{(-)}$ increases only on $\{t;Eh(t,X_t)={u}_t\}$. If the barriers $l,u\in\ensuremath{{D}(\Rp,\R)}$ satisfy the condition \begin{equation}\label{eq2.8} \inf_{t\leq q}({u}_t-{l}_t)>0,\quad q\in\ensuremath{\R^+}, \end{equation} then it is possible to estimate the variation of $k$ using Proposition \ref{prop5}.
\begin{corollary}\label{cor2} If $(X, k)=\mathbb{SP}^{u}_{l}(h,Y)$, then for any $q\in\ensuremath{\R^+}$ and $\eta$ such that $0<2\eta\leq\inf_{t\leq q}({u}_t-{l}_t)/3$ we have
\begin{equation}\label{eq2.9}|k|_q\leq 6(N_{\eta}(y,q)+N_{\eta}(\bar l,q)+N_{\eta}(\bar u,q)+1)(\sup_{
t\leq q}|y_t|+\sup_{ t\leq q}\max(|\bar l_t|,|\bar u_t|)),\end{equation} where $y_t=EY_t$, $\bar l_t=H^{-1}(t,l_t,Y_t)$ and $\bar u_t=H^{-1}(t,u_t,Y_t)$, $t\in\ensuremath{\R^+}$. \end{corollary}
Similarly to the case of the deterministic Skorokhod problem with $u=+\infty$ or $l=-\infty$ the definitions of solutions and forms of $k$ are much simpler. We start with the case of one lower barrier. \begin{definition} \label{def2} Let $Y$ be an $({\cal F}_t)$-adapted process with trajectories in $\ensuremath{{D}(\Rp,\R)}$,
$l\in\ensuremath{{D}(\Rp,\R)}$ and let $h:\ensuremath{\R^+}\times\ensuremath{\mathbb R}\rightarrow \ensuremath{\mathbb R}$ satisfy (\ref{eq2.1}) and $Eh(0,Y_0)\geq l_0$. We say that a pair $(X,k)\in{\cal D}\times\ensuremath{{D}(\Rp,\R)}$ with $k_0=0$ is a solution to the Skorokhod problem (with mean minimality condition and a lower barrier) associated with $h,Y,l$ ($(X,k)=\SPl{Y}$ for short) if \begin{enumerate}[\bf(i)] \item $X_t=Y_t+k_t$, \item $Eh(t,X_t)\geq l_t,\quad t\in\ensuremath{\R^+}$, \item $k$ is nondecreasing and \[\int_{0}^t [Eh(s,X_s)-l_s]\,dk_s=0,\quad t\in\ensuremath{\R^+}.\] \end{enumerate} \end{definition}
Using the arguments from the proof of Theorem \ref{thm1} and (\ref{eq5.7}) we obtain the following corollary. \begin{corollary} Assume \mbox{\rm{(H)}}. If $Y=(Y_t)\in{\cal D}$, $l\in\ensuremath{{D}(\Rp,\R)}$ and $Eh(0,Y_0)\geq l_0$, then there exists a unique solution of the Skorokhod problem $(X,k)=\SPl{Y}$. Moreover, \begin{equation}\label{eq2.10} k_t=\sup_{s\leq t}(\bar l_s-EY_s)^+,\quad t\in\ensuremath{\R^+},\end{equation} where $\bar l_t=H^{-1}(t,l_t,Y_t)$, $t\in\ensuremath{\R^+}$. \label{cor3} \end{corollary} Note that \[(\bar l_s-EY_s)^+=\inf\{x\geq0;E h(s,Y_s+x)\geq l_s\},\quad s\in\ensuremath{\R^+},\] which means that our formula for $k$ coincides with the formula considered in \cite{briand:2018,briand:2018a}.
\begin{definition} \label{def3} Let $Y$ be an $({\cal F}_t)$-adapted process with trajectories in $\ensuremath{{D}(\Rp,\R)}$,
$u\in\ensuremath{{D}(\Rp,\R)}$, and let$h:\ensuremath{\R^+}\times\ensuremath{\mathbb R}\rightarrow \ensuremath{\mathbb R}$ satisfy (\ref{eq2.1}) and $Eh(0,Y_0)\leq u_0$. We say that a pair $(X,k)\in{\cal D}\times\ensuremath{{D}(\Rp,\R)}$ with $k_0=0$ is a solution to the Skorokhod problem (with mean minimality condition and a upper barrier) associated with $h,Y,u$ ($(X,k)=\SPu{Y}$ for short) if \begin{enumerate}[\bf(i)] \item $X_t=Y_t+k_t$, \item $Eh(t,X_t)\leq u_t,\quad t\in\ensuremath{\R^+}$, \item $k$ is nonincreasing and \[\int_{0}^t [u_s-Eh(s,X_s)]\,dk_s=0,\quad t\in\ensuremath{\R^+}. \] \end{enumerate} \end{definition} By the arguments from the proof of Theorem \ref{thm1} and (\ref{eq5.8}) we obtain the the following counterpart to Corollary \ref{cor3}.
\begin{corollary} Assume \mbox{\rm{(H)}}. If $Y=(Y_t)\in{\cal D}$, $u\in\ensuremath{{D}(\Rp,\R)}$ and $Eh(0,Y_0)\leq u_0$, then there exists a unique solution of the Skorokhod problem $(X,k)=\SPu{Y}$. Moreover, \begin{equation}\label{eq2.11} k_t=-\sup_{s\leq t}(\bar u_s-EY_s)^-,\quad t\in\ensuremath{\R^+}, \end{equation} where $\bar u_t=H^{-1}(t,u_t,Y_t)$, $t\in\ensuremath{\R^+}$. \label{cor4} \end{corollary} One can note that \[(\bar u_s-EY_s)^-=\inf\{x\geq0;E h(s,Y_s-x)\leq u_s\},\quad s\in\ensuremath{\R^+}. \]
\section{Stability of solutions}
In this section, we consider a sequence of processes $\{Y^n\}$ such that for every $q\in\ensuremath{\R^+}$, \begin{equation} \label{eq3.1} \{Y^n_t;\, t\leq q,\,n\in\ensuremath{{\mathbb N}}\}\,\,\mbox{\rm is uniformly integrable.} \end{equation} In the sequel, we will use the notion of the convergence in $\ensuremath{{D}(\Rp,\R^{d})}$ for different $d\in\ensuremath{{\mathbb N}}$. We recall that $(x^{n,1},...,x^{n,d})\longrightarrow(x^1,...,x^d)$ in $\ensuremath{{D}(\Rp,\R^{d})}$ if
for every $t\in\ensuremath{\R^+}$ there exists a sequence $t_n\to t$ such that for all $t'_n\to t$ and $t''_n\to t$ satisfying $t'_n<t_n\leq t''_n$, $ n\in\ensuremath{{\mathbb N}}$, we have \begin{equation}\label{eq3.2} x^{n,i}_{t'_n} \longrightarrow x^i_{t-}\quad\mbox{\rm and} \quad x^{n,i}_{t''_n}\longrightarrow x^i_t,\,\,i=1,\dots,d. \end{equation} Note that (\ref{eq3.2}) implies in particular that in the case of the jump in $t\in\ensuremath{\R^+}$ in the limit there exists a common sequence $t_n\to t$ such that $\Delta x^{n,i}_{t_n}\longrightarrow\Delta x^i_t$, $i=1,\dots,d$.
\begin{lemma} Assume \mbox{\rm{(H)}} and \mbox{\rm(\ref{eq3.1})}. Let $\{\bar z^n\}\subset\ensuremath{{D}(\Rp,\R)}$ and $z^n=(z^n_t=H(t,\bar z^n_t,Y^n_t))$, $n\in\ensuremath{{\mathbb N}}$. If $(Y^n,y^n,\bar z^n)\mathop{\rightarrow}_{\cal D} (Y,y,\bar z)$ in $\ensuremath{{D}(\Rp,\R^{3})}$, then \[ (Y^n,y^n,\bar z^n,z^n)\mathop{\rightarrow}_{\cal D} (Y,y,\bar z,z) \quad in\,\,\ensuremath{{D}(\Rp,\R^{4})}.\] Similarly, let $\{z^n\}\subset\ensuremath{{D}(\Rp,\R)}$ and $\bar z^n=(\bar z^n_t=H^{-1}(t,z^n_t,Y^n_t))$, $n\in\ensuremath{{\mathbb N}}$. If $(Y^n,y^n, z^n)\mathop{\rightarrow}_{\cal D} (Y,y, z)$ in $\ensuremath{{D}(\Rp,\R^{3})}$, then \[ (Y^n,y^n, z^n,\bar z^n)\mathop{\rightarrow}_{\cal D} (Y,y,z,\bar z) \quad in\,\,\ensuremath{{D}(\Rp,\R^{4})}.\] \label{lem2} \end{lemma} \begin{proof} Assume (\ref{eq3.2}) for the sequence $\{(Y^n,y^n,\bar z^n)\}$, i.e. assume that for every $t\in\ensuremath{\R^+}$ there exists a sequence $t_n\to t$ such that for all $t'_n\to t$, $t''_n\to t$ satisfying $t'_n<t_n\leq t''_n$, $ n\in\ensuremath{{\mathbb N}}$, we have \[ (Y^{n}_{t'_n},y^n_{t'_n},\bar z^n_{t'_n})\mathop{\rightarrow}_{\cal D} (Y_{t-},y_{t-},\bar z_{t-})\quad\mbox{\rm and}\quad (Y^{n}_{t''_n},y^n_{t''_n},\bar z^n_{t''_n})\mathop{\rightarrow}_{\cal D} (Y_{t},y_{t},\bar z_{t}). \] We will check that the same condition holds true for the sequence $\{(Y^n,y^n,\bar z^n,z^n)\}$. Fix $t\in\ensuremath{\R^+}$ and assume that there exists a sequence $\{r_n\}$ such that $r_n\to t$ and \[(Y^n_{r_n},y^n_{r_n},\bar z^n_{r_n})\mathop{\rightarrow}_{\cal D} (Y',y',\bar z')\quad\mbox{\rm in}\,\, \ensuremath{\mathbb R}^3. \] By the Lebesgue dominated convergence theorem, \[ z^n_{r_n}=Eh(r_n,Y^n_{r_n}-EY^n_{r_n}+\bar z^n_{r_n}) \longrightarrow Eh(t,Y'-EY'+\bar z'). \] Clearly the limit is equal to $z_t$ if $r_n\geq t_n$ for all sufficiently large $n$ (resp. $z_{t-}$ if $r_n<t_n$ for all sufficiently large $n$).
To prove the second part of the lemma we first show that for every $q\in\ensuremath{\R^+}$ the sequence $\{\sup_{t\leq q}|\bar z^n_t)|\}$ is bounded. To get contradiction, suppose that there is a sequence $r_n\leq q$ such that $\bar z^n_{r_n}\nearrow+\infty $ (resp. $\searrow-\infty$). Then there exists a subsequence $(n')\subset(n)$ such that $r_{n'}\to t$ and $Y^{n'}_{r_{n'}}\mathop{\rightarrow}_{\cal D} Y$ for some $t\le q$ and random variable $Y$. Consequently, \[ z^{n'}_{r_{n'}} =Eh(r_{n'},Y^{n'}_{r_{n'}}-EY^{n'}_{r_{n'}} +\bar z^{n'}_{r_{n'}})\longrightarrow+\infty \mbox{ (resp.}\,\,-\infty), \] which contradicts the fact that $\{z^{n'}_{r_{n'}}\}$ has two possible limit points $z_t$ and $z_{t-}$. Now we assume (\ref{eq3.2}) for the sequence $\{(Y^n,y^n, z^n)\}$. We are going to check that the same condition holds true for $\{(Y^n,y^n, z^n,\bar z^n)\}$. Let $\{r_n\}$ be such that $r_n\to t\leq q$. First we assume that $r_n\geq t_n$ for all sufficiently large $n$. In this case $Y^{n}_{r_{n}}\mathop{\rightarrow}_{\cal D} Y_t$ and $z^n_{r_n}\to z_t$. Moreover, there is a finite $z'$ such that for some subsequence $(n')\subset(n)$, $\bar z_{r_{n'}}\to z'$. Then using the Lebesgue dominated convergence theorem shows that $z'$ satisfies the equation \[z_t=Eh(t,Y_t-EY_t+z').\] Since the solution of the above equation is unique, by the same arguments as above we check that in fact $z'$ is the limit of the sequence $\{\bar z^n_{r_n}\}$. Hence $\bar z^n_{r_n}\to \bar z_t$. Similarly we show that if $r_n<t_n$ for all sufficiently large $n$, then $\bar z_{r_n}\longrightarrow \bar z_{t-}$ being a solution to the equation $z_{t-}=Eh(t,Y_{t-}-EY_{t-}+\bar z_{t-})$. \end{proof}
\begin{remark} {\rm Since the sequences $\{z^n\}$, $\{\bar z^n\}$ are deterministic, it is clear that in the statement of Lemma \ref{lem2} one can replace the convergence in law by the convergence in probability in $J_1$ or by the convergence $P\text{-a.s.}$ in $J_1$. }\label{rem4} \end{remark}
\begin{theorem} Assume \mbox{\rm({H})}. Let $\{Y^n\}$ be a sequence of processes satisfying \mbox{\rm(\ref{eq3.1})} and $\{l^n\},\{u^n\}$ be sequences of c\`adll\`ag functions such that $l^n\leq u^n$, $l^n_0\leq Eh(0,Y_0)\leq u^n_0$, $n\in\ensuremath{{\mathbb N}}$. Let $(X^n, k^n)=\mathbb{SP}^{u^n}_{l^n}(h,Y^n)$, $n\in\ensuremath{{\mathbb N}}$. \begin{enumerate}[\bf(i)] \item If $(Y^n,y^n,l^n,u^n)\mathop{\rightarrow}_{\cal D} (Y,y,l,u)$ in $\ensuremath{{D}(\Rp,\R^{4})}$, then \[(X^n,k^n,Y^n,y^n,l^n,u^n)\mathop{\rightarrow}_{\cal D} (X,k,Y,y,l,u) \quad\mbox{ in}\,\,\ensuremath{{D}(\Rp,\R^{6})},\] \item if $(Y^n,y^n,l^n,u^n)\mathop{\rightarrow}_{\cal P} (Y,y,l,u)$ in $\ensuremath{{D}(\Rp,\R^{4})}$, then \[(X^n,k^n,Y^n,y^n,l^n,u^n)\mathop{\rightarrow}_{\cal P} (X,k,Y,y,l,u) \quad\mbox{ in}\,\,\ensuremath{{D}(\Rp,\R^{6})}, \] \item if $(Y^n,y^n,l^n,u^n)\longrightarrow (Y,y,l,u)$ $P\text{-a.s.}$ in $\ensuremath{{D}(\Rp,\R^{4})}$, then \[(X^n,k^n,Y^n,y^n,l^n,u^n)\longrightarrow(X,k,Y,y,l,u)\quad P\text{-a.s.} \mbox{ in}\,\,\ensuremath{{D}(\Rp,\R^{6})},\] \end{enumerate} where $(X, k)=\mathbb{SP}^{u}_{l}(h,Y)$.\label{thm2} \end{theorem} \begin{proof}By Lemma \ref{lem2} and (\ref{eq3.2}) it is clear that \[(Y^n,y^n,l^n,\bar l^n,u^n,\bar u^n)\mathop{\rightarrow}_{\cal D} (Y,y,l,\bar l,u, \bar u)\quad\mbox{\rm in}\,\,\ensuremath{{D}(\Rp,\R^{6})}.\] By this and (\ref{eq5.4}), \[ (x^n,k^n,Y^n,y^n,l^n,\bar l^n,u^n,\bar u^n) \mathop{\rightarrow}_{\cal D} (x,k,Y,y,l,\bar l,u, \bar u)\quad\mbox{\rm in}\,\,\ensuremath{{D}(\Rp,\R^{8})},\] where $(x^n,k^n)=SP_{\bar l^n}^{\bar u^n}(y^n)$, $n\in\ensuremath{{\mathbb N}}$ and $(x,k)=SP_{\bar l}^{\bar u}(y)$, Since $X^n=Y^n+k^n$ and $X=Y+k$, assertion (i) easily follows. To prove (ii) and (iii) it is sufficient to use Remark \ref{rem4} and previously used arguments with the convergence in probability in $J_1$ or $P\text{-a.s.}$ in $J_1$ in place of the convergence in law. \end{proof}
In the above theorem joint convergence of $\{(Y^n,y^n)\}$ in $\ensuremath{{D}(\Rp,\R^{2})}$ is assumed. This assumption is much stronger than the convergence of the initial sequence $\{Y^n\}$.
If $Y^n\mathop{\rightarrow}_{\cal D} Y$ in $\ensuremath{{D}(\Rp,\R)}$, then it is clear only that $y^n_t=EY^n_t\to y_t=EY_t$ provided that $E|\Delta Y_t|=0$. Unfortunately, the functional convergence $y^n\to y$ in $\ensuremath{{D}(\Rp,\R)}$ need not hold.
\begin{example} \label{ex1} Let $V$ be an arbitrary nondeterministic random variable. For some $c>0$ put $Y^n_t={\bf 1}_{\{t\geq c+V/n\}}$, $t\in\ensuremath{\R^+}$, $n\in\ensuremath{{\mathbb N}}$. Then of course $Y^n\longrightarrow Y$ $P\text{-a.s.}$ in $\ensuremath{{D}(\Rp,\R)}$, where $Y_t={\bf 1}_{\{t\geq c\}}$, $t\in\ensuremath{\R^+}$. On the other hand $y^n$ do not tend to $y$ in $\ensuremath{{D}(\Rp,\R)}$ because there in no sequence $t_n\to c$ such that $\Delta y^n_{t_n}\to\Delta y_c=1$. \end{example}
However, in important cases one can deduce joint convergence of $\{(Y^n,y^n)\}$ in $\ensuremath{{D}(\Rp,\R^{2})}$
from the convergence of $\{Y^n\}$ in $\ensuremath{{D}(\Rp,\R)}$. \begin{proposition}\label{prop4} Let $\{Y^n\}$ be a sequence of processes satisfying \mbox{\rm(\ref{eq3.1})} and such that $Y^n\mathop{\rightarrow}_{\cal D} Y$ in $\ensuremath{{D}(\Rp,\R)}$. If $P(\Delta Y_t=0)=1$, $t\in\ensuremath{\R^+}$, or $\{Y^n\}$ is a sequence of processes with independent increments, then \begin{equation}\label{eq3.3} (Y^n,y^n)\mathop{\rightarrow}_{\cal D}(Y,y)\quad\mbox{ in}\,\,\ensuremath{{D}(\Rp,\R^{2})}. \end{equation} \end{proposition} \begin{proof} First assume that $P(\Delta Y_t=0)=1$, $t\in\ensuremath{\R^+}$. We will show that in this case $y^n$ tends to $y$ uniformly on compact sets. By (\ref{eq3.1}), for every $t\in\ensuremath{\R^+}$, $\Delta y_t=E\Delta Y_t=0$, which means that the map $t\mapsto y_t$ is continuous. Therefore, in order to finish the proof of (\ref{eq3.3}), it is sufficient to observe that for every $t$ and every sequence $\{t_n\}$ such that $t_n\to t$, \[y^n_{t_n}=EY^n_{t_n}\longrightarrow EY_t=y_t.\] The above property is a consequence of the fact that $Y^n_{t_n}\mathop{\rightarrow}_{\cal D} Y_t$, which holds true provided that $P(\Delta Y_t=0)=1$.
To prove the second case we will use Jacod's theorem \cite[Theorem 1.21]{jacod} which says that the convergence $Y^n\mathop{\rightarrow}_{\cal D} Y$ in $\ensuremath{{D}(\Rp,\R)}$ is equivalent to the joint convergence of their characteristics $(B^{h,n},\tilde C^{h,n},f*\nu^n)\longrightarrow(B^h,\tilde C^h,f*\nu)$ in $\ensuremath{{D}(\Rp,\R^{3})}$, $f\in C_s$. It is important here that $B^{h,n}_t=EY^{h,n}_t$, $t\in\ensuremath{\R^+}$,
$n\in\ensuremath{{\mathbb N}}$ and $h:\ensuremath{\mathbb R}\to\ensuremath{\mathbb R}$ is a continuous function depending on the parameter $a>0$ such that $|h|\leq a$ and if $|x|\leq a/2$
(resp. $|x|\geq a$) then $h(x)=x$ (resp. $h(x)=0$) and \[Y^{h,n}_t=Y^n_t-\sum_{s\leq t}(\Delta Y^n_s-h(\Delta Y_s)) =Y^n_t-J^{n,a}_t,\quad t\in\ensuremath{\R^+},\,n\in\ensuremath{{\mathbb N}}.\] Since $\{Y^{h,n}\}$ is a sequence of weakly convergent processes with independent increments with bounded jumps, the family of random variables
$\{\sup_{t\leq q}|Y^{h,n}_t|\}$ is uniformly integrable. By this and (\ref{eq3.1}),
$\lim_{a\to\infty}\limsup_{n\to\infty}\sup_{t\leq q}E|J^{n,a}_t|=0$. Since $y^n_t=B^{h,n}_t+EJ^{n,a}_t$, from \cite[Theorem 1.21]{jacod} we deduce that $y^n\longrightarrow y$ in $\ensuremath{{D}(\Rp,\R)}$. To prove (\ref{eq3.3}) it is sufficient to observe that for every $t\in\ensuremath{\R^+}$ and every sequence $\{t_n\}$ such that $t_n\to t$ and \[(B^{h,n}_{t_n},\tilde C^{h,n}_{t_n},f*\nu^n_{t_n})\longrightarrow(B^h_t,\tilde C^h_t,f*\nu_t),\quad (B^{h,n}_{t_n-},\tilde C^{h,n}_{t_n-},f*\nu^n_{t_n-}) \longrightarrow(B^h_{t-},\tilde C^h_{t-},f*\nu_{t-})\] the convergences $Y^n_{t_n}\mathop{\rightarrow}_{\cal D} Y_t$ and $Y^n_{t_n-}\mathop{\rightarrow}_{\cal D} Y_{t-}$ also hold true. \end{proof}
\begin{corollary}\label{cor5} Let $\{Y^n\}$ be a sequence of processes satisfying \mbox{\rm(\ref{eq3.1})} and such that $Y^n\mathop{\rightarrow}_{\cal D} Y$ in $\ensuremath{{D}(\Rp,\R)}$. If for every $q$ and all sequences $\{s_n\}$, $\{r_n\}$ such that $0\leq s_n,r_n\leq q$ and $s_n-r_n\to0$ we have \begin{equation}\label{eq3.4}Y^n_{s_n}-Y^n_{r_n}\arrowp0, \end{equation} then \mbox{\rm(\ref{eq3.3})} holds true. \end{corollary} \begin{proof} It is sufficient to observe (\ref{eq3.4}) implies that function $y$ is continuous. \end{proof}
Condition (\ref{eq3.3}) is also satisfied in the case where $Y^n$, $n\in\ensuremath{{\mathbb N}}$, are discretizations of $Y$. This allows us to approximate
the solution $(X,k)=\SP{Y}$ by simple discretization method. As above set $y=(y_t=EY_t)$ and $\rho^n_t=k/n$ for $t\in[k/n,(k+1)/n)$. Let $Y^n,y^n,l^n,u^n$ be discretizations of $Y,y,l,u$, that is $Y^n_t=Y_{\rho^n_t}$, $y^n_t=y_{\rho^n_t}$, $l^n_t=l_{\rho^n_t}$ and $u^n_t=u_{\rho^n_t}$ $t\in\ensuremath{\R^+}$. Set \begin{equation}\label{eq3.6} \left\{\begin{array}{ll} k^n_0&=0,\qquad X^n_0=Y_0,\\[2mm] k^n_{{(k+1)}/{n}}&= \max\big[\min\big[k^n_{{k}/{n}}, \bar u^{n}_{(k+1)/{n}}-EY_{(k+1)/n}\big], \bar l^n_{(k+1)/n}-EY_{(k+1)/n})\big],\\[2mm] X^n_{{(k+1)}/{n}}&= Y_{(k+1)/n}+k^n_{(k+1)/n}, \end{array} \right. \end{equation} where $\bar l_{(k+1)/n}=H^{-1}((k+1)/n,l_{(k+1)/n},Y_{(k+1)/n})$, $\bar u_{(k+1)/n}=H^{-1}((k+1)/n,u_{(k+1)/n},Y_{(k+1)/n})$ and
$k^n_t=k^n_{k/n}$, $X^n_t=X^n_{k/n}$ for $t\in[k/n,(k+1)/n)$, $k\in\ensuremath{{\mathbb N}}\cup\{0\}$, $n\in\ensuremath{{\mathbb N}}$.
\begin{theorem}\label{thm3}Assume \mbox{\rm({H})}. Let $Y\in{\cal D}$ and $l,u\in\ensuremath{{D}(\Rp,\R)}$ be such that $l\leq u$ and $l_0\leq Eh(0,Y_0)\leq u_0$. Let $\{Y^n\}$ and $\{l^n\}$, $\{u^n\}$ be sequences of discretizations of $Y$ and $l,u$, respectively, and $(X^n, k^n)$ be defined by \mbox{\rm(\ref{eq3.6})}, $n\in\ensuremath{{\mathbb N}}$. Then $(X^n, k^n)=\mathbb{SP}^{u^n}_{l^n}(h,Y^n)$, $n\in\ensuremath{{\mathbb N}}$ and \begin{enumerate}\item[{\bf (i)}] $\displaystyle{(X^n,k^n,Y^n,y^n,l^n,u^n)\longrightarrow (X,k,Y,y,l,u)\quad P\text{-a.s.}\,\,\mbox{ in}\,\,\ensuremath{{D}(\Rp,\R^{6})}},$ \item[{\bf (ii)}] for every $q\in\ensuremath{\R^+}$,
\[\max_{k;\,k/n\leq q}(|X^n_{k/n}-X_{k/n}|+|k^n_{k/n}-k_{k/n}|) \lra0\quad P\text{-a.s.},\] \end{enumerate} where $(X,k)=\SP{Y}$. \end{theorem}
\begin{proof} First note that in the case where $y,l$ or $u$ have a jump in $t$ there is a common sequence $\{t^*_n=\inf\{k/n;k/n\geq t\}\}$ such that $t^*_n\to t$ and $\Delta Y^n_{t^*_n}\to\Delta Y_t$, $\Delta y^n_{t^*_n}\to\Delta y_t$, $\Delta l^n_{t^*_n}\to\Delta l_t$ and $\Delta u^n_{t^*_n}\to\Delta u_t$. Consequently, \begin{equation}\label{eq3.7} (Y^n,y^n,l^n,u^n)\longrightarrow (Y,y,l,u)\quad P\text{-a.s.}\,\,\mbox{\rm in}\,\,\ensuremath{{D}(\Rp,\R^{4})}.\end{equation} Let $(x^n,k^n)$ be the solution of the deterministic Skorokhod problem associated with $y^n$, $\bar l^n$ and $\bar u^n$, $n\in\ensuremath{{\mathbb N}}$. Then $k^n$ and $X^n=Y^n+k^n$ have the form given in (\ref{eq3.6}). Moreover, from (\ref{eq3.7}) and Theorem \ref{thm2}(iii) the assertion (i) easily follows. To prove (ii) observe that for every $t\in\ensuremath{\R^+}$ the convergences $\Delta \hat X^n_{t^*_n}\to\Delta X_t$, $\Delta \hat k^n_{t^*_n}\to\Delta k_t$, hold true, where $\hat X^n$, $\hat k^n$ are discretizations of $X$, $k$ i.e. $\hat X^n_t=X_{k/n}$, $\hat k^n_t=k_{k/n}$ for $t\in[k/n,(k+1)/n)$, $k\in\ensuremath{{\mathbb N}}\cup\{0\}$, $n\in\ensuremath{{\mathbb N}}$. Therefore \[( X^n,\hat X^n,k^n,\hat k^n)\longrightarrow (X,X,k,k)\quad P\text{-a.s.}\,\,\mbox{ in}\,\,\ensuremath{{D}(\Rp,\R^{4d})},\] which implies that $X^n-\hat X^n\to0$, $k^n-\hat k^n\to0$ $P\text{-a.s.}$ in $\ensuremath{{D}(\Rp,\R)}$. This completes the proof. \end{proof}
\section{SDEs with mean reflection}
Let $ l,u\in\ensuremath{{D}(\Rp,\R)}$ be such that $l\le u$ and $(\Omega, {\cal F}, ({\cal F}_t), P)$ be a filtered probability space.
Let $X_0$ be an ${\cal F}_0$-measurable integrable random variable, $V$ be an $({\cal F}_t)$-adapted process with trajectories in $\ensuremath{{D}(\Rp,\R)}$ such that
$E|V|_q<\infty$ for every $q\in\ensuremath{\R^+}$, and let $M$ be a square integrable $({\cal F}_t)$-martingale.
We consider equation with mean reflection of the form (\ref{eq1.1}). \begin{definition}\label{def_solutiion} We say that a pair $(X,k)$ of $({\cal F}_t)$ adapted processes with trajectories in $\ensuremath{{D}(\Rp,\R)}$ is a strong solution of (\ref{eq1.1}) if $(X,k)=\SP{Y}$, where \[ Y_t=X_0+\int_0^tf(s,X_{s-})\,dM_s+\int_0^tg(s,X_{s-})\,dV_s,\quad t\in\ensuremath{\R^+}. \] \end{definition}
We will need the following conditions on the coefficients $f,g$. \begin{enumerate} \item[(A1)] $f$ and $g$ are continuous and there exists $\mu>0$ such that
\[|f(t,x)|+|g(t,x)|\leq \mu(1+|x|),\quad t\in\ensuremath{\R^+},\,\,x,\,y\in\ensuremath{\mathbb R}.\] \item[(A2)] There exists $c>0$ such that
\[|f(t,x)-f(t,y)|+|g(t,x)-g(t,y)|\leq c|x-y|,\quad t\in\ensuremath{\R^+},\,\,x,\,y\in\ensuremath{\mathbb R}.\] \end{enumerate} We will also assume boundedness of predictable characteristics of $M,V$ in the following sense. \begin{enumerate} \item[(M)] There exists a nondecreasing c\`adl\`ag function $m:\ensuremath{\R^+}\rightarrow\ensuremath{\mathbb R}$ with $m_0=0$ such that
\[\max(\langle M\rangle_t,\widetilde{|V|}_t)\le m_t,\quad t\in\ensuremath{\R^+}.\] \end{enumerate}
Below we give simple examples of processes $M,V$ satisfying (M). \begin{example}\label{ex2} (a) Let $Z$ be a semimartingale with independent increments and bounded jumps (i.e. there exists $a>0$
such that $|\Delta Z|\leq a$). Then $Z$ is of the form
\begin{equation}\label{eq4.0}Z_t=B_t +M_t,\quad t\in\ensuremath{\R^+},\end{equation} where $B=(B_t=EZ_t)$ is a deterministic function with locally bounded variation, $|\Delta B|\leq a$ and
$M=(M_t=Z_t-EZ_t)$ is a square integrable martingale, $|\Delta M|\leq 2a$. In this case $\widetilde{|B|}=|B|$ and $\langle M\rangle$ are deterministic nondecreasing functions starting from
$0$. Clearly, $M$ and $B$ satisfy (M) with $m_t=\max(|B|_t,\langle M\rangle_t)$, $t\in\ensuremath{\R^+}$.
(b) Let $M,V$ satisfy (M) and $H^1,H^2$ be two bounded predictable processs. Then $\int_0^\cdot H^1_s\,dM_s$, $\int_0^\cdot H^2_s\,dV_s$ also satisfy (M) with slightly modified function $m$. \end{example}
We will use the following version of the estimate \cite[Chapter 1, Section 9 Theorem 5]{LS}: for every square integrable martingale $M$ and every $t>0$,
\begin{equation}\label{eq4.1}E\sup_{s<t}|M_s|\leq 3E(\langle M\rangle_{t-})^{1/2}. \end{equation}
\begin{theorem} \label{thm4} Assume \mbox{\rm(H)}, \mbox{\rm(A1)}, \mbox{\rm(A2)} and \mbox{\rm (M)}. If $l_0\leq Eh(0,X_0)\leq u_0$, then there exists a unique strong solution $(X,k)$ of \eqref{eq1.1} such that
$E\sup_{t\leq q} |X_t|<+\infty$, $q\in\ensuremath{\R^+}$. \end{theorem} \begin{proof} Set $t_1=\inf\{t>0:(C_h+1)c\max(m_t,3(m_t)^{1/2})>1/2\}$, where $C_h$ is the constant from Proposition \ref{prop1}. In the first step of the proof we show the existence and uniqueness of solutions of \eqref{eq1.1} on the interval $[0,t_1)$. Set \[ {\mathcal S}^1=\{Y:\, Y\text{ is }(\ensuremath{{\cal F}}_t)\text{-adapted, }
Y_0=X_0,\,\,Y=Y^{t_1-},\,\,E\sup_{t<t_1}|Y_t|<\infty\} \] and define the map $\Phi:\, {\mathcal S}^1\rightarrow{\mathcal S}^1$ by putting $\Phi(Y)$ to be the first coordinate of the solution of the Skorokhod problem $\SP{Z}$ with $Z= X_0+\int_0^\cdot f(s,Y_{s-})\,dM^{t_1-}_s+\int_0^\cdot g(s,Y_{s-})\,dV^{t_1-}_s$ stopped in $t_1-$. First we prove that if $Y\in{\mathcal S}^1$, then $\Phi(Y)\in{\mathcal S}^1$. Since $k$ is deterministic it is sufficient to prove that $Z\in{\mathcal S}^1$. By (\ref{eq4.1}), (M) and (A1), \begin{eqnarray*}
E\sup_{t<t_1}|Z_t|&\leq& E|X_0|
+E\sup_{t<t_1}\int_0^t|f(s,Y_{s-})\,dM_s|
+E\int_0^{t_1-}| g(s,Y_{s-})|\,d|V|_s\\
&\leq&E|X_0|+3E(\int_0^{t_1-}|f(s,Y_{s-})|^2\,d\langle M\rangle_s)^{1/2}+E\int_0^{t_1-}| g(s,Y_{s-})|\,d\widetilde{|V|}_s\\
&\leq&E|X_0|+3(m_{t_1-})^{1/2}E\sup_{s< t_1}|f(s,Y_{s-})|+m_{t_1-}E\sup_{s< t_1}|g(s,Y_{s-})|\\
&\leq&E|X_0|+\max(3(m_{t_1-})^{1/2},m_{t_1-})\mu(1+E\sup_{s<t_1}|Y_s|)<+\infty. \end{eqnarray*} By Proposition \ref{prop1}, for any $Y,Y'\in{\mathcal S}^1$ we have \begin{align*}
E\sup_t |\Phi(Y)_t-\Phi(Y')_t| & \le (C_h+1)E\sup_{t<t_1} \left|\int_0^{t} f(s,Y_{s-})-f(s,Y'_{s-})\,dM_s\right. \\
&\quad+\left.\int_0^{t_1-}| g(s,Y_{s-})-g(s,Y'_{s-})|\,d|V|_s\right|\\
&\le (C_h+1)3E\left(\int_0^{t_1-}|f(s,Y_{s-})-f(s,Y'_{s-})|^2\,d\langle M\rangle_s\right)^{1/2}\\
&\quad+(C_h+1)E\int_0^{t_1-}|g(s,Y_{s-})-g(s,Y'_{s-})|\,d\widetilde{|V|}_s)\\
&\le (C_h+1)c\max(3(m_{t_1-})^{1/2},m_{t_1-})E\sup_t|Y_t-Y'_t|\\
&\le \frac 12 E\sup_t|Y_t-Y'_t|. \end{align*} By the Banach contraction principle, there exists a unique solution $(X,k)$ of \eqref{eq1.1} on the interval $[0,t_1)$. By Remark \ref{rem_dwie} (b) we can get a unique solution on $[0,t_1]$ by putting \[k_{t_1}=\max(\min(k_{t_1-},\bar u_{t_1}-EY_{t_1}),\bar l_{t_1}-EY_{t_1})),\quad X_{t_1}=Y_{t_1}+k_{t_1},\] where $Y_{t_1}=X_{t_1-}+f(t_1,X_{t_1-})\Delta V_{t_1}+g(t_1,X_{t_1-})\Delta M_{t_1}$, $\bar l_{t_1}=H^{-1}(t_1,l_{t_1},Y_{t_1})$ and $\bar u_{t_1}=H^{-1}(t_1,u_{t_1},Y_{t_1})$. Now we define the sequence $\{t_k\}$ by setting $t_{k+1}=t_k+\inf\{t>0:(C_h+1)c\min(\hat m_t,3(\hat m_t)^{1/2})>1/2\}$, where $\hat m_t=m_{t_k+t}-m_{t_k}$, $k\in\ensuremath{{\mathbb N}}$. Arguing as above, one can obtain a solution of \eqref{eq1.1} on $[t_k,t_{k+1}]$. Since $t\mapsto m_t$ is c\`adl\`ag, $t_k\uparrow\infty$ and the unique solution on $\ensuremath{\R^+}$ we obtain by putting together the solutions on intervals $[t_k,t_{k+1}]$, $k\in\ensuremath{{\mathbb N}}$. \end{proof} The following example shows that solutions of (\ref{eq1.1}) can be applied in investment models with constraints.
\begin{example} Consider an insurance company with initial capital $x>0$, whose risk reserve process is given by a semimartingale $J$ with independent increments and bounded jumps. The company is allowed to invest the risk reserve into the financial market consisting of the riskless bond $B_t=1$, $t\in\ensuremath{\R^+}$, and the risky stock whose price $S$ evolves according to the stochastic differential equation \[S_t=S_0+\int_0^tS_ub\,du+\int_0^tS_u\sigma \, dW_u,\quad t\in\ensuremath{\R^+},\] where $W$ is a Wiener process, $\sigma>0$ and $b\in\ensuremath{\mathbb R}$. Let $V^{k,\pi}$ denote the company wealth portfolio under investment strategy $(-k,\pi)$, that is \begin{equation}\label{eq_kapital} V^{k,\pi}_t=-k_t+\pi_tS_t,\quad t\in\ensuremath{\R^+}. \end{equation} In \cite{ChM} (see also \cite{BR}) the authors suggest that it is reasonable to assume that the strategy is deterministic. Following this suggestion we restrict ourselves to strategies $(-k,\pi)$ such that $k$ is deterministic and $(-k,\pi)$ is self-financing, that is the dynamics of $V^{k,\pi}$ is given by the following equation \begin{equation}\label{eq_samo}V^{k,\pi}_t=x+\int_0^t\pi_{u}\,dS_u+\int_0^t\,dJ_u,\quad t\in\ensuremath{\R^+}. \end{equation} Let $X_t=\pi_tS_t$, $t\in\ensuremath{\R^+}$, be the amount of money invested in stock at time $t\in\ensuremath{\R^+}$. By \eqref{eq_kapital} and \eqref{eq_samo}, $X$ satisfies the equation \[ X_t=x+\int_0^t X_{s}b\,ds+\int_0^tX_{s}\sigma\,dW_s +J_t+k_t,\quad t\in\ensuremath{\R^+}. \]
Let $l,u\in\ensuremath{{D}(\Rp,\R)}$, $l\le u$ and $h:\ensuremath{\R^+}\times \ensuremath{\mathbb R}\rightarrow\ensuremath{\mathbb R}$ be a concave function satisfying (H). We consider risk measure imposing some restriction on the class of admissible strategies. Namely, a portfolio is considered admissible if and only if for every $t\in\ensuremath{\R^+}$ the amount of money $X_t$ invested in the risky stock satisfies the following constraints \begin{equation}\label{eq_admis} L_t(X_t)\le 0\le U_t(X_t),\quad t\in\ensuremath{\R^+}, \end{equation} where $(L_t)$ and $(U_t)$ are collections of convex risk measures given by the formulas \[ L_t(X_t)=\inf\{k\in \ensuremath{\mathbb R}:Eh(t,X_t+k)\ge l_t\},\quad U_t(X_t)=\sup\{k\in\ensuremath{\mathbb R}:Eh(t,X_t+k)\le u_t\}. \] (recall that a map $\rho:\ensuremath{{\mathbb L}}^1\rightarrow\ensuremath{\mathbb R}$ is a convex risk measure iff for all $a\in\ensuremath{\mathbb R}$, $\lambda\in[0,1]$ and $X$, $Y\in\ensuremath{{\mathbb L}}^1$ such that $X\le Y$ we have that $\rho(X+k)=\rho(X)-k$, $\rho(X)\ge \rho(Y)$ and $\rho(\lambda X+(1-\lambda) Y)\le \lambda\rho(X)+(1-\lambda)\rho(Y)$).
Note that for every $t\in\ensuremath{\R^+}$ we have $L_t(X_t)\le U_t(X_t)$, so one can say that for every $t\in\ensuremath{\R^+}$ the risk measure $U_t$ is more restrictive than $L_t$. The interpretation of \eqref{eq_admis} is the following. If position $X$ is very risky (i.e. $L_t(X_t)>0$), then the company have to borrow some money from bank account and buy some amount of stock. On the other hand, if position $X$ is safe enough (i.e. $U_t(X_t)<0$), then the company sells some amount of stock and reduces the debt in bank account. The company is looking for the minimal strategy in the sense that if \eqref{eq_admis} is satisfied then there is no money flow from the bank account to the stock market or in the opposite direction.
Since for every $\ensuremath{\R^+}$ the condition $L_t(X_t)\le 0$ (resp. $U_t(X_t)\ge 0$) is equivalent to the condition $Eh(t,X_t)\ge l_t$ (resp. $Eh(t,X_t)\le u_t$), by Theorem \ref{thm4} there exists a unique minimal admissible strategy $(-k,\pi)$. More precisely, there exists a unique solution $(X,k)=\SP{x+\int_0^\cdot X_{s}b\,ds+\int_0^\cdot X_{s}\sigma\,dW_s+J}$ and if we set $\pi _t=S^{-1}_tX_t$, $t\in\ensuremath{\R^+}$, then the wealth portfolio $V^{k,\pi}$ satisfies \eqref{eq_admis}. \end{example}
We are able to approximate
the solution $(X,k)$ of (\ref{eq1.1}) by a simple discretization method being a counterpart to the Euler scheme (see, e.g.,\cite{s4}). The scheme for the SDE (\ref{eq1.1}) is given by the following recurrent formula. Set \begin{equation}\label{eq4.2} \left\{\begin{array}{ll} k^n_0&=0,\qquad X^n_0=Y^n_0=Y_0,\\[2mm] Y^n_{(k+1)/n}&=Y^n_{k/n}+f((k+1)/n,X^n_{k/n})(M_{(k+1)/n}-M_{k/n})\\[2mm] &\qquad\quad+g((k+1)/n,X^n_{k/n})(V_{(k+1)/n}-V_{k/n}),\\[2mm] k^n_{{(k+1)}/{n}}&= \max\big[\min\big[k^n_{{k}/{n}}, \bar u^{n}_{(k+1)/{n}}-EY^n_{(k+1)/n}\big], \bar l^n_{(k+1)/n}-EY^n_{(k+1)/n})\big],\\[2mm] X^n_{{(k+1)}/{n}}&= Y^n_{(k+1)/n}+k^n_{(k+1)/n}, \end{array} \right. \end{equation} where $\bar l_{(k+1)/n}=H^{-1}((k+1)/n,l_{(k+1)/n},Y^n_{(k+1)/n})$, $\bar u_{(k+1)/n}=H^{-1}((k+1)/n,u_{(k+1)/n},Y^n_{(k+1)/n})$ and $Y^n_t=Y^n_{k/n}$, $k^n_t=k^n_{k/n}$, $X^n_t=X^n_{k/n}$ for $t\in[k/n,(k+1)/n)$, $k\in\ensuremath{{\mathbb N}}\cup\{0\}$, $n\in\ensuremath{{\mathbb N}}$.
\begin{theorem}\label{thm5} Assume \mbox{\rm (H)}, \mbox{\rm(A1)}, \mbox{\rm(A2)} and \mbox{\rm(M)}. If $X_0$ is an integrable random variable such that $l_0\leq Eh(0,X_0)\leq u_0$, $\{l^n\},\{u^n\}$ are sequences of discretizations of $l,u$ and $(X^n, k^n)$ and $Y^n$ are defined by \mbox{\rm(\ref{eq4.2})}, then $(X^n, k^n)=\mathbb{SP}^{u^n}_{l^n}(h,Y^n)$, $n\in\ensuremath{{\mathbb N}}$ and \begin{enumerate}\item[{\bf (i)}] $\displaystyle{(X^n,k^n,Y^n,l^n,u^n)\mathop{\rightarrow}_{\cal P} (X,k,Y,l,u)\quad \,\,\mbox{ in}\,\,\ensuremath{{D}(\Rp,\R^{5d})},}$ \item[{\bf (ii)}] for every $q\in\ensuremath{\R^+}$,
\[\max_{k;\,k/n\leq q}(|X^n_{k/n}-X_{k/n}|+|k^n_{k/n}-k_{k/n}|)\arrowp0,\] \end{enumerate} where $(X,k)$ is a unique strong solution of \mbox{\rm(\ref{eq1.1})}. \end{theorem}
\begin{proof} We know that $(X,k)=\SP{Y}$, where $Y_t=X_0+\int_0^tf(s,X_{s-})\,dM_s+\int_0^tg(s,X_{s-})\,dV_s$, $t\in\ensuremath{\R^+}$. Let $\hat X^n$, $\hat k^n$, $\hat Y^n$ be discretizations of $X,k,Y$, that is $\hat X^n_t=X_{\rho^n_t}$, $\hat k^n_t=k_{\rho^n_t}$, $\hat Y^n_t=Y_{\rho^n_t}$, $t\in\ensuremath{\R^+}$. Let $(\bar X^n,\bar k^n)$ be a solution of the Skorokhod problem associated with $\hat Y^n$, $ l^n$ and $u^n$, i.e. $(\bar X^n, \bar k^n)=\mathbb{SP}^{u^n}_{l^n}(h,\hat Y^n)$, $n\in\ensuremath{{\mathbb N}}$. Then, by Theorem \ref{thm3}(ii), for every $q\in\ensuremath{\R^+}$, \[
\sup_{t\leq q}|(\bar X^n_t-\hat X^n_t|+|\bar k^n_t-\hat k^n_t|)\lra0. \]
Fix $q\in\ensuremath{\R^+}$. Since $E\sup_{t\leq q}|X_t|<+\infty$ and $E\sup_{t\leq q}|Y_t| <+\infty$, the sequences
$\{\sup_{t\leq q}|\hat X^n_t|\}$ and $\{\sup_{t\leq q}|\bar X^n_t|\}$ are uniformly integrable, which implies that
\begin{equation}\label{eq4.3}\epsilon^n_1=E \sup_{t\leq q}|\bar X^n_t-\hat X^n_t|\lra0.\end{equation} Let $\{M^n\},\{V^n\}$ be sequences of discretizations of $M,V$, respectively. By (A1) and the Lebesgue dominated convergence theorem, \begin{eqnarray*}\nonumber
\epsilon^n_2&=&E \sup_{t\leq q}(|\int_0^{{t}}f(s,\hat X^n_{s-})\,dM^n_s-\int_0^{\rho^n_{t}}f(s, X_{s-})\,dM_s|\\
&&\qquad\qquad+ E|\int_0^{{t}}g(s,\hat X^n_{s-})\,dV^n_s-
\int_0^{\rho^n_{t}}g(s, X_{s-})\,dV_s |)\nonumber \\
&=&E (\sup_{t\leq q}(|\int_0^{{\rho^n_{t}}} (f(\rho^n_{s},\hat X^n_{s-})-f(s, X_{s-}))\,dM_s
+|\int_0^{{\rho^n_{t}}}(g(\rho^n_{s},\hat X^n_{s-})-
g(s, X_{s-}))\,dV_s |)\nonumber\\
&\leq&3E (|\int_0^{{\rho^n_{q}}}|f(\rho^n_{s},\hat X^n_{s-})-f(s, X_{s-})|^2\,dm_s)^{1/2} \nonumber\\&&\qquad\qquad+E\int_0^{{\rho^n_{q}}}|g(\rho^n_{s},\hat X^n_{s-})-
g(s, X_{s-})|\,dm_s |\lra0 \end{eqnarray*} because $\rho^n_s\to s$ and $\hat X^n_{s-}\to X_{s-}$ $P\text{-a.s.}$ for $s\in[0,q]$. On the other hand, by Theorem \ref{thm3}, the pair $(X^n, k^n)$ defined by (\ref{eq4.2}) is a solution of mean reflected SDEs with barriers $l^n$, $u^n$ of the form \[X^n_t=X_0+\int_0^{{t}}f(s, X^n_{s-})\,dM^n_s+\int_0^{{t}}g( s,X^n_{s-})\,dV^n_s +k^n_t,\quad t\in\ensuremath{\R^+},\] i.e. $(X^n, k^n)=\mathbb{SP}^{u^n}_{l^n}(h,Y^n)$, where $Y^n_t=X_0+\int_0^{{t}}f(s, X^n_{s-})\,dM^n_s+\int_0^{{t}}g( s,X^n_{s-})\,dV^n_s$, $t\in\ensuremath{\R^+}$, $n\in\ensuremath{{\mathbb N}}$. From the above and Proposition \ref{prop1}(ii), for every $t\leq q$,
\begin{eqnarray*}\nonumber E\sup_{s\leq t}|\hat X^n_s-X^n_s|&\leq&E\sup_{s\leq t}|\hat X^n_s-\bar X^n_s|+E\sup_{s\leq t}|\bar X^n_s-X^n_s|\\
&\leq&\epsilon^n_1+(C_h+1)E\sup_{s\leq t}|\hat Y^n_s-Y^n_s|\nonumber\\
&\leq&\epsilon^n_1+(C_h+1)\epsilon^n_2+(C_h+1)E\sup_{s\leq t}|\int_0^sf(u,\hat X^n_{u-})-f(u,X^n_{u-})\,dM^n_u|\nonumber\\
&&\quad+(C_h+1)E\sup_{s\leq t}|\int_0^sg(u,\hat X^n_{u-})-g(u,X^n_{u-})\,dV^n_s|.\nonumber\\
&=&\epsilon^n_1+(C_h+1)\epsilon^n_2+(C_h+1)E\sup_{s\leq t}|\int_0^{\rho^n_s}f(\rho^n_u,\hat X^n_{u-})-f(\rho^n_u,X^n_{u-})\,dM_u|\nonumber\\
&&\quad+(C_h+1)E\sup_{s\leq t}|\int_0^{\rho^n_s}g({\rho^n_u},\hat X^n_{u-})-g({\rho^n_u},X^n_{u-})\,dV_s|. \end{eqnarray*} Let $m^n$ be a discretization of the function $m$, i.e. $m^n_t=m_{\rho^n_t}$, $t\in\ensuremath{\R^+}$, $n\in\ensuremath{{\mathbb N}}$. Set $t^n_1=\inf\{t>0:(C_h+1)c\max(m^n_t,3(m^n_t)^{1/2})>1/2\}$ and observe that arguing similarly to the proof of Theorem \ref{thm3} shows that \begin{eqnarray*}
E\sup_{s< t^n_1}|\hat X^n_s-X^n_s|&\leq&\epsilon^n_1+(C_h+1)\epsilon^n_2+(C_h+1)E\sup_{s< t^n_1}|\int_0^{\rho^n_s}f(\rho^n_u,\hat X^n_{u-})-f(\rho^n_u,X^n_{u-})\,dM_u|\\
&&\quad+(C_h+1)E\sup_{s< t^n_1}|\int_0^{\rho^n_s}g({\rho^n_u},\hat X^n_{u-})-g({\rho^n_u},X^n_{u-})\,dV_s|.\\
&\leq&\epsilon^n_1+(C_h+1)\epsilon^n_2+(C_h+1)3cE(\int_0^{\rho^n_{t^n_1-}}|\hat X^n_{s-}-X^n_{s-}|^2\,d\langle M\rangle_s)^{1/2}\\
&&\quad+(C_h+1)cE\int_0^{\rho^n_{t^n_1-}}|\hat X^n_{s-}-X^n_{s-}|\,d\widetilde{|V|}_s\\
&\leq& \epsilon^n_1+(C_h+1)\epsilon^n_2+(C_h+1)3c (m^n_{t^n_1-})^{1/2}E\sup_{s<t^n_1}|\hat X^n_{s-}-X^n_{s-}|\\
&&\qquad+(C_h+1)c m^n_{t^n_1-}E\sup_{s<t^n_1}|\hat X^n_{s-}-X^n_{s-}|\\
&\leq& \epsilon^n_1+(C_h+1)\epsilon^n_2+\frac 12 E\sup_{s<t^n_1}|\hat X^n_{s}-X^n_{s}|, \end{eqnarray*} which implies that \begin{equation} \label{eq4.6}
E\sup_{s<t^n_1}|\hat X^n_{s}-X^n_{s}|\leq 2(\epsilon^n_1+(C_h+1)\epsilon^n_2)\lra0. \end{equation} In fact the convergence holds true on the closed interval $[0,t^n_1]$. To check this we first observe that for a sufficiently large $n$, $t^n_1=t^*_n=\inf\{k/n;k/n\geq t_1\}\}\longrightarrow t_1$, where $t_1$ was defined in the proof of Theorem \ref{thm3}. Hence $(\hat X^n_{t^n_1-},X^n_{t^n_1-},Y^n_{t^n_1-})\longrightarrow(X_{t_1-},X_{t_1-},Y_{t_1-})$. Moreover, \[ k^n_{t^n_1}=\max(\min(k^n_{t^n_1-},\bar u^n_{t^n_1}-EY^n_{t^n_1}),\bar l^n_{t^n_1}-EY^n_{t^n_1})),\] where $Y^n_{t^n_1}=Y^n_{t^n_1-}+f(t^n_1,X^n_{t^n_1-})\Delta V^n_{t^n_1}+g(t^n_1,X^n_{t^n_1-})\Delta M^n_{t^n_1}$, $\bar l^n_{t^n_1}=H^{-1}(t^n_1,l^n_{t^n_1},Y^n_{t^n_1})$ and $\bar u^n_{t^n_1}=H^{-1}(t^n_1,u^n_{t^n_1},Y^n_{t^n_1})$, which implies that $\Delta X^n_{t^n_1}=\Delta Y^n_{t^n_1}+\Delta k^n_{t^n_1}\mathop{\rightarrow}_{\cal P}\Delta X_{t_1}$. On the other hand, $\Delta \hat X^n_{t^n_1}\longrightarrow\Delta X_{t_1}$ $P\text{-a.s.}$, which implies that $\hat X^n_{t^n_1}-\Delta X^n_{t^n_1}\arrowp0$. Therefore, by
(\ref{eq4.6}) and uniform integrability of jumps $\{\Delta \hat X^n_{t^n_1}\}$, $\{\Delta X^n_{t^n_1}\}$ we have $ E\sup_{s\leq t^n_1}|\hat X^n_{s}-X^n_{s}|\lra0$.
It is easy to see that we can repeat the previously used arguments on next intervals $[t^n_k,t^n_{k+1}]$, where $t^n_{k+1}=t^n_k+\inf\{t>0:(C_h+1)c\max(\hat m^n_t,3(\hat m^n_t)^{1/2})>1/2\}$ and $\hat m^n_t=m^n_{t^n_k+t}-m^n_{t^n_1}$, $k\in\ensuremath{{\mathbb N}}$. Since for a sufficiently large $n$, $t^n_k=\inf\{k/n:k/n\geq t_k\}\}$ ($t_k$ was defined in the proof of Theorem \ref{thm3}), for every $q\in\ensuremath{\R^+}$ in finitely many steps we are able to prove that \begin{equation}
\label{eq4.7} E\sup_{s\leq q}|\hat X^n_{s}-X^n_{s}|\lra0,\quad q\in\ensuremath{\R^+}. \end{equation} From (\ref{eq4.7}) and the observation that $(\hat X^n,M^n,V^n,l^n,u^n)\to (X,M,V,l,u)$ $P\text{-a.s.}$ in $\ensuremath{{D}(\Rp,\R^{5d})}$ the theorem follows. \end{proof}
We say that equation \eqref{eq1.1} has a {weak solution} if there exist a filtered probability space $(\bar{\Omega},\bar{\ensuremath{{\mathcal F}}},(\bar{\cal F}_t),\bar{P})$, adapted processes $(\bar X,\bar k)$, $\bar M,\bar V$ defined on $(\bar{\Omega},\bar{\ensuremath{{\mathcal F}}},\bar{P})$ such that ${\cal L}(\bar{X}_0,\bar M,\bar V)=$ ${\cal L}({X}_0, M, V)$ and $(\bar X,\bar k)$ is a solution of mean reflected SDE \begin{equation}\label{eq4.8} \bar X_t = \bar X_0 +\int_0^t f(s,\bar X_{s-})\,d\bar M_s+\int_0^tg(s,\bar X_{s-})\, d\bar V_s + \bar k_t,\quad t\in\ensuremath{\R^+}. \end{equation} If the laws ${\cal L}(\bar{X},\bar k,\bar M,\bar V)$ and ${\cal L}(\bar{X}',\bar k',\bar M',\bar V')$ of any two weak solutions of \eqref{eq1.1}, possibly defined on different probability spaces, are the same we say that the weak uniqueness holds for \eqref{eq1.1}.
Let $\{M^n\}$ and $\{V^n\}$ be sequences of square integrable $({\cal F}^n_t)$-martingales and $({\cal F}^n_t)$-adapted process with trajectories in $\ensuremath{{D}(\Rp,\R)}$ such that $E|V^n|_q<\infty$ for every $q\in\ensuremath{\R^+}$. We will consider a sequence of mean reflected SDEs of the form (\ref{eq4.9}). We assume that the starting points $X^n_0$ are integrable ${\cal F}^n_0$-measurable random variables and for every $n\in\ensuremath{{\mathbb N}}$ the processes $M^n$, $V^n$ satisfy the condition \begin{enumerate} \item[(Mn)] there exists a nondecreasing c\`adl\`ag function $m^n:\ensuremath{\R^+}\rightarrow\ensuremath{\mathbb R}$ with $m^n_0=0$ such that
\[\max(\langle M^n\rangle_t,\widetilde{|V^n|}_t)\le m^n_t,\quad t\in\ensuremath{\R^+}.\] \end{enumerate} Clearly, if $\sup_n m^n_t<+\infty$, $t\in\ensuremath{\R^+}$, then the sequences $\{M^n\}$ and $\{V^n\}$ satisfy the so called condition (UT) introduced by Stricker\cite{st}: \begin{description} \item[{(UT)}]\index{Condition (UT)} For every $q\in\ensuremath{\R^+}$ the family of random variables \[ \{\int_{[0,q]} U^n_s\,dZ^n_s;\,n\in\ensuremath{{\mathbb N}}\,,\,U^n\in\mbox{\bf U}^n_q\} \] is bounded in probability. Here $ \mbox{\bf U}^n_q$ is the class of discrete predictable processes of the form
$U^n_s=U^n_0+\sum_{i=0}^kU^n_i\mbox{\bf 1}_{\{t_i<s\leq t_{i+1}\}}$, where $0=t_0<t_1<...<t_k=q$ and $U^n_i$ is ${\cal F}^n_{t_i}$ measurable, $|U^n_i|\leq 1$ for $i\in\{0,...,k\}\,,\,n,k\in\ensuremath{{\mathbb N}}.$ \end{description} Condition (UT) proved to be very useful in the theory of limit theorems for stochastic integrals and for solutions of SDEs (see, e.g., \cite{jmp, kp, ms,s4, Sl1}).
We will also assume that \begin{enumerate} \item[(A3)] the coefficients $f^n$, $g^n$ satisfy (A1) for every $n\in\ensuremath{{\mathbb N}}$ and there exists $f,g$ such that
\[||f^n-f||_{[0,q]\times K}+||g^n-g||_{[0,q]\times K}\lra0\] for every $q\in\ensuremath{\R^+}$ and every compact subset $K\subset \ensuremath{\mathbb R}$. \end{enumerate}
\begin{theorem}\label{thm6} Assume \mbox{\rm(H)}, \mbox{\rm(A1)} and \mbox{\rm(A3)}. Let $\{l^n\},\{u^n\}$ be sequences of c\`adl\`ag functions such that
$l^n\leq u^n$, $\{M^n\}$, $\{V^n\}$ be sequences of processes satisfying \mbox{\rm(Mn)} and $\{(X^n,k^n)\}$ be a sequence of solutions of $\mbox{\rm(\ref{eq4.9})}$, $n\in\ensuremath{{\mathbb N}}$. If $\sup_n m^n_t<+\infty$, $t\in\ensuremath{\R^+}$, $\sup_n E|X^n_0|<+\infty$, $l^n_0\leq Eh(0,X^n_0)\leq u^n_0$, $n\in\ensuremath{{\mathbb N}}$, and \begin{equation}\label{eq4.10} (X^n_0,M^n,V^n,v^n,l^n,u^n)\mathop{\rightarrow}_{\cal D} (X_0,M,V,v,l,u) \quad in \,\,\ensuremath{\mathbb R}\times\ensuremath{{D}(\Rp,\R^{5d})},\end{equation} where $v^n_t=EV^n_t$, $v_t=EV_t$, $t\in\ensuremath{\R^+}$, then \begin{equation}\label{eq4.15}\{(X^n,k^n,M^n, V^n,v^n,l^n,u^n)\}\quad is \,\,tight\,\,in\,\,\ensuremath{{D}(\Rp,\R^{7})} \end{equation} and its each limit point is a weak solution of the mean reflected SDE \mbox{\rm(\ref{eq1.1})}. \end{theorem} \begin{proof} First we show that \begin{equation}
\label{eq4.11} \sup_nE\sup_{t\leq q}|X^n_t|,\quad q\in\ensuremath{\R^+}. \end{equation} Fix $q\in\ensuremath{\R^+}$ and set $t^n_1=\inf\{t>0:(C_h+1)\mu\max(m^n_t,3(m^n_t)^{1/2})>1/2\}\wedge q$. By (\ref{eqcor1}), for every $t\leq q$, \begin{eqnarray*}
E\sup_{s< t}|X^n_s|&\leq&E|X^n_0| +(C_h+1)E\sup_{s< t}|Y^n_s-Y^n_0 |\\ &&\qquad+C_h(\lambda_ht+
\sup_{s<t}\max(|l^n_s-l^n_0|,|u^n_s-u^n_0|)\big), \end{eqnarray*}
where $Y^n=X^n_0+\int_0^\cdot f^n(u,X^n_{u-})\,d M^n_u+\int_0^\cdot g^n(u,X^n_{u-})\, dV^n_u$, $n\in\ensuremath{{\mathbb N}}$. Since $\sup_nE|X^n_0|<+\infty$
and $\sup_n\sup_{t\leq q}\max(|l^n_s|,|u^n_s|)<+\infty$, it follows from (Mn) and (A1) that
\begin{eqnarray*}E\sup_{s< t^n_1}|X^n_s|&\leq&C(h,q) +(C_h+1)\max(3(m^n_{t^n_1-})^{1/2},m^n_{t^n_1-})\mu(1+E\sup_{s<t^n_1}|X^n_s|)\\ &\leq& C(h,q) +(C_h+1)\max(3(m^n_{t^n_1-})^{1/2},m^n_{t^n_1-})\mu
+\frac12E\sup_{s<t^n_1}|X^n_s|, \end{eqnarray*} and hence that \[
\sup_nE\sup_{s< t^n_1}|X^n_s|<+\infty.
\] Since $\Delta Y^n_{t^n_1}=f^n(t^n_1,X^n_{t^n_1-})\Delta M^n_{t^n_1}+g(t^n_1,X^n_{t^n_1}-)\Delta V^n_{t^n_1}$, this implies that $\sup_nE|\Delta Y^n_{t^n_1}|<+\infty$. Hence also
$\sup_n|\Delta k^n_{t^n_1}|<+\infty$. Consequently,
$\sup_nE\sup_{s\leq t^n_1}|X^n_s|<+\infty$.
Similarly to the proof of Theorem \ref{thm4} we can repeat the previously used arguments on the next intervals $[t^n_k,t^n_{k+1}]$, where $t^n_{k+1}=t^n_k+\inf\{t>0:(C_h+1)\mu\max(\hat m^n_t,3(\hat m^n_t)^{1/2})>1/2\}$ and $\hat m^n_t=m^n_{t^n_k+t}-m^n_{t^n_1}$, $k\in\ensuremath{{\mathbb N}}$. What is left is to show that $\sup_nj(n)=\inf\{k:t^n_k=q\}<+\infty$. To check this set $r^n(t)= \max(m^n_t,3(m^n_t)^{1/2})$, $r=\sup_n r^n(q)$ and observe that $r^n_0=0$, $r^n$ is nondecreasing and for every $k$, $\frac12\le (C_h+1)\mu(r(t^n_{k+1})-r(t^n_{k}))$, which implies that for every $n\in\ensuremath{{\mathbb N}}$, \[j(n)\frac12\le(C_h+1)\mu r^n(q)\leq(C_h+1)\mu r,\] which completes the proof of (\ref{eq4.11}).
In next step we show (\ref{eq4.15}). By (\ref{eq4.11}) and (A1), for every $q\in\ensuremath{\R^+}$ the sequences $\{\sup_{t\leq q}|f^n(t,X^n_t)|\}$ and $\{\sup_{t\leq q}|g^n(t,X^n_t)|\}$ are bounded in probability. Therefore, by (Mn) and \cite[Lemma 1.6]{ms}, $\{Y^n\}$ satisfies (UT). Moreover, for every $\epsilon>0$, \begin{equation}\label{eq4.13} \{\bar V_{2+\epsilon}(Y^{n})_q\}\,\,\mbox{\rm is bounded in probability.} \end{equation} Indeed, if we set
$\tau^N_n=\inf\{t:\min(|f^n(t,X^n_t)|,|g^n(t,X^n_t)|>N)\}$ for $n,N\in\ensuremath{{\mathbb N}}$, then \[ \lim_{N\to\infty}\limsup_{n\to \infty}P(\tau_n^N<q)=0,\quad q\in\ensuremath{\R^+}, \] and by the estimate from \cite{kubilius:2009} (see also \cite[Remark 2.4]{FS}), \begin{eqnarray*} P(\bar V_{2+\epsilon}(\int_0^\cdot f^n(s,X^{n}_{s-})dM^n_s)_q>K)&\leq& P(\bar V_{2+\epsilon}(\int_0^\cdot f^n(s,X^{n}_{s-})dM^{n,\tau_n^N}_s)_q>K)+P(\tau_n^N<q)\\ &\leq&\frac{E\bar V_{2+\epsilon}(\int_0^\cdot f^n(s,X^{n}_{s-})dM^{n,\tau_n^N}_s)_q}{K}+P(\tau_n^N<q)\\
&\leq&\frac{C(\epsilon)E(\int_0^q |f^n(s,X^{n}_{s-})|^2d\langle M^{n,\tau_n^N}\rangle_s)}{K}+P(\tau_n^N<q)\\ &\leq&\frac{C(\epsilon)N^2 m^{n}_q}{K}+P(\tau_n^N<q), \end{eqnarray*} which implies that $\{\bar V_{2+\epsilon}(\int_0^\cdot f^n(s,X^{n}_{s-})dM^n_s)_q\}$ is bounded in probability. Since the similar property for integrals driven by processes with uniformly bounded variation is straightforward, the proof of (\ref{eq4.13}) is completed. Unfortunately, the sequence of solutions $\{(X^n, k^n)=\mathbb{SP}^{u^n}_{l^n}(h,Y^n)\}$ need not satisfy (UT) and it is not even clear if is possible to approximate them by sequences satisfying (UT). Consequently, standard tightness criterions from \cite{jmp, kp, ms} in the case of solutions of \eqref{eq4.9} are useless. In the present paper, we use new tightness results recently proved in \cite[Proposition 4.1]{FS}. We will show that it is possible to approximate solutions of \eqref{eq4.9} by processes having uniformly bounded in probability $2+\epsilon$\,-variation for $\epsilon>0$.
Set $\gamma ^{i}_0=0$, $\gamma ^{i}_{k+1}= \min(\gamma^{i}_{k}+\delta^i_k, \inf \{ t > \gamma ^{i}_{k}:
\min(|\Delta l_t |, |\Delta u_t|)> \delta^i \})$ and $\gamma ^{ni}_0=0$, $\gamma ^{ni}_{k+1}= \min(\gamma ^{ni}_{k}+\delta^i_k,
\inf \{ t > \gamma ^{ni}_{k}: \min(|\Delta l^n_t |, |\Delta u^n_t|)> \delta^i \})$, where $\{ \delta^i \}$, $\{ \{ \delta^i_k \} \}$ are families of positive constants such that $\delta^i
\downarrow 0$, $\delta^i/2 \leq \delta^i_k \leq \delta^i$ $|\Delta l_t|\neq\delta^i$, $|\Delta u_t|\neq\delta^i$, $t \in \ensuremath{\R^+}$, $\Delta l_{\gamma_k^{i}+\delta^i_k} =0$, $\Delta u_{\gamma_k^{i}+\delta^i_k} =0$. For every $i\in\ensuremath{{\mathbb N}}$ define new sequences $\{l^{n,(i)}\}$, $\{u^{n,(i)}\}$ of functions by putting $l^{n,(i)}_t=l^n_{\gamma ^{ni}_{k}}$, $u^{n,(i)}_t=u^n_{\gamma ^{ni}_{k}}$, $t \in [\gamma ^{ni}_{k}, \gamma ^{ni}_{k+1} )$, $k \in \ensuremath{{\mathbb N}\cup\{0\}} $, $n\in \ensuremath{{\mathbb N}}$, $l^{(i)}_t=l_{\gamma ^{i}_{k}}$, $u^{(i)}_t=u_{\gamma ^{i}_{k}}$, $t \in [\gamma ^{i}_{k}, \gamma ^{i}_{k+1} )$, $k \in \ensuremath{{\mathbb N}\cup\{0\}} $. Then using the continuous mapping argument, we have $(l^n,l^{n,(i)},u^n,u^{n,(i)}) \rightarrow (l,l^{(i)},u,u^{(i)})$ in $\ensuremath{{D}(\Rp,\R^{4})}$, which implies that \begin{equation} \label{ni1} \lim_{i \rightarrow \infty }\limsup_{n \rightarrow \infty }
\sup_{t \leq q}\max(|l^{n,(i)}_t-l^n_t|,|u^{n,(i)}_t-u^n_t|) \lra0, \quad q \in \ensuremath{\R^+}.
\end{equation}
Moreover, for every $i\in\ensuremath{{\mathbb N}}$, \begin{equation} \label{ni2}
\sup_n\max(|l^{n,(i)}|_q, |u^{n,(i)}|_q)<+\infty, \quad q \in \ensuremath{\R^+}.
\end{equation} Set $\{(X^{n,(i)},k^{n,(i)}) =\mathbb{SP}^{u^{n,(i)}}_{l^{n,(i)}}(h,Y^n)\}$, $n,i\in\ensuremath{{\mathbb N}}$, and observe that by (\ref{ni1}) and Proposition \ref{prop1}, \begin{equation}\label{eq4.12} \lim_{i \rightarrow \infty }\limsup_{n
\rightarrow \infty } E\sup_{t \leq q}|X^{n,(i)}_t-X^n_t| =0,\quad
q\in\ensuremath{\R^+}.\end{equation} On the other hand, by simple calculations, for all $q\in\ensuremath{\R^+}$, $i\in\ensuremath{{\mathbb N}}$ there exists $C(h,q,i)>0$ such that for every $n\in\ensuremath{{\mathbb N}}$, \begin{eqnarray*} v_2(k^{n,(i)})_q&=&\sup_\pi \sum_{j=1}^m
|k_{t_j}-k_{t_{i-j}}|^2\\&\leq& 45 C_h^2\mu^2(1+E\sup_{t\leq q}|X^n_t|)m_q+5 C_h^2\mu^2(1+E\sup_{t\leq q}|X^n_t|)(m_q)^2\\
&&\qquad+5C_h^2\lambda_h^2(q^2+|l^{n,(i)}|_q^2+|u^{n,(i)}|_q^2)\leq C(h,q,i). \end{eqnarray*} Since
\[\bar V_{2+\epsilon}(X^{n,(i)})_q\leq \bar V_{2+\epsilon}(Y^{n})_q+(C(h,q,i)^{1/2},\]
it follows from (\ref{eq4.13}) that also the sequence
$\{\bar V_{2+\epsilon}(X^{n,(i)})_q\}$ is bounded in probability.
Furthermore, it is well known that for continuous $f: \ensuremath{\R^+}\times\ensuremath{\mathbb R} \longrightarrow \ensuremath{\mathbb R} $ one can construct a sequence $\{ f^{(i)} \}$ of functions such that
$f^{(i)} $ satisfies (A2), for $i\in\ensuremath{{\mathbb N}}$ and $||f^{(i)} -
f||_{[0,q]\times K} \longrightarrow 0$
for any compact subset $K \subset \ensuremath{\mathbb R}$ and $q\in\ensuremath{\R^+}$. Similarly one can construct a sequence of functions $\{ g^{(i)} \}$
satisfying (A2) for $i\in\ensuremath{{\mathbb N}}$ and approximating $g$. Since $f^{(i)},g^{(i)} $ satisfy (A2), $\{X^{n,(i)}\}$ has bounded $2+\epsilon$\,-variation and $\{ f^{(i)}(X^{n,(i)}) \}$, $\{ g^{(i)}(X^{n,(i)}) \}$ have bounded $2+\epsilon$\,-variation as well. Using \cite[Proposition 4.1]{FS} and the fact that the sequences $\{M^n\}$ and $\{V^n\}$ satisfy (UT) shows that \[ \{(X^n_0,M^n, \int_0^\cdot f^{(i)}(s,X^{n,(i)}_{s-})\,dM^n_s, V^n,\int_0^\cdot g^{(i)}(s,X^{n,(i)}_{s-})\,dV^n_s,v^n,l^n,u^n)\} \]
is tight in $\ensuremath{\mathbb R}\times\ensuremath{{D}(\Rp,\R^{7})}$. Therefore, by (\ref{eq4.12}), also \[\{(X^n_0,M^n, \int_0^\cdot f^n(s,X^{n}_{s-})\,dM^n_s, V^n,\int_0^\cdot g^n(s,X^{n}_{s-})\,dV^n_s,v^n,l^n,u^n)\} \] is tight in $\ensuremath{\mathbb R}\times\ensuremath{{D}(\Rp,\R^{7})}$, which together with Theorem \ref{thm2} implies (\ref{eq4.15}).
Assume that there exists a subsequence $(n') \subset (n )$ such that \begin{equation}\label{eq4.14} (X^{n'},k^{n'},M^{n'}, V^{n'},v^{n'},l^{n'},u^{n'}) \mathop{\rightarrow}_{\cal D}(\bar X,\bar k,\bar M,\bar V,v,l,u) \quad\,\,\mbox{\rm in}\,\,\ensuremath{{D}(\Rp,\R^{7})}, \end{equation} where ${\cal L}(\bar X^n_0,\bar M,\bar V)={\cal L}(\bar X_0,M,V)$. Then, by \cite[Proposition 4.1(ii)]{FS}, \[(X^{n'},k^{n'},M^{n'}, V^{n'},Y^{n'},l^{n'},u^{n'})\mathop{\rightarrow}_{\cal D}(\bar X,\bar k,\bar M,\bar V,\bar Y,l,u)\quad\,\,\mbox{\rm in}\,\,\ensuremath{{D}(\Rp,\R^{8})}.\] Now set $y^{n'}_t=EY^{n'}_t=E\int_0^t g(s,X^{n'}_{s-})\,dV^{n'}_s$, $\bar y_t=E\bar Y_t=E\int_0^t g(s,\bar X_{s-})\,d\bar V_s$, $t\in\ensuremath{\R^+}$. Since $\{t:\Delta \bar y_t\neq0\}\subset\{t:\Delta v_t\neq0\}$, from the above convergence and (\ref{eq4.10}) we deduce that \begin{equation} \label{4.16}(X^{n'},k^{n'},M^{n'}, V^{n'},Y^{n'},y^{n'},l^{n'},u^{n'}) \mathop{\rightarrow}_{\cal D}(\bar X,\bar k,\bar M,\bar V,\bar Y,\bar y,l,u)\quad\,\, \mbox{\rm in}\,\,\ensuremath{{D}(\Rp,\R^{8})}. \end{equation} Therefore from Theorem \ref{thm3}(i) it follows that $(\bar X, \bar k)=\mathbb{SP}^{u}_{l}(h,\bar Y)\}$ with $\bar Y$ defined as $\bar Y = \bar X_0 + \int_0^{\cdot}\;f(s,\bar X_{s-})\,d\bar M_s+\int_0^{\cdot}\;g(s,\bar X_{s-})\,d\bar V_s$, which implies that $(\bar X,\bar k)$ is a weak solution of the SDE (\ref{eq1.1}). \end{proof}
In the proof of existence of weak solutions of (\ref{eq1.1}) we will use the approximation scheme defined in (\ref{eq4.2}).
\begin{corollary}\label{cor4.1}Assume \mbox{\rm(H)}, \mbox{\rm(A1)} and \mbox{\rm (M)}. If $l_0\leq Eh(0,X_0)\leq u_0$, then there exists a weak solution $(X,k)$ of \eqref{eq1.1}. \end{corollary} \begin{proof} Let $\{M^n\},\{V^n\}$ and $\{l^n\},\{u^n\}$ be sequences of discretizations of $M,V$ and $l,u$. Clearly \begin{equation}\label{eq411}(M^n,V^n,v^n,l^n,u^n) \longrightarrow (M,V,v,l,u)\quad P\text{-a.s.}\,\,\mbox{\rm in}\,\,\ensuremath{{D}(\Rp,\R^{5})}. \end{equation} If $(X^n, k^n)$ and $Y^n$ are defined by \eqref{eq4.2}, then $(X^n, k^n)=\SPp{Y^n}{l^n}{u^n}$, where $Y^n_t=X_0+\int_0^{{t}}f(s, X^n_{s-})\,dM^n_s+\int_0^{{t}}g(s,X^n_{s-})\,dV^n_s$, $t\in\ensuremath{\R^+}$. This means that $(X^n, k^n)$, $n\in\ensuremath{{\mathbb N}}$, are solutions of (\ref{eq4.9}) with $f^n=f$ and $g^n=g$. Now observe that using (M) we are able to estimate the predictable characteristics of $M^n$ and $V^n$. Indeed, for every $t\in\ensuremath{\R^+}$, \begin{eqnarray*}
\langle M^n\rangle_t&=&\sum_{k;k/n\leq t}E(M_{k/n}-M_{(k-1)/n})^2|{\cal F}_{(k-1)/n})\\
&=&\sum_{k;k/n\leq t}E(\langle M\rangle^{k/n}_{(k-1)/n}|{\cal F}_{(k-1)/n})\leq m^n_t
\end{eqnarray*} and\begin{eqnarray*}
\widetilde{|V^n|}_t &=&\sum_{k;k/n\leq t}E(|V_{k/n}-V_{(k-1)/n}||{\cal F}_{(k-1)/n})\\
&=&\sum_{k;k/n\leq t}E(\widetilde{|V|}^{k/n}_{(k-1)/n}|{\cal F}_{(k-1)/n})\leq m^n_t,
\end{eqnarray*} where $m^n$ is a discretization of the function $m$, i.e. $m^n_t=m_{\rho^n_t}$, $t\in\ensuremath{\R^+}$. Consequently, the condition (Mn) is satisfied and the existence of a weak solution follows from Theorem \ref{thm6}. \end{proof}
\begin{corollary}\label{cor4.2} Under the assumptions of Theorem \ref{thm6}, if moreover \mbox{\rm(A2)} is satisfied and the processes $M,V$ have independent increments, \[ (X^{n},k^{n},M^{n}, V^{n},l^{n},u^{n})\mathop{\rightarrow}_{\cal D}(X,k,M,V,l,u) \quad\,\,\mbox{\rm in}\,\,\ensuremath{{D}(\Rp,\R^{6})}, \] where $(X,k)$ is the unique weak solution of \mbox{\rm(\ref{eq1.1})}. \end{corollary} \begin{proof} It is sufficient to apply Theorem \ref{thm6} and observe that in the case where \mbox{\rm(A2)} is satisfied and $M,V$ have independent increments then weak uniqueness for (\ref{eq1.1})
holds true. To check this assume that $(\bar{X},\bar k,\bar M,\bar V)$ and $(\bar{X}',\bar k',\bar M',\bar V')$ are two weak solutions of (\ref{eq1.1}) and ${\cal L}(\bar{X}_0,\bar M,\bar V)=$ ${\cal L}(\bar{X}'_0,\bar{ M}', \bar{V}')$ $={\cal L}({X}_0,M, V)$. Since $M,V$ have independent increments, the processes $\bar M,\bar V$ and $\bar{M}',\bar{V}'$ also have independent increments. Moreover, they have the same deterministic predictable characteristics $\langle M\rangle,\widetilde{|V|}$. Let $(\bar X^n, k^n)=\mathbb{SP}^{u^n}_{l^n}(h,\bar Y^n)$, $n\in\ensuremath{{\mathbb N}}$, and $(\bar{X^n}', k^n)=\mathbb{SP}^{u^n}_{l^n}(h,\bar{Y^n}')$, $n\in\ensuremath{{\mathbb N}}$, be the approximations of $(\bar{X},\bar k)$ and $(\bar{X}',\bar k')$, respectively, considered in Theorem \ref{thm5}. By Theorem \ref{thm5}, $ (\bar X^n,k^n,\bar Y^n,l^n,u^n)\mathop{\rightarrow}_{\cal P} (\bar X,\bar k,\bar Y,l,u)$ in $\ensuremath{{D}(\Rp,\R^{5d})}$ and $(\bar{ X^n}',k^n,\bar {Y^n}',l^n,u^n)\mathop{\rightarrow}_{\cal P} (\bar{ X}',\bar k',\bar Y,l,u)$ in $\ensuremath{{D}(\Rp,\R^{5d})}$. Clearly, the equality of laws ${\cal L}(\bar{X}_0,\bar M,\bar V)=$ ${\cal L}(\bar{X}'_0,\bar{ M}', \bar{V}')$ implies the equality of laws of the approximations, i.e. ${\cal L}(\bar X^n,k^n,\bar Y^n,l^n,u^n)={\cal L}(\bar {X^n}',k^n,\bar {Y^n}',l^n,u^n)$, $n\in\ensuremath{{\mathbb N}}$, which completes the proof. \end{proof}
\begin{remark} {\rm From Proposition \ref{prop4} one can deduce that in case the processes $\{V^n\}$ have independent increments or $P(\Delta V_t=0)=1$, $t\in\ensuremath{\R^+}$, in Theorem \ref{thm6} and Corollary \ref{cor4.2} in place of (\ref{eq4.10}) it suffices to assume that \[ (X^n_0,M^n,V^n,l^n,u^n)\longrightarrow (X_0,M,V,l,u) \quad \mbox{\rm in} \,\,\ensuremath{\mathbb R}\times\ensuremath{{D}(\Rp,\R^{4d})}.\] }\end{remark}
\section{Appendix. Skorokhod problem, deterministic case}
We begin with a general definition of the Skorokhod problem with time-dependent reflecting barriers for c\`adl\`ag functions considered in \cite{BKR}. \begin{definition}{\rm Let $y,l,u\in\ensuremath{{D}(\Rp,\R)}$ with \label{def5} $l\leq u$ and $l_0\leq y_0\leq u_0$. We say that a~pair $(x,k)\in\ensuremath{{D}(\Rp,\R^{2})}$ with $k_0=0$ is a solution of the Skorokhod problem associated with $y$ and barriers $l,u$ (and we write $(x, k)=SP_l^u(y)$) if \begin{description} \item[(i)]$ x_t = y_t+k_t\in[l_t,u_t]$, $t\in\ensuremath{\R^+}$, \item[(ii)]for every $0\leq t\leq q$, \begin{eqnarray*} k_q-k_t\geq0,&&\quad\mbox{\rm if}\,\,x_s<u_s\,\,\mbox{\rm for all}\,\,s\in(t,q],\\ k_q-k_t\leq0,&&\quad\mbox{\rm if}\,\,x_s>l_s\,\,\mbox{\rm for all}\,\,s\in(t,q],\end{eqnarray*}and for every $t\in\ensuremath{\R^+}$, $ \Delta k_t\geq0$ if $x_t<u_t$ and $\Delta k_t\leq0$ if $x_t>l_t$. \end{description} } \end{definition} \begin{theorem}(\cite[Theorem 2.6]{BKR}) \label{thm5.1} There exists a unique solution of the Skorokhod problem $(x,k)=SP_l^u(y)$. Moreover, \begin{equation}\label{eq5.1} k_t=-\max(0\wedge\inf_{0\leq u\leq t}(y_u-l_u), \sup_{0\leq s\leq t}[(y_s-u_s)\wedge\inf_{s\leq u\leq t}(y_u-l_u)]), \quad t\in\ensuremath{\R^+}. \end{equation} \end{theorem} One can also show Lipschitz continuity of the mapping $(y,l,u)\mapsto(x,k)$ in supremum norm. \begin{theorem}(\cite[Theorem 2.6]{sl-wo/13}) \label{thm5.2} If $(x^i,k^i)$ is a solution associated with ${y^i}\in\ensuremath{{D}(\Rp,\R)}$ and barriers $l^i,u^i$, $i=1,2$, then for every $q\in\ensuremath{\R^+}$,
\begin{equation}\label{eq5.2}\sup_{t\leq q}|x^1_t-x^2_t|\leq2\sup_{t\leq q}|y^1_t-y^2_t| +\sup_{t\leq q}\max(|l^1_t-l^2_t|,|u^1_t-u^2_t|)\end{equation} and \begin{equation}\label{eq5.3}
\sup_{t\leq q}|k^1_t-k^2_t|\leq\sup_{t\leq q}|y^1_t-y^2_t|
+\sup_{t\leq q}\max(|l^1_t-l^2_t|,|u^1_t-u^2_t|).\end{equation} \end{theorem}
From (\ref{eq5.2}), (\ref{eq5.3}) it is easy to deduce stability of the mapping $(y,l,u)\mapsto(x,k)$ in the Skorokhod topology $J_1$. More precisely, if $(y^n,l^n,u^n)\longrightarrow(y,l,u)$ in $\ensuremath{{D}(\Rp,\R^{3})}$, then \begin{equation}\label{eq5.4}
(x^n,k^n,y^n,l^n,u^n)\longrightarrow(x,k,y,l,u)\quad in\,\, \ensuremath{{D}(\Rp,\R^{5})}.\end{equation} It is worth adding that in \cite[Definition 2.1]{sl-wo/13} condition (ii) is replaced by \begin{itemize} \item for every $0\leq t\leq q$ such that $\inf_{s\in[t,q]}(u_s-l_s)>0$ the function $k$ has bounded variation on $[t,q]$ and \[\int_{[t,q]}(x_s-l_s)\,dk_s\leq0\quad\mbox{\rm and}\quad\int_{[t,q]}(x_s-u_s)\,dk_s\leq0.\] \end{itemize} However, simple calculations shows that in fact these two definitions are equivalent.
In the classical Skorokhod problem it is assumed that the function $k$ has bounded variation on any bounded interval $[0,q]$, or, equivalently, $k=k^{(+)}-k^{(-)}$,
where $k^{(+)}$, $k^{(-)}$ are nondecreasing right continuous functions with $k_0=k^{(+)}_0=k^{(-)}_0=0$ such that $k^{(+)}$ increases only on $\{t:x_t={l}_t\}$ and $k^{(-)}$ increases only on $\{t:x_t={u}_t\}$. In \cite{rutkowski:1980,ChL} and \cite{DN} it is observed that the above property holds true in the case where the barriers $l,u\in\ensuremath{{D}(\Rp,\R)}$ satisfy the additional condition \begin{equation}\label{eq5.5} \inf_{t\leq q}({u}_t-{l}_t)>0,\quad q\in\ensuremath{\R^+}. \end{equation} Under (\ref{eq5.5}) it is possible to estimate the variation of the function $k$ compensating the reflections with the use of $\eta$-oscillations of $y,l,u$.
\begin{proposition}(\cite[Proposition 2.11]{sl-wo/10}) \label{prop5}For any $q\in\ensuremath{\R^+}$ and $\eta$ such that $0<2\eta\leq\inf_{t\leq q}({u}_t-{l}_t)/3$ we have
\begin{equation}\label{eq5.6}|k|_q\leq 6(N_{\eta}(y,q)+N_{\eta}(l,q)+N_{\eta}(u,q)+1)(\sup_{ t\leq q}|y_t|+\sup_{ t\leq q}\max(|l_t|,|u_t|)).\end{equation} \end{proposition}
We close this section with simple remark showing that in the special case of the Skorokhod problems with one barrier, i.e. when $u=+\infty$ or $l=-\infty$, the function $k$ has much simpler form. Indeed, in case of one lower barrier $l$ one can check that $k$ is a nondecreasing function of the form \begin{equation}\label{eq5.7} k_t=\sup_{s\leq t}(y_s-l_s)^-=\sup_{s\leq t}(l_s-y_s)^+,\quad t\in\ensuremath{\R^+}. \end{equation} Similarly, in case of one upper barrier $u $ one can check that $k$ is a nonincreasing function of the form \begin{equation}\label{eq5.8} k_t=-\sup_{s\leq t}(y_s-u_s)^+=-\sup_{s\leq t}(u_s-y_s)^-,\quad t\in\ensuremath{\R^+}.\end{equation}
\noindent{\bf Acknowledgements}\\ {This work was supported by the Polish National Science Centre under Grant \\ 2016/23/B/ST1/01543).}
\end{document} | arXiv |
Improving anomaly detection in SCADA network communication with attribute extension
Mahwish Anwar1,
Lars Lundberg1 &
Anton Borg1
Network anomaly detection for critical infrastructure supervisory control and data acquisition (SCADA) systems is the first line of defense against cyber-attacks. Often hybrid methods, such as machine learning with signature-based intrusion detection methods, are employed to improve the detection results. Here an attempt is made to enhance the support vector-based outlier detection method by leveraging behavioural attribute extension of the network nodes. The network nodes are modeled as graph vertices to construct related attributes that enhance network characterisation and potentially improve unsupervised anomaly detection ability for SCADA network. IEC 104 SCADA protocol communication data with good domain fidelity is utilised for empirical testing. The results demonstrate that the proposed approach achieves significant improvements over the baseline approach (average \(F_{1}\) score increased from 0.6 to 0.9, and Matthews correlation coefficient (MCC) from 0.3 to 0.8). The achieved outcome also surpasses the unsupervised scores of related literature. For critical networks, the identification of attacks is indispensable. The result shows an insignificant missed-alert rate (\(0.3\%\) on average), the lowest among related works. The gathered results show that the proposed approach can expose rouge SCADA nodes reasonably and assist in further pruning the identified unusual instances.
Critical infrastructure is under constant threat (Tariq et al. 2019). A critical infrastructure (CI) is a system or part of a system that maintains vital societal functions. Examples of CI sectors include; energy, oil and gas, water and waste treatment, and transportation. The disruption or destruction of such a system would result in failure for the society to function and can negatively affect its economy and safety.
CIs widely rely on supervisory control and data acquisition (SCADA) systems to manage and control CI operations (Tariq et al. 2019). For example, the SCADA system in the energy power grid would be responsible for the transmission and distribution of electricity. A SCADA system performs centralised monitoring and control for geographically distributed remote units, often scattered over thousands of square kilometers. The gathered data results in automated or operator-driven supervisory commands for the field units, e.g., open and close valves/breakers, share sensor data or monitor the local environment for alarm conditions (Zhu et al. 2011). Since the SCADA system is an essential element within the CI, it becomes vital to protect it from the threats that exist in the cyber-landscape. As Industrial Control System (ICS) / SCADA system security experts warn, "it is not a matter of if it (ICS/SCADA system) will fail, but when it will fail" (Assante and Lee 2015).
SCADA systems usually are zoned out from the external cyber-threats through air-gapping, intrusion detection, and prevention systems, and firewalls (Pliatsios et al. 2020). However, by exploiting SCADA-specific protocol vulnerabilities and launching a successful malware attempt, the intruder can bypass the security measures and gain unauthorised access to the critical network (Assante and Lee 2015; Pliatsios et al. 2020). Stuxnet and BlackEnergy attacks on control systems highlighted the lack of awareness of the security of these systems. It showed that the hacker could passively listen to the SCADA communication and deliver the attack successfully once inside the network (Assante and Lee 2015).
On the one hand, the SCADA systems have become intelligent, real-time, and inter-connected with the integration of the Internet of Things and Cloud. On the other hand, these advancements have made the SCADA system more prone to network vulnerabilities (Tariq et al. 2019). It is, therefore, imperative to detect anomalies proactively in SCADA networks and meet the growing security challenges. Thus, continuous effort is required by industry and academia alike to monitor and safeguard SCADA networks.
Generally, SCADA intrusion detection systems rely on the network traffic data, the host process data, or the data related to the physical event or operation. The approaches to intrusion detection include signature-based detection, machine learning-based anomaly detection, and deep learning-based anomaly detection. Suricata is an example of signature-based detection that utilises SCADA network traffic data to detect cyberattacks (Wong et al. 2017). In Robles-Durazno et al. (2018), various machine learning-based anomaly detection methods are applied to classify signal deviations in a water supply system. Whereas, in Gaggero et al. (2020), the undesired working conditions of the distributed energy control system are identified using a deep learning-based anomaly detection technique. We also find a hybrid intrusion detection approach that applies both network protocol traffic data and physical behaviour characteristics to isolate SCADA network anomalies (Yang et al. 2016). Our work focusses on SCADA network traffic data and the application of a machine learning-based anomaly detection approach.
Canonical data-driven approaches for CI can detect new anomalies at the cost of a high error rate (Rakas et al. 2020; Panagiotis et al. 2021). This is because of the overlapping nature of the normal and anomalous communication packets, making it difficult for the detector to separate the network anomalies effectively. To reduce machine learning-based network anomaly detection errors, we approach the issue by extending the input set (or attribute set) of a standard SCADA communication protocol.
We see the application of composing advanced attributes for IEC 61870 SCADA protocol in Linda et al. (2009), where the authors propose neural networks to extract the trends in network communication to perform intrusion detection. In Mantere et al. (2013), an analysis of IP traffic traces in SCADA is presented, and an intrusion detection system using machine learning-based techniques is suggested as future work. To the best of our knowledge, attribute extension has not been investigated for anomaly detection in the IEC 60870-5-104 (or IEC 104) SCADA protocol. Hence, in this study, we investigate the possibility of analysing the SCADA network through topological behaviour and extending the attribute space for improving anomaly detection performance.
The intuition behind attribute extension is to represent the SCADA network behavior by modeling the relationship between interacting SCADA nodes. We perform the detection of attacks for IEC 60870-5-104 communication protocol, both with and without attribute extension. IEC 104 is a widely implemented telecontrolling protocol and is prone to vulnerabilities (György and Holczer 2020). In this study, we focus on IEC 104 SCADA protocol and derive new attributes to improve one-class SVM anomaly detection performance.
One-class support vector machine (SVM) algorithm is a popular machine learning intrusion detection algorithm (Tsai et al. 2009; Thakkar and Lohiya 2021). The learning algorithm is also an acknowledged choice for intrusion detection in the SCADA network (Rakas et al. 2020). Furthermore, recent works on standard SCADA-specific protocol (IEC 104) relayed the algorithm's stable performance for detecting different attacks (Egger et al. 2020; Anwar et al. 2021). Egger et al. (2020) compared intrusion detection of the signature-based method with machine learning methods. Supervised and semi-supervised (with one-class SVM) learning performed better intrusion detection, while Snort signature-based gave worse (Egger et al. 2020). The same protocol dataset is systematically evaluated with other learning algorithms in Anwar et al. (2021). Mahwish et al. evaluated the SCADA network intrusion detection ability of distance-based, density-based, and kernel-based learning methods in an unsupervised setting for IEC 104 communication protocol. The comparison of detection methods revealed that on average one-class SVM method performs steadily for the given SCADA protocol data in reference to other candidate learning methods. In the current work, we draw a comparison with study (Anwar et al. 2021). Realising the predictable and steady performance of one-class SVM for SCADA protocol and its ability to segregate communication network data, we intend to amplify the outlier detection capability of one-class SVM for the IEC 104 protocol. Briefly, we explore the following research question: To what extent can attribute extension improve one-class SVM anomaly detection in IEC 104 protocol communication within SCADA network?
More explicitly, we make the following contributions:
A method for extending the attributes to project the SCADA network behaviour is presented.
The impact of the extended attribute set is evaluated using machine learning-based anomaly detection technique with the Support Vector Machine algorithm. The study describes the improved machine learning model design and implementation; and compares the performance with the baseline results and previous research.
Background and related work
Introduction to SCADA system
Abstraction of SCADA system (Zhu et al. 2011; Maynard et al. 2018)
SCADA system (Fig. 1) is instilled in critical infrastructure architecture. It is a process monitoring and controlling system that perform geographically distributed operations. One of the system's main components is the remote terminal unit (RTU). The RTU is an intermediate node between field devices and a master unit that connects with the SCADA human machine interface. RTU exchanges sensory data with the master unit and sends specified control commands to the field devices. The human machine interface (HMI) sits between the SCADA operator and RTUs. The master unit gathers the data, which the HMI translates to enable interaction with the operator. The operator monitors the system via SCADA HMI, troubleshoots alerts, and performs the necessary control operations. The operator can access the SCADA HMI remotely, or through the Internet (Zhu et al. 2011).
SCADA network communication protocols are adopted to facilitate continuous and reliable communication within the SCADA system. These communication protocols consider the processing capabilities of SCADA nodes and the communication requirements of industrial applications. Standard protocols used in electrical applications and power system automation for remote control and monitoring include a set of IEC 60870 protocols, Modbus and DNP3 (more common outside of Europe) (Zhu et al. 2011; Pliatsios et al. 2020). IEC 60870-5-104 (part of the IEC 60870 standard) is a widely adopted protocol for telecontrol in European power transmission, distribution, and control systems, despite its security vulnerabilities, which include lack of authentication, integrity checking, and encryption (Matousek 2017; Radoglou-Grammatikis et al. 2019; György and Holczer 2020). Due to the widespread interconnectivity and complexity of IEC 60870-5-104 (or IEC 104), the vendors and utility operators are reluctant to roll-out its successor IEC 62351, which is more secure than IEC 104.
Here we focus on the IEC 104 protocol. IEC 104 operates using the client-server communication model. Under the protocol, every node in the network is either a controlling station (master) or a controlled station (slave) (Matousek 2017). The transmission happens in the monitor direction, i.e., from the controlled station (typically an RTU) to the controlling station (e.g., SCADA HMI). Or the control direction, i.e., from the controlling station to the controlled station (Matousek 2017).
The IEC 104 protocol defines the Application layer of the OSI model and uses Ethernet technology for the link layer. IEC 104 enables the communication between the controlling node and the controlled nodes via a TCP/IP communication network. The IEC 104 protocol data transmits in either of the following three frame formats: (i) format-U is for the control functions, e.g., the controlling node issues START and STOP commands to control the data transfer from a controlled node. (ii) format-S is for supervisory commands, e.g., to indicate time-out in case of longer data transmission. (iii) format-I is to transfer information in both directions, e.g., interrogation command in the control direction or to send measured value in the monitor direction (Matousek 2017).
Anomaly detection in SCADA communication network
It is common for SCADA system operators to protect the SCADA communication network by gathering and parsing the communication protocol packets, e.g., from SCADA nodes, and forward them to the intrusion detection system (IDS). The IDS takes parsed SCADA protocol packets (i.e., packets where key fields from the protocol frames have been identified and dissected, e.g., payload length, IP address, port, etc.) and performs monitoring and detection based on the predefined signatures. In the case of a flag, the filtering of the protocol packet is performed. Such IDSs perform active monitoring and deep packet inspection often on the edge due to high processing requirements, thus, limiting the scope to external threats (Pliatsios et al. 2020). However, the internal SCADA network goes unchecked. Similarly, firewalls and anti-virus software shield the critical SCADA system only partially from security threats (ENISA 2017).
The European Union Agency for Cybersecurity (ENISA) emphasises the need to monitor the internal and external SCADA communications in the following words, "without active network monitoring, it is very difficult to detect suspicious activity, identify potential threats, and quickly react to cyber attacks" (ENISA 2017). An adversary in the past accessed the SCADA system by sending an email with malware to an authorised SCADA user. The malware helped the attackers to listen to SCADA communication, plot and execute the attack and create a backdoor. The attack caused an outage of 6 h and affected over 200,000 customers (CISA 2016). Similar other incidents (Assante and Lee 2015; Pliatsios et al. 2020) stresses the need to monitor the internal SCADA protocol communication traffic regularly.
Regular analysis of the internal SCADA communication can enrich the operators to get visibility of the SCADA traffic which in turn can aid in understanding the routine network behavior, thus, enabling outlier identification (Mahmood et al. 2010; Matousek et al. 2019). A SCADA architecture to monitor inside and outside network traffic is presented in Mahmood et al. (2010). A similar course of action is proposed in Matousek et al. (2019) where the research highlights gaining visibility of the network characteristics and operations (such as transmission data, connected nodes, malfunctioning nodes, etc.) through analysing network traffic. The authors later extended their work by proposing an anomaly detection approach based on an analysis of SCADA protocol communication to point resource scanning, rogue devices, and unusual traffic (Matousek et al. 2020). They employ finite state automata to infer the IEC 104 communication flow profile of two SCADA nodes. If the probability of the candidate nodes is lower than the defined threshold, the detector will flag it as an unknown communication sequence (Matousek et al. 2020).
The sequence attacks in IEC 104 protocol are detected with the use of Discrete-time Markov Chains in Ferling et al. (2018). To identify malicious IEC 104 communications in SCADA networks a signature-based method is given in Yang et al. (2013), where the authors propose inspection of the incoming communication packets based on the customised rules and correlations between different protocol fields that represent usual SCADA communication flow. Robert Udd et al. (2016) suggest a hybrid approach for anomaly detection, where initially, the SCADA protocol packet passes through allowlists (node pairs, TCP control, etc.). If no alert is issued from the initial step, a statistical analysis of the packet's timing characteristics is executed. Their work resulted in anomaly detection for IEC 104 zero-day attacks. The use of spontaneous packet analysis for IEC 104 SCADA protocol is utilised for anomaly detection in Lin and Nadjm-Tehrani (2019), where the authors investigate the inter-arrival time of the packet. If the inter-arrival time value is outside the training interval boundary, an alarm is raised, and the second detection phase begins. The time-interval flags for individual SCADA nodes are correlated to create a time-series in the second phase. The system labels the corresponding node anomalous if the threshold exceeds the warning-threshold. This approach is more suitable for intermittent anomalies.
An IEC 104 intrusion detection approach similar to Udd et al. (2016) can be seen in a recent publication (Grammatikis et al. 2020). In the latest work, an access control mechanism is enforced initially to filter unknown ports, Internet Protocol, and Media Access Control addresses. Afterward, based on 7 aggregated features (e.g., total packets in the forward direction, the total size of the packets in the backward direction, standard deviation size of the packets in the forward direction, etc.) based on different flow intervals, outlier analysis is discharged. At higher flow intervals, \(F_{1}\) score slightly increases. In the absence of outlier ratio and error rates, it is ambiguous to contextualise the detector's true capability.
Anomaly detection for IEC 104 protocol with supervised machine learning methods, such as Decision Tree, Nearest-neighbour, etc., is given in Hodo et al. (2017). Egger et al. compared Snort-based intrusion detection with machine learning-based intrusion detection methods (Egger et al. 2020). The supervised and semi-supervised machine learning methods for IEC 104 SCADA protocol outperformed signature-based intrusion detection, and unsupervised learning (Egger et al. 2020). Later, systematic performance evaluation of IEC 104 anomaly detection with unsupervised learning approaches was accomplished in Anwar et al. (2021). Both studies (Egger et al. 2020; Anwar et al. 2021), utilised the same IEC 104 dataset. However, the dataset lacks multiple SCADA nodes; therefore, additional exploration is required to assess the performance of the unsupervised learning approach and, if required enhance the outcome. In the present research, we address the same knowledge gap.
Evaluations from Egger et al. (2020); Anwar et al. (2021); Grammatikis et al. (2020) reveal that for IEC 104 protocol communication, the Support Vector Machine method offers stable results for unsupervised anomaly detection. Accordingly, we focus on improving SVM unsupervised anomaly detection performance for IEC 104 SCADA protocol.
Attribute processing
Machine learning-based network anomaly detection solutions (Tsai et al. 2009) often manipulate the attributes to make better predictions and sometimes to reduce computational costs of processing large datasets (Flach 2012; Thakkar and Lohiya 2021). To underline the need for attribute processing, we diverge it into four modus operandi (Fig. 2). Peter Flach defines the observations or instance space as a set of all possible objects of interest in machine learning. The instance space can be inconceivably expansive; therefore, a fraction of instances commonly formulate a dataset. Since each instance in the dataset is described by a fixed number of attributes, we refer to it as attribute set (Flach 2012).
Modus operandi of attribute processing in machine learning network anomaly detection
The attributes in the attribute set can be manipulated or processed in one or a few of the following methods. Attribute decomposition enables the creation of new attributes that are linear combinations of available attributes, e.g., through principal component analysis (Flach 2012). The same projection method is named differently in Thakkar and Lohiya (2021), but both explain the notion of projecting higher-dimensional or sparse attribute space to a lower-dimensional attribute space. Attribute transformation includes various mechanisms with which the attribute kind is transformed. For example, thresholding transforms a quantitative attribute into a Boolean attribute by finding a split threshold value. Discretisation transforms a continuous attribute into an ordinal attribute kind. The transformation mechanisms, generally, are required to scale the attribute values, indicate the presence of a certain attribute, or make the attribute meaningful for prediction task (Flach 2012). Attribute reduction mainly involves attribute selection (Flach 2012; Thakkar and Lohiya 2021), for example based on attribute importance or relevance. It also includes dropping redundant attributes based on correlation analysis. We append a fourth method to the list called Attribute extension, which forms the basis of improving the machine learning prediction ability of one-class SVM within the context of the IEC 104 protocol. We define it as a method to construct new attributes based on domain knowledge to enrich the available attribute set with more representative attributes.
Topological attribute extension
Graph-based anomaly detection approaches are a branch of data mining and machine learning techniques that capture and analyse the interactions between data objects of a network or graph to detect potential anomalies (Pourhabibi et al. 2020). Such approaches can analyse the connectivity patterns and graph object behaviour in communication networks to flag suspicious graph nodes, irregular connectivity between nodes, or unusual subgraphs by drawing intra-graph comparisons (Pourhabibi et al. 2020). Our approach to model SCADA networks resembles the structural-based graph method, as described in Pourhabibi et al. (2020), where we exploit topological graph structure and characterise the SCADA network nodes with node and edge attributes, thus extending the attribute space.
In Akoglu et al. (2010) the authors exploit graph node and node-neighbourhood characteristics to model the egonet laws and to identify nodes violating the laws. Topological and temporal graph attributes are measured in Henderson et al. (2010) to analyse volatile network behaviour. The work uses a multi-level approach, where the network is analysed from a topological global-level, such that if an unusual event is discovered, the analysis moves to the next level (node-level). Application of graph node characteristics to group similar nodes was put-forth in Henderson et al. (2012), where the nodes with similar degrees and edges were unified under a single role. The role assigned to each node can be utilised to find strange nodes within a network.
We exploit the structural attribute discovery for the SCADA network nodes participating in the communication network. The characterisation of the SCADA nodes and node neighbours yields additional attributes—attribute extension—that enrich the attribute space.
This section describes the research process undertaken (Fig. 3), starting from the extraction of the original IEC 104 protocol attributes through the machine learning experiment's design choices. We describe the reconstruction of IEC 104 attribute set and the application of the single-class SVM learning algorithm to cluster anomalous exchanges in the SCADA protocol communication (step 1 through 11).
Step 1: Data extraction
We extract IEC 104 instances from a simulated IEC 104 protocol communication (Maynard et al. 2018). The authors in Maynard et al. (2018) generate the protocol communication data from their standard compliant implementation of testbed framework that mimics a real SCADA system. Furthermore, they simulate attacks and make the complete dataset openly available. The log file of the packet capture encapsulates IEC 104 attributes, including the application layer fields. Due to these strengths, the authors in Maynard et al. (2018) recommend using the provided dataset to verify the effectiveness of a network-based intrusion detection for SCADA networks, thus, making the dataset suitable for our study.
Other than comprehensiveness and imitation of real-world deployment of SCADA networks, the chosen dataset is suited for the work since the SCADA network protocol, IEC 104, adheres to a shared network master-slave topology (Maynard et al. 2018), making it possible to apply the approach and attributes to other IEC 104 datasets and without prior network details.
The initial protocol attribute set (Table 1) is elicited from previous work (Egger et al. 2020) and knowledge gathered from the domain experts.
Table 1 IEC 104 attributes
The main limitation with IEC 104 dataset in Egger et al. (2020) is that it does not define the direction of the transmission of IEC 104 packets, nor does it provide Ethernet address information of the nodes in the SCADA network. IEC 104 communication logs include these data and are deemed helpful by domain experts when designing a network anomaly detector. Thus, to build our dataset, we consider the transmission direction along with the time difference between two transmitted packets, source, and destination Ethernet addresses.
Step 2: Instance labelling
The emulated IEC 104 protocol communication includes two attack simulations: Man-in-the-Middle (MITM) and Reconnaissance (Maynard et al. 2018). The protocol logs represent 150 min of IEC 104 communication (44373 packets) between 8 nodes: 1x Controlling Node; 5x Controlled Nodes; and 2x Attackers.
Since Maynard et al. (2018) described the attacks, we are able to label the individual frames: Label 0: normal; Label 1: MITM intrusion, and Label 2: Reconnaissance intrusion. However, the main objective is to segregate normal IEC 104 packets from anomalous frames for anomaly detection. Therefore, all attack instances are regrouped as anomalous.
After manual labelling of the packets, the dataset contains 41948 normal IEC 104 protocol packets and 2425 packets with anomalies (Table 2).
Table 2 Dataset class composition
Step 3: Attribute reduction
Our IEC 104 dataset has both categorical and continuous data. For missing categorical values (TypeID and CauseTx), 'none' is substituted. It indicates the transmission does not have a format-I frame. There are no missing continuous attributes.
We perform correlation analysis for non-categorical attributes, and find Length and tcpPduSize have near-perfect correlation. Hence, only Length is retained. Also, this analysis reveals, ipTtl and tcpHdrLen have no coorelation with any other attribute nor with the target class. Further exploration indicates that values for both, ipTtl and tcpHdrLen, are constant throughout the simulation, and therefore, are dropped.
Step 4: Attribute transformation
The remaining continuous attributes (Length, tcpWinSize and deltaTime) are discritised using ordinal uniform binning. The process transforms the attribute values into ordinal values, such that each ordinal value or bin corresponds to an interval of the actual quantitative values. All the attributes are encoded as dummy variables before implementing the next steps.
Step 5: Cross-validation
The dataset is sliced into two parts: train and test sets to measure the machine learning models' detection ability. The first set is used to fit the detection solution, while the second set is used to realise if the model will function on new or unseen data. To circumvent over-fitting, we split the entire dataset 10-times using 5-fold cross-validation. We summarise the working of 5-fold cross-validation in the following steps: (i) the entire dataset is resampled into 5-folds (Fig. 4), where one fold becomes the test fold and the remaining folds are used for training one-class SVM anomaly detection model. (ii) the detection performance on each test fold is calculated. For endline approach the extended features obtained from the training set are used when classifying the test set instances. When new nodes appear in the test set, we assign a default value of zero to the respective attribute. (iii) after cross-validating 5 test folds, the dataset is again split into 5-folds. We perform k-fold cross-validation 10 times, where k is equal to 5. This process is also known as 10x5 fold cross-validation.
5-Fold cross-validation on IEC 104 dataset
Evaluating performance on the test folds indicates if the built models will generalise. We enforce the class composition in all the split folds to retain the normal to anomalous ratio.
Step 6: Attribute extension
We examine training sets from cross-validation to capture the extended features. To improve the network anomaly detection ability of one-class SVM for IEC 104 protocol communication, we extend the original IEC 104 attributes (Table 3). We propose characterisation of the SCADA communication network such that the topology of the participating nodes is represented in the form of extended attributes. The intuition of utilising the topological features is founded on the knowledge gained from domain experts and the fact that IEC 104 SCADA communication network complies with a common master-slave topology (Matousek 2017).
Since the IEC 104 protocol adheres to the standard network master-slave topology, it is possible to process other IEC 104 packet capture files without prior network details. We automatically extract the graph attributes using the network node's source and destination Ethernet addresses. A similar approach should be applicable to other IEC 104 SCADA protocol datasets.
Table 3 Extended IEC 104 attributes
Considering the SCADA network topology knowledge, we regard the SCADA network as a graph—a structure composed of connected vertices. The vertices are the nodes in the SCADA network. Since two-way communication exists in IEC 104 protocol, a node can be a sender and a receiver. Thus, we model each node from the perspective of the sender and receiver of the communication packet. We represent each vertex (or node) of the graph (or network) by a measure of its neighbouring nodes, attributed as deg (node degree). For example, the volume of neighbours for node A when node A is the packet receiver is equal to 6 (Fig. 5).
Example SCADA network nodes with corresponding degree attributes
Another attribute that explains the participation of a node within the network is node weight. Node weight demonstrates the distinct behavior of the given node in terms of its relative IEC 104 packet frequency. The attribute allows modelling the respective node participation with the communication frequency of other nodes in the network. Like node degree attribute, each node will be featured as wt (node weight). In IEC 104 protocol communication, the slave nodes typically fall under the same frequency interval. For example (Fig. 5), weight (wt) of nodes E, F, G and H, ideally, would have less variance.
The logical assumption is to secure the master or the controlling nodes in the SCADA network (which also resonates with the domain expert's preference). We model the point-to-point communication between two nodes assuming the receiving node in the communication frame is potentially a master or serving node and is vulnerable to attacks. Correspondingly, we assume the opposite node in the communication is a slave or the client node and is passively gathering network knowledge or actively attempting a MITM. In regular circumstances, the IEC 104 network topology would ideally show the slave nodes to follow a similar participation behaviour. Thus, distinguishing an unusual participating node behaviour could be interesting to isolate. Therefore, we consider the source node participation behaviour (wt) and the corresponding node's neighbourhood volume (deg) to complement the communication between a pair. Both these attributes will model two-way point-to-point communication between the nodes in the network.
Hundreds of IEC 104 protocol packets are exchanged within the SCADA network daily. We capture the evidence of two-way communication between two nodes by adding an attribute called pairEx that records the existence of a response packet. For example, if node A sends format-I packet to node B, node B sends IEC 104 packet to node A with the same type ID, we can establish that pair communication exists. This attribute has binary values. Such extended behavioural node attributes characterise the SCADA network's function, thereby enriching the dimensional space for unary SVM based anomaly detection.
Step 7: Extended attribute reduction
After we project the network nodes with extended attributes, the node addresses are disregarded. This is because each node has been modeled with newly constructed behavioural features. In a real SCADA network, where hundreds of nodes are present, such reduction would reduce computational costs, besides maintaining the characteristics of the respective node.
Afterward, we perform correlation analysis, similar to Step 3. This is done to understand the relationship between newly constructed attributes. The analysis reveals that new attributes have a strong linear correlation.
Step 8: Extended attribute transformation
We transform the new node degree attribute using the user-defined threshold (considering the network architecture knowledge). Consider a SCADA network with one controller node and three monitoring nodes. In an intruder-free example scenario, the node degree threshold becomes two. If any node has a degree greater than half of the nodes in the network, then it may be anomalous, implying possible master impersonation.
The node weight attribute is transformed using a user-defined threshold of the 40th percentile. This indicates that if the nodes in the SCADA network have a relative weight less than the relative mode weight (weight of the majority of nodes in the network), they should be segmented. More than half of the network nodes (\(60\%\)) will have relative participation of at least threshold node weight in a normal network. At last, the extended binary attribute and all the transformed attributes are encoded as dummy variables.
Step 9: One-class SVM anomaly detection
To identify potential anomalies, we opt for one-class support vector machine (SVM) learning algorithm because it has been systematically evaluated to be stable and better when classifying anomalies in IEC 104 communication in an unsupervised setting (Anwar et al. 2021). The results (Anwar et al. 2021) show the potential of the chosen algorithm and call for attention to boost its usefulness for the given context.
The algorithm separates the instances by computing the relationship between each pair of observations using the Radial Basis Kernel. This function projects the observations in a higher dimension and then dissects the projection with a hyperplane (Schölkopf et al. 1999). The algorithm uses the default parameter settings with PyOD (Zhao et al. 2019).
We execute one-class SVM anomaly detection learning in an unsupervised setting for binary prediction; for original and reconstructed IEC 104 protocol attributes.
Step 10: Experimental evaluation
The test folds give detection performance of the 50 candidate one-class SVM anomaly detection models resulting from 10x5 cross-validations. Subsequently, we average the performance of the candidate models and calculate the standard deviation over all folds. We report the evaluation of IEC 104 data on cross-validated test folds with the help of below described metrics.
False negative rate (FNR);
False positive rate (FPR);
\(F_{1}\) score;
Matthews correlation coefficient (MCC);
AUC score.
False negative rate (FNR) and False positive rate (FPR) indicate the incorrect decisions of the anomaly detection approach, also known as costs. Therefore, it is essential to gauge the skill of the approach in reference to the errors. Ideally, the anomaly detector should have no errors. Due to the criticality of the context, our focus is drawn toward FNR.
False negative rate (FNR) gives an insight into miss-classifications. It is the error ratio of the number of packets that are misclassified as normal (FN) to the sum of false negative and true positive (TP) values (Eq. 1). This is a crucial metric since it tells how well the model detects anomalies. FNR close to 0 means the model is good at detecting the anomalies.
$$\begin{aligned} {FNR = FN / (FN + TP) } \end{aligned}$$
False positive rate (FPR) also gives insight into positive miss-classifications. It is the ratio of the number of packets that are misclassified as anomalous (FP) to the sum of false positive and true negative (TN) values (Eq. 2). In other words, it is the rate of normal packets that are incorrectly labelled as anomalous. FPR close to 0 is indicative of good detection performance. However, it is common for unsupervised machine learning-based anomaly detection systems to suffer from high FPR, where often each false positive case requires human investigation. For our evaluation, FPR lower than \(5\%\) and at the cost of non-existent missed anomalous packets is considered satisfactory.
$$\begin{aligned} {FPR = FP / (FP + TN) } \end{aligned}$$
Identifying the attack class is crucial in the current context, which is represented by the recall—ratio of correctly identified attacks to total (actual) attacks. At the same time, the result produced by the detector should reflect the precision—ratio of correctly identified attacks to total predicted attacks. To capture a balanced view of recall and precision, we rely on the \(F_{1}\) score—harmonic mean of recall and precision (Eq. 3). Considering the need to represent the ability of the detector in terms of both classes, we measure the \(F_{1}\) score for both classes and then average them (referred to as macro-average \(F_{1}\) score). An acceptable macro-average \(F_{1}\) score value for the given context is greater than 0.8 (where 1 is maximum and worst is 0).
$$\begin{aligned} { F_1 score= 2*TP / (2*TP + FP + FN)} \end{aligned}$$
Macro-average \(F_{1}\) score gives equal weight to both classes but ignores true negatives, i.e., the correctly separated routine IEC 104 packets. Additionally, its magnitude bents toward true positives, i.e., the correctly separated rare IEC 104 packets. To overcome this deficiency of macro-average \(F_{1}\) score, we also calculate the Matthews correlation coefficient (MCC). MCC is an educative score to evaluate binary classifications as compared to accuracy and \(F_{1}\) score (Chicco and Jurman 2020).
MCC is a correlation coefficient between the actual values and the values the detector outputs. To do so, it considers errors (missed classifications) and correct classifications as well as the variable composition of classes (Eq. 4). It ranges from \(+1\) to \(-1\); where a coefficient of \(+1\) indicates perfect classification, a coefficient of 0 indicates average classification, and \(-1\) indicates worst classification.
$$\begin{aligned} {MCC = \frac{TP*TN - FP*FN}{\sqrt{(TP + FP)(TP + FN) * (TN + FP)(TN + FN)}}} \end{aligned}$$
Another measure that demonstrates the skill of the anomaly detection system is a receiver operating curve plot or ROC plot, where the rate of true positives (TPR), i.e., the rate of correctly detected anomalies, is plotted in contrast to FPR. The performance measure is generally represented as an area under curve (AUC) score. Simply put, AUC score gives insight into the trade-off between correctly detected attacks (true positives) and errors of miss-classifying attacks (false positives). A good AUC score is close to 1.
Step 11: Performance comparison
To discern if endline approach performs better than the baseline, or vice versa, we perform significance hypothesis testing. Though we can consume any (or all) of the evaluation metrics for reporting comparative evaluation, we consider two—macro-average \(F_{1}\) score and MCC, mainly for their comprehensiveness but also for brevity. The normality test on macro-average \(F_{1}\) score and MCC samples over 50 candidate test-folds for both approaches yield that the samples are likely drawn from Gaussian distributions. The preceding deduction converges our choice to Student's t-test that outputs p-value (Flach 2012). The p-value is compared with the significance level to establish evidence for the null hypothesis. The null hypothesis of Student's t-test state that two related samples have identical average values, which in our case refers to both macro-average \(F_{1}\) score and average MCC values for both approaches. The level of significance is set to \(1\%\) as a criterion for accepting the null hypothesis.
The significance test is followed by Cohen's d effect size test with the intent to quantify the significance of the magnitude of difference between the two approaches. Cohen classifies effect sizes as small, medium and large, where d \(> 0.8\) indicates large effect size (Sullivan and Feinn 2012).
Initially, we executed the experiment with original IEC 104 attributes, which provides a baseline unary SVM anomaly detection evaluation. In the second run, the machine learning experiment performs anomaly detection using the reconstructed attribute set, which results in endline performance evaluation. Ultimately, the performance summary of IEC 104 one-class SVM anomaly detection in an unsupervised setting for both approaches is compared (Fig. 3).
We perform 10x5-fold cross-validations to assess the ability of the one-class SVM anomaly detector for the IEC 104 protocol communication. Each cross-validation model comprises a test set of approximately 8390 typical and 485 anomalous IEC 104 protocol packets. The result of the candidate anomaly detection models is represented in the form of an interval confusion matrix (Fig. 6). Each contingency matrix quadrant indicates the minimum and maximum IEC 104 packets for correct predictions and errors.
Resulting confusion matrices from 10x5 cross-validations of one-class SVM for anomaly detection on IEC 104 test sets for baseline and endline approaches
Table 4 Baseline and endline experimental evaluation results for IEC 104 protocol with unsupervised one-class SVM anomaly detection on cross-validated test folds
The results from the confusion matrices are used to calculate values for the evaluation metrics (Table 4). The table represents the models' performance on all test folds for both approaches. Each row reports the average error rates—false negative rate (FNR) and false positive rate (FPR), along with mean detection ability in terms of macro-average \(F_{1}\) score, Matthews correlation coefficient (MCC), and area under the receiver operation characteristic curve score, shortened as AUC score.
Baseline one-class SVM anomaly detection results for IEC 104 protocol data
Upon testing the detection ability of the approach with initial IEC 104 attributes (Table 4), the FNR lingers between \(45\%\) (0.45) and \(54\%\) (0.54), which indicates, on average, almost half of the anomalous IEC 104 packets (\(49\%\)) are undetected. The FPR on average remained around \(6.8\%\), i.e., out of approximately 8390 normal IEC 104 protocol packets in each cross-validation test fold, 524 - 635 frames were falsely categorised as anomalous.
The macro-average \(F_{1}\) score of 0.6 for the given imbalanced IEC 104 communication shows the initial approach is separating the normal as well as anomalous IEC 104 packets poorly. Correspondingly, the average MCC (0.3) and average AUC (0.7) relay the same inefficiency of baseline one-class SVM anomaly detection models for the given IEC 104 protocol data.
Endline one-class SVM anomaly detection results for IEC 104 protocol data
For our evaluation, an FPR is satisfactory only when there is non-existent FNR and when the FPR remains lower than \(5\%\). Both hold for our approach. The FNR remains between 0 and 0.8% throughout the cross-validation folds (Table 4). The average false alert rate reduced to \(2.8\%\) from the baseline average of \(6.8\%\). The overall endline FPR is about \(3\%\), i.e., \(2\%\) less than the threshold of \(5\%\).
A good anomaly detector for this context should correctly isolate malicious IEC 104 protocol traffic and, at the same time, produce fewer false alerts. The macro-average \(F_{1}\) score metric reflects this behavior of the detector. The macro-average \(F_{1}\) score for all the folds remained above 0.88, indicating better performance than the baseline approach. The AUC score of 0.98 on average, shows the detector is skillfully discriminating the IEC 104 protocol packets in the given dataset. To understand the detection performance of correct predictions while considering the errors, FNR and FPR, we calculate MCC. The average MCC value of 0.8 depicts near perfect detection performance for the endline case.
Performance comparison results
The results are analysed with the Student t-test and reveal that baseline has a mean macro-average \(F_{1}\) score of 0.6 and MCC of 0.3; and that endline have a mean macro-average \(F_{1}\) score of 0.9 and MCC of 0.8. The p-value close to 0 indicates that the average performance of both approaches over 50 candidate models is not identical, failing to accept the null hypothesis at a \(1\%\) significance level.
We calculate the magnitude of difference between the performance of baseline and endline approaches with the help of Cohen's d test. The test is carried on macro-average \(F_{1}\) scores and MCCs values. The test result indicates the existence of a large effect size of over 20 Standard Deviations between the two configurations of one-class SVM algorithm. Hence, we establish the endline anomaly detection for IEC 104 has significant improvement over baseline approach.
When one-class SVM is applied to IEC 104 dataset (Egger et al. 2020) in an unsupervised setting, an AUC score of 0.49 (default algorithm setting) was reported (Table 5) and 0.64 (after parameter tuning) on unseen data when final cross-validated candidate models are used for training (Anwar et al. 2021). Also, both instances have ensued a meager correct classification rate (Anwar et al. 2021) and are plagued with prediction errors (Table 5). In comparison to the aforementioned previous work, this study presents an improved one-class SVM anomaly detection approach for IEC 104 protocol communication. The average cross-validated AUC score for endline approach is 0.98, higher than the average cross-validated baseline AUC score. Other associated metrics, FNR, FPR, and MCC, show similar trends and are relayed for comparative purposes. Crucial criteria to assess the anomaly detector's ability is to isolate suspicious IEC 104 protocol packets correctly and not miss any suspicious IEC 104 packets. Both criteria for the given context are crucial and are satisfied in the endline approach, providing an average TPR of \(99.6\%\) and an average FPR of \(2\%\). The endline approach does not miss attack communications for some test folds, as seen from the FNR, i.e., the best among other values (Table 5).
Table 5 Comparison of related one-class SVM anomaly detection results on 2 IEC 104 protocol datasets
The anomaly detection algorithm in the learning phase forms a boundary for the given data. The SVM hyperplane cannot form an optimal decision boundary because our training data is polluted (to replicate a real scenario). Having some sanitised data for the learning phase may reduce prediction errors. For example, the case of semi-supervised learning where prior knowledge about some datapoints is used to train the classifier. However, this additional processing may require more effort as compared to our approach.
Prediction errors require additional analysis, which can be a hassle in production anomaly detection systems. We perform a preliminary analysis on the 50th candidate model of the endline approach to highlight how the approach can assist in further analysis of the anomalous IEC 104 packets. Further analysis reveals that the detected anomalous exchanges are mainly between four SCADA nodes. Two of the identified nodes, of which one is a MITM attacker, are transmitting information to a high-degree node (third node). The MITM attacker tries to synchronise clock times like other nodes, possibly RTUs. It goes undetected as the protocol does not verify senders. Upon interrogation request from the high-degree node (possible attack target), the attacker replaces the cause of transmission with invalid data and terminates the connection.
The second isolated node is a legitimate RTU but is separated as it demonstrates low participation in the network. Further analysis and opinion of domain expert are crucial to investigate the reasons behind low participation. If low participation is acceptable for the particular RTU, the analyst can ignore the identified node. This falsely identified SCADA node constitutes about \(99.6\%\) of the FPs in the last candidate model.
The rare participation behaviour can help detect reconnaissance attackers. We see that the endline approach separated the reconnaissance attacker (fourth node). Reconnaissance attack nodes are passively observing the network and can contribute to advance persistent threats (Assante and Lee 2015); hence, their isolation can potentially delay or disrupt the following attack sequence.
Due to the lack of new nodes appearing in the test sets, it is difficult to confirm or deny the detection performance of the approach. As an alternative, we intentionally added two new nodes such that they only appear in the test set. The communication frames were flagged as an anomaly due to their rare characteristic, for example, the absence of communicating nodes.
Graph-based attribute extension of SCADA network nodes with one-class SVM algorithm has the potential to isolate the rouge network nodes in IEC 104 protocol communication. The work extracts meaningful relations between the network nodes to model the behavior of the network. Consequently, the representation allows isolating strange nodes, e.g., passive intruders trying to ping neighbouring nodes. Since it is possible to classify a new instance immediately when it arrives without considering other instances, it is feasible to use the approach for active detection in real-time. We compare the potential of attribute extension by presenting baseline and endline results. The cross-validation models retain the highest average \(F_{1}\) score (0.90), MCC (0.80), and AUC score (0.98), while giving modest false-alert and miss-rates in comparison to related works, as well as the baseline detection method.
Keeping miss-alerts and false-alerts to a minimum is crucial for deploying an anomaly detector for critical infrastructures. The endline results produce fewer errors overall. The missed-alerts are almost negligible, with a drastic drop in the false alerts, depicting a holistic boost in the endline method significantly over the baseline scores. Hence, through topological attribute extension of IEC 104 protocol features, one-class SVM can likely identify anomalies in the SCADA network.
One-class SVM is a popular choice for anomaly detection in communication networks (Tsai et al. 2009; Thakkar and Lohiya 2021; Rakas et al. 2020). Moreover, it demonstrated stable outcomes when assessed on a SCADA network communication dataset (Anwar et al. 2021). Future work will benefit by including other classifying methods, such as Auto-encoders or neural networks.
It is necessary to iterate that the approach is implemented and evaluated as an unsupervised learning method and that the detection models are created in the presence of routine and anomalous data. For future work, sanitised SCADA protocol attributes could be used for modelling the detector. We can also test the presented approach for similar isolated topological networks to identify eavesdroppers, for example, in the Modbus SCADA protocol.
Often attribute processing is dependent on human intervention, creating scalability concerns. The proposed approach relies on automatic extraction of attributes and, thus, is possible to be scaled. Notable advances are made towards graph embedding techniques and stacked auto-encoders to reveal hidden and intricate attributes, i.e., without manual effort, to model complex networks (Pourhabibi et al. 2020; Corizzo et al. 2021). For this reason, we expect that the acceptance of other methods to infer network behaviour will continue to grow.
We utilise the dataset that is available under the following link: https://figshare.com/articles/dataset/dataset-v1_pcap/6133457/1
DNP3:
Distributed network protocol 3
FNR:
False negative rate
HMI:
IEC:
IEC 104:
IEC 60870-5-104
MCC:
Matthews correlation coefficient
OSI:
Open systems interconnection
ROC-AUC (or AUC):
Receiver operating curve area under curve
RTU:
Remote terminal unit
SCADA:
Supervisory control and data acquisition
SVM:
TCP/IP:
Transmission control protocol/internet protocol
Akoglu L, McGlohon M, Faloutsos C (2010) Oddball: Spotting anomalies in weighted graphs. In: Zaki MJ, Yu JX, Ravindran B, Pudi V (eds) Advances in knowledge discovery and data mining. Pacific-Asia conference on knowledge discovery and data mining (PAKDD). Lecture notes in computer science, Vol 6119. Springer, Berlin, Heidelberg, pp. 410–421
Anwar M, Borg A, Lundberg L (2021) A comparison of unsupervised learning algorithms for intrusion detection in IEC 104 SCADA protocol. In: 20th International conference on machine learning and cybernetics (ICMLC), IEEE, pp. 1–8. https://doi.org/10.1109/ICMLC54886.2021.9737267
Assante MJ, Lee RM (2015) The industrial control system cyber kill chain. Technical report, SANS Institute InfoSec Reading Room
Chicco D, Jurman G (2020) The advantages of the Matthews correlation coefficient (MCC) over F1 score and accuracy in binary classification evaluation. BMC Genomics 21(1):6. https://doi.org/10.1186/s12864-019-6413-7
CISA (2016) ICS Alert (IR-ALERT-H-16-056-01). Cyber-attack against Ukrainian critical infrastructure; Cybersecurity and Infrastructure Security Agency. Cybersecurity and Infrastructure Security Agency (CISA). https://www.cisa.gov/uscert/ics/alerts/IR-ALERT-H-16-056-01. Accessed 9 May 2022
Corizzo R, Ceci M, Pio G, Mignone P, Japkowicz N (2021) Spatially-aware autoencoders for detecting contextual anomalies in geo-distributed data. In: International conference on discovery science, pp. 461–471. Springer
Egger M, Eibl G, Engel D (2020) Comparison of approaches for intrusion detection in substations using the IEC 60870–5-104 protocol. Energy Inform 3(S1):1–17. https://doi.org/10.1186/s42162-020-00118-4
ENISA (2017) Communication Network Dependencies for ICS/SCADA Systems. European Network and Information Security Agency, Athens. https://doi.org/10.2824/397676
Flach P (2012) Machine learning: the art and science of algorithms that make sense of data. Cambridge University Press, Cambridge
Book MATH Google Scholar
Ferling B, Chromik J, Caselli M, Remke A (2018) Intrusion detection for sequence-based attacks with reduced traffic models. In: International conference on measurement, modelling and evaluation of computing systems, pp. 53–67. Springer
Gaggero GB, Rossi M, Girdinio P, Marchese M (2020) Detecting system fault/cyberattack within a photovoltaic system connected to the grid: a neural network-based solution. J Sens Actuator Netw 9(2):20. https://doi.org/10.3390/jsan9020020
Grammatikis PR, Sarigiannidis P, Sarigiannidis A, Margounakis D, Tsiakalos A, Efstathopoulos G (2020) An anomaly detection mechanism for iec 60870-5-104. In: 9th International conference on modern circuits and systems technologies (MOCAST), IEEE, pp. 1–4. https://doi.org/10.1109/MOCAST49295.2020.9200285
György P, Holczer T (2020) Attacking iec 60870-5-104 protocol. In: 1st Conference on Information Technology and Data Science (CITDS), pp. 140–150. http://ceur-ws.org/Vol-2874/paper13.pdf
Henderson K, Gallagher B, Eliassi-Rad T, Tong H, Basu S, Akoglu L, Koutra D, Faloutsos C, Li L (2012) Rolx: structural role extraction & mining in large graphs. In: 18th ACM SIGKDD International conference on knowledge discovery and data mining, pp. 1231–1239. https://doi.org/10.1145/2339530.2339723
Henderson K, Eliassi-Rad T, Faloutsos C, Akoglu L, Li L, Maruhashi K, Prakash BA, Tong H (2010) Metric forensics: a multi-level approach for mining volatile graphs. In: 16th ACM SIGKDD international conference on knowledge discovery and data mining, pp. 163–172. ACM, NY. https://doi.org/10.1145/1835804.1835828
Hodo E, Grebeniuk S, Ruotsalainen H, Tavolato P (2017) Anomaly detection for simulated IEC-60870-5-104 Trafiic. In: 12th International conference on availability, reliability and security, pp. 1–7. ACM, Reggio Calabria. https://doi.org/10.1145/3098954.3103166. Accessed 1 Jul 2021
Lin C-Y, Nadjm-Tehrani S (2019) Timing patterns and correlations in spontaneous scada traffic for anomaly detection. In: 22nd International symposium on research in attacks, intrusions and defenses (RAID), pp. 73–88
Linda O, Vollmer T, Manic M (2009) Neural network based intrusion detection system for critical infrastructures. In: 2009 International joint conference on neural networks, IEEE, pp. 1827–1834
Mahmood AN, Leckie C, Hu J, Tari Z, Atiquzzaman M (2010) Network traffic analysis and scada security. Handbook of Information and Communication Security. Springer, Berlin, pp 383–405. https://doi.org/10.1007/978-3-642-04117-4
Matousek P (2017) Description and analysis of IEC 104 Protocol. Technical Report FIT-TR-2017-1, Faculty of Information Technology, Brno University of Technology, Brno, Czech Republic. https://www.fit.vut.cz/research/publication-file/11570/TR-IEC104.pdf
Matousek P, Ryšavỳ O, Grégr M (2019) Increasing visibility of iec 104 communication in the smart grid. In: 6th International Symposium for ICS & SCADA Cyber Security Research, pp. 21–30. https://doi.org/10.14236/ewic/icscsr19.3
Mantere M, Sailio M, Noponen S (2013) Network traffic features for anomaly detection in specific industrial control system network. Future Internet 5(4):460–473. https://doi.org/10.3390/fi5040460
Matousek P, Ryšavỳ O, Grégr M, Havlena V (2020) Flow based monitoring of ics communication in the smart grid. J Inf Secur Appl 54:1–16. https://doi.org/10.1016/j.jisa.2020.102535
Maynard P, McLaughlin K, Sezer S (2018) An open framework for deploying experimental scada testbed networks. In: 5th International symposium for ICS & SCADA cyber security research 2018, pp. 92–101. https://doi.org/10.14236/ewic/ICS2018.11
Panagiotis F, Taxiarxchis K, Georgios K, Maglaras L, Ferrag MA (2021) Intrusion detection in critical infrastructures: a literature review. Smart Cities 4(3):1146–1157. https://doi.org/10.3390/smartcities4030061
Pliatsios D, Sarigiannidis P, Lagkas T, Sarigiannidis AG (2020) A survey on SCADA systems: secure protocols, incidents, threats and tactics. IEEE Commun Surv Tutor 22(3):1942–1976. https://doi.org/10.1109/COMST.2020.2987688
Pourhabibi T, Ong K-L, Kam BH, Boo YL (2020) Fraud detection: a systematic literature review of graph-based anomaly detection approaches. Decis Support Syst 133:113303. https://doi.org/10.1016/j.dss.2020.113303
Radoglou-Grammatikis P, Sarigiannidis P, Giannoulakis I, Kafetzakis E, Panaousis E (2019) Attacking iec-60870-5-104 SCADA systems. In: IEEE World Congress on Services (SERVICES), IEEE, pp. 41–46. https://doi.org/10.1109/SERVICES.2019.00022
Rakas SVB, Stojanović MD, Marković-Petrović JD (2020) A review of research work on network-based scada intrusion detection systems. IEEE Access 8:93083–93108. https://doi.org/10.1109/ACCESS.2020.2994961
Robles-Durazno A, Moradpoor N, McWhinnie J, Russell G (2018) A supervised energy monitoring-based machine learning approach for anomaly detection in a clean water supply system. In: 2018 International Conference on Cyber Security and Protection of Digital Services (Cyber Security), IEEE, pp. 1–8. https://doi.org/10.1109/CyberSecPODS.2018.8560683
Schölkopf B, Williamson RC, Smola A, Shawe-Taylor J, Platt J (1999) Support vector method for novelty detection. In: 12th International conference on neural information processing systems (NIPS), pp. 582–588. MIT Press, Colorado. https://doi.org/10.5555/3009657.3009740. https://proceedings.neurips.cc/paper/1999/file/8725fb777f25776ffa9076e44fcfd776-Paper.pdf
Sullivan GM, Feinn R (2012) Using effect size-or why the p value is not enough. J Grad Med Educ 4(3):279–282. https://doi.org/10.4300/JGME-D-12-00156.1
Tariq N, Asim M, Khan FA (2019) Securing scada-based critical infrastructures: challenges and open issues. Proc Comput Sci 155:612–617. https://doi.org/10.1016/j.procs.2019.08.086
Thakkar A, Lohiya R (2021) A survey on intrusion detection system: feature selection, model, performance measures, application perspective, challenges, and future research directions. Artif Intell Rev. https://doi.org/10.1007/s10462-021-10037-9
Tsai C-F, Hsu Y-F, Lin C-Y, Lin W-Y (2009) Review: intrusion detection by machine learning: a review. Expert Syst Appl 36(10):11994–12000. https://doi.org/10.1016/j.eswa.2009.05.029
Udd R, Asplund M, Nadjm-Tehrani S, Kazemtabrizi M, Ekstedt M (2016) Exploiting bro for intrusion detection in a scada system. In: 2nd ACM international workshop on cyber-physical system security, pp. 44–51. ACM, Xian. https://doi.org/10.1145/2899015.2899028
Wong K, Dillabaugh C, Seddigh N, Nandy B (2017) Enhancing suricata intrusion detection system for cyber security in SCADA networks. In: 2017 IEEE 30th Canadian Conference on Electrical and Computer Engineering (CCECE), IEEE, pp. 1–5. https://doi.org/10.1109/CCECE.2017.7946818
Yang Y, McLaughlin K, Littler T, Sezer S, Pranggono B, Wang H (2013) Intrusion detection system for iec 60870-5-104 based scada networks. In: 2013 IEEE power & energy society general meeting, IEEE, pp. 1–5. https://doi.org/10.1109/PESMG.2013.6672100
Yang Y, Xu H-Q, Gao L, Yuan Y-B, McLaughlin K, Sezer S (2016) Multidimensional intrusion detection system for iec 61850-based scada networks. IEEE Trans Power Deliv 32(2):1068–1078. https://doi.org/10.1109/TPWRD.2016.2603339
Zhao Y, Nasrullah Z, Li Z (2019) Pyod: a python toolbox for scalable outlier detection. J Mach Learn Res 20(96):1–7
Zhu B, Joseph A, Sastry S (2011) A taxonomy of cyber attacks on scada systems. In: 2011 International Conference on Internet of Things and 4th International Conference on Cyber, Physical and Social Computing, IEEE, pp. 380–388. https://doi.org/10.1109/iThings/CPSCom.2011.34
The authors would like to thank Henrik Johansson and Anders Karlberg at Techinova for sharing their domain knowledge.
Open access funding provided by Blekinge Institute of Technology.
Department of Computer Science, Blekinge Institute of Technology, 371 79, Karlskrona, Sweden
Mahwish Anwar, Lars Lundberg & Anton Borg
Mahwish Anwar
Lars Lundberg
Anton Borg
Conceptualisation: MA and LL.; methodology: MA, AB and LL.; software: MA.; validation: MA, LL, AB; formal analysis: MA, AB and LL.; investigation: MA; resources: MA.; data curation, MA; writing—original draft preparation: MA.; writing—review and editing, MA, AB and LL.; visualization, MA, LL and AB.; supervision, LL and AB. All authors have read and approved the final manuscript.
Correspondence to Mahwish Anwar.
Anwar, M., Lundberg, L. & Borg, A. Improving anomaly detection in SCADA network communication with attribute extension. Energy Inform 5, 69 (2022). https://doi.org/10.1186/s42162-022-00252-1
Attribute extension | CommonCrawl |
Problem G
Horror List
Photo by Janet Hudson
It was time for the 7th Nordic Cinema Popcorn Convention, and this year the manager Ian had a brilliant idea. In addition to the traditional film program, there would be a surprise room where a small group of people could stream a random movie from a large collection, while enjoying popcorn and martinis.
However, it turned out that some people were extremely disappointed, because they got to see movies like Ghosts of Mars, which instead caused them to tear out their hair in despair and horror.
To avoid this problem for the next convention, Ian has come up with a solution, but he needs your help to implement it. When the group enters the surprise room, they will type in a list of movies in a computer. This is the so-called horror list, which consists of bad movies that no one in the group would ever like to see. Of course, this list varies from group to group.
You also have access to the database Awesome Comparison of Movies which tells you which movies are directly similar to which. You can assume that movies that are similar to bad movies will be almost as bad. More specificly, we define the Horror index as follows:
\[ HI = \left\{ \begin{array}{ll} 0 & \textrm{if movie is on horror list. This overrides the other definitions.} \\ Q+1 & \textrm{if the worst directly similar movie has $HI = Q$} \\ +\infty & \textrm{if not similar at all to a horrible movie} \end{array} \right. \]
The first line of input contains three positive integers $N$, $H$, $L$ ($1 \leq H < N \leq 1\, 000,0 \leq L \leq 10\, 000$), where $N$ is the number of movies (represented by IDs, ranging from $0$ to $N-1$), $H$ is the number of movies on the horror list and $L$ is the number of similarities in the database.
The second line contains $H$ unique space-separated integers $x_ i$ ($0 \leq x_ i <N$) denoting the ID of the movies on the horror list.
The following $L$ lines contains two space-separated integers $a_ i,b_ i$ ($0 \leq a_ i < b_ i < N$), denoting that movie with ID $a_ i$ is similar to movie with ID $b_ i$ (and vice versa).
Output the ID of the movie in the collection with the highest Horror Index. In case of a tie, output the movie with the lowest ID.
Problem ID: horror
Author: Matias Holte
Source: Nordic Collegiate Programming Contest (NCPC) 2012 | CommonCrawl |
\begin{document}
\title{On exponential functionals of L\'evy processes} \author{Anita Behme\thanks{Technische Universit\"at M\"unchen, Institut f\"ur Mathematische Statistik, Boltzmannstra\ss e 3, D-85748 Garching bei M\"unchen, Germany, email: [email protected], tel.: +49/89/28917424, fax:+49/89/28917435}$\,$ and Alexander Lindner\thanks{Technische Universit\"at Braunschweig, Institut f\"ur Mathematische Stochastik, Pockelsstr. 14, D-38106 Braunschweig, Germany, email: [email protected], tel.:+49/531/3917575, fax:+49/531/3917564} } \date{\today} \maketitle
\begin{abstract} Exponential functionals of L\'evy processes appear as stationary distributions of generalized Ornstein-Uhlenbeck (GOU) processes. In this paper we obtain the infinitesimal generator of the GOU process and show that it is a Feller process. Further we use these results to investigate properties of the mapping $\Phi$, which maps two independent L\'evy processes to their corresponding exponential functional, where one of the processes is assumed to be fixed. We show that in many cases this mapping is injective and give the inverse mapping in terms of (L\'evy) characteristics. Also, continuity of $\Phi$ is treated and some results on its range are obtained. \end{abstract}
2000 {\sl Mathematics subject classification.} 60G10, 60G51, 60J35.\\ {\sl Key words and phrases.} generalized Ornstein-Uhlenbeck process, L\'evy process, Feller process,
infinitesimal generator, integral mapping, stationarity
\section{Introduction}\label{S1} \setcounter{equation}{0}
The {\it exponential functional} of a bivariate L\'evy process $(\xi,\eta)^T = ((\xi_t,\eta_t)^T)_{t\geq 0}$ is defined as \begin{equation} \label{eq-integral} V_\infty = \int_{(0,\infty)}e^{-\xi_{t-}}d\eta_t. \end{equation} Necessary and sufficient conditions for the convergence of integrals of the form $\int_{(0,t]}e^{-\xi_{s-}}d\eta_s$ as $t\to \infty$ for a bivariate L\'evy process $(\xi,\eta)^T$ were given by Erickson and Maller~\cite[Thm. 2]{ericksonmaller05}. Distributional properties of exponential functionals have been studied in various articles throughout the years by e.g. Paulsen \cite{paulsen93}, Yor \cite{yor}, Bertoin et al.~\cite{bertoinlindnermaller08}, Kondo et al.~\cite{kondomaejimasato06}, Lindner and Sato \cite{lindnersato09}, Behme \cite{behme2011} and Kuznetsov et al.~\cite{savovetal} to name just a few.
Denote by $\mathcal{L}(X)$ the law of a random variable $X$. In this paper, for a given one-dimensional L\'evy process $\xi$, we will consider mappings like \begin{eqnarray*} \Phi_\xi : D_\xi & \to & \mbox{set of probability distributions on $\mathbb{R}$} ,\\ \mathcal{L}(\eta_1) & \mapsto & \mathcal{L} \left( \int_0^\infty e^{-\xi_{s-}} \, d\eta_s \right) \end{eqnarray*} defined on $D_\xi := \{ \mathcal{L} (\eta_1) : \eta$ L\'evy process, independent of $\xi,$ such that $\int_0^\infty e^{-\xi_{s-}} \, d\eta_s $ converges a.s.$\}$ and we will examine injectivity and continuity of such mappings and gather information about their ranges.
In the case that $\xi_t=at$ is deterministic, it is well known that $D_\xi=\mbox{ID}_{\log}(\mathbb{R})$ is the set of real-valued infinitely divisible distributions with finite $\mbox{log}^+$-moment and that $\Phi_\xi$ is an algebraic isomorphism between $\mbox{ID}_{\log}(\mathbb{R})$ and $L(\mathbb{R})$, the set of real-valued selfdecomposable distributions \cite[Prop. 3.6.10]{jurekmason}.
We start with a short example of a special case to illustrate the kind of results we obtain, as well as the occuring problems.
\begin{example} \label{example-start} Suppose $(\xi_t)_{t\geq 0}$ is a compound Poisson process with intensity rate $\lambda$ and jump heights measure $\tau$. Let $\eta$ be a L\'evy process independent of $\xi$ such that $\mathcal{L}(\eta_1)\in D_\xi$. Define $T_i$ to be the time of the $i$th jump of $\xi$ with $T_0 := 0$. Then $$V_\infty=\int_{(0,\infty)} e^{-\xi_{s-}} \, d\eta_s = \sum_{i=0}^\infty \int_{(T_i,T_{i+1}]}e^{-\xi_{T_i}}d\eta_t =
\sum_{i=0}^\infty \left(\prod_{k=1}^i e^{-\Delta \xi_{T_k}} \right) (\eta_{T_{i+1}}-\eta_{T_i}).$$ Since $(e^{-\Delta \xi_{T_i}}, \eta_{T_{i+1}}-\eta_{T_i})_{i=0,1,2,\ldots}$ is an i.i.d. sequence, as e.g. in \cite{behme-et-al} we obtain from this the distributional fixed point equation $$V_\infty\overset{d}= X V_\infty' + H$$ where $(X,H)\overset{d}= (e^{-\Delta \xi_{T_i}},\eta_{T_{i+1}}-\eta_{T_i})$ for $i=1,2,\ldots$ and $V_\infty\overset{d}=V_\infty'$ where $V_\infty'$ is independent of $(X,H)$. In terms of characteristic functions this yields $\phi_{V_\infty}(u)= \phi_{XV_\infty'}(u)\phi_H(u)$ and adding the fact that the characteristic function $\phi_\eta$ of the L\'evy process $(\eta_t)_{t\geq 0}$ and the corresponding exponentially subordinated process $(H_t)_{t\geq 0}=(\eta_{\tau(t)})_{t\geq 0}$ with $\tau\sim\mbox{Exp}(\lambda)$ fulfill the equation $$\phi_H(u)=\frac{\lambda}{\lambda -\log (\phi_\eta(u))}$$ (see e.g. \cite[p.10]{steutelvanharn}) we have \begin{equation} \label{eq-motivation} \log(\phi_\eta(u)) \phi_{V_\infty}(u)= \lambda \left( \phi_{V_\infty}(u)- \phi_{XV_\infty'}(u)\right)= \lambda \int_{\mathbb{R}} \left(E\left[e^{iuV_\infty} \right]- E\left[e^{iue^{-y}V_\infty} \right] \right) \tau(dy). \end{equation} \end{example}
Now, in the setting of the example {\it if we knew} that the characteristic function of $V_\infty$ is non-zero on a dense subset of $\mathbb{R}$ this gave us a formula for the characteristic exponent of $\eta$ and thus injectivity of the mapping $\Phi_\xi$. But in general the quotient of two characteristic functions does not necessarily yield a unique solution as has already been remarked in \cite{loeve}. Examples for non-uniqueness of such quotients are also given in \cite{levy61}.
To obtain formulas like \eqref{eq-motivation} for general L\'evy processes $(\xi, \eta)^T$ we will strongly make use of the fact that GOU processes are Markov processes. So, in Section \ref{sec-markov} we first compute the infinitesimal generator of the GOU process and show that it is actually a Feller process. In Section \ref{sec-relations} these results will be used to obtain formulas of the form \eqref{eq-motivation} for general, independent L\'evy processes $\xi$ and $\eta$. Hereby we obtain a general formula for $\log(\phi_{\eta}(u)) \phi_{V_\infty}(u)$ in terms of the characteristic triplet of $\xi$ and $\mathcal{L}(V_\infty)$ as given in Theorem \ref{thm-generatorlimit} and Corollary \ref{cor-generatorlimit} and on the other hand in Theorem \ref{thm-generatorlimit-2} we express
$\log(\phi_{-\xi}(u)) \phi_{\log|V_\infty|}(u)$ in terms of the characteristic triplet of $\eta$ and $\mathcal{L}(V_\infty)$.\\ Further, Section \ref{sec-inj} is devoted to the study of injectivity, which - in view of the results of Section \ref{sec-relations} -
now reduces to an examination of when either $\phi_{V_\infty}(u)$ or $\phi_{\log|V_\infty|}(u)$ are non-zero on a dense subset of $\mathbb{R}$. We give various examples of when the mapping $\Phi_\xi$ or its counterpart $\tilde{\Phi}_\eta$ (which maps $\mathcal{L}(\xi_1)$ to $\mathcal{L}(V_\infty)$ for $\eta$ fixed) are injective and argue why injectivity cannot be obtained if $\xi$ and $\eta$ are allowed to exhibit a dependence structure.\\ Section \ref{sec-ranges} then uses the previous results to obtain information on the ranges of $\Phi_\xi$ and $\tilde{\Phi}_\eta$. In particular, Theorem \ref{thm-nonormal} shows that centered Gaussian distributions can only be obtained in the setting of (standard) OU processes, i.e. for $\xi$ being deterministic and $\eta$ being a Brownian motion.\\ Finally, in Section \ref{sec-cont} we give conditions for continuity (in a weak sense) of the mappings $\Phi_\xi$ and $\tilde{\Phi}_\eta$ and give an example of $\Phi_\xi$ being not continuous.
\section{Some background on GOU processes and Notations} \setcounter{equation}{0}
By the L\'evy-Khintchine formula (e.g. \cite[Thm. 8.1]{sato}) the {\it characteristic exponent} of an $\mathbb{R}^d$-valued L\'evy process $X=(X_t)_{t\geq 0}$ is given by \begin{eqnarray*}
\psi_X(u)&:=&\log \phi_X(u) := \log E\left[e^{i\langle u, X_1 \rangle } \right]\\
&=& i \langle \gamma_X, u\rangle - \frac{1}{2} \langle u, A_X u \rangle + \int_{\mathbb{R}^d} (e^{i \langle u, x \rangle} -1 -i \langle u, x \rangle \mathds{1}_{|x|\leq 1}) \nu_X(dx) \end{eqnarray*}
where $(\gamma_X, A_X, \nu_X)$ is the {\it characteristic triplet} of $X$. In case that $X$ is real valued we will usually replace $A_X$ by $\sigma^2_X$. To simplify notations, we set $\nu(\{0\})=0$ for any L\'evy measure $\nu$. If the L\'evy measure $\nu_X$ satisfies the condition $\int_{|x|\leq 1} |x|\nu_X(dx)<\infty$ we may also use the L\'evy-Khintchine formula in the form \begin{eqnarray*}
\psi_X(u)&=& i \langle \gamma^0_X, u\rangle - \frac{1}{2} \langle u, A_X u \rangle + \int_{\mathbb{R}^d} (e^{i \langle u, x \rangle} -1 ) \nu_X(dx) \end{eqnarray*} and call $\gamma_X^0$ the {\it drift} of $X$. We refer to \cite{sato} for any further information on L\'evy processes. We write $\Delta Y_t = Y_t - Y_{t-}$ for any c\`adl\`ag process $Y$.
Given a bivariate L\'evy process $((\xi_t,\eta_t)^T)_{t\geq 0}$ and a random variable $V_0$ on the same probability space, \begin{equation} V_t=e^{-\xi_t} \left( \int_0^t e^{\xi_{s-}}d\eta_s +V_0 \right),\quad t\geq 0, \label{GOUdef} \end{equation} defines the {\it generalized Ornstein-Uhlenbeck (GOU) process driven by $(\xi,\eta)^T$ with starting random variable $V_0$}. In the case that $\xi_t=at$ is deterministic, the process $V_t$ is usually called {\it Ornstein-Uhlenbeck-type process}, while if $(\xi_t, \eta_t)=(at,B_t)$ for $B$ a Brownian motion, $V_t$ is known as {\it Ornstein-Uhlenbeck (OU) process}.
The GOU process driven by $(\xi,\eta)^T$ is the unique solution of the stochastic differential equation \begin{equation}\label{SDEGOU} dV_t=V_{t-}dU_t+dL_t,\quad t\geq 0, \end{equation} for the bivariate L\'evy process $((U_t,L_t)^T)_{t\geq 0}$ given by \begin{equation} \label{eq-def-UL} \left( \begin{array}{c} U_t \\ L_t \end{array} \right) = \left( \begin{array}{l} -\xi_t + \sum_{0< s \leq t} \left(e^{- \Delta \xi_s} -1 + \Delta \xi_{s}\right) + t \,\sigma_{\xi}^2/2 \\
\eta_t + \sum_{0 < s \leq t} (e^{-\Delta \xi_s}-1) \Delta \eta_s - t\,\sigma_{\xi,\eta} \end{array}\right),\quad t\geq 0, \end{equation} where $\sigma^2_{\xi}$ and $\sigma_{\xi,\eta}$ denote the $(1,1)$ and $(1,2)$ elements of the Gaussian covariance matrix $A_{(\xi,\eta)}$. Equation \eqref{eq-def-UL} defines a bijection between all bivariate L\'evy processes $(\xi,\eta)^T$ and all bivariate L\'evy process $(U,L)^T$ such that $\nu_U((-\infty,-1]) = 0$. The upper line of \eqref{eq-def-UL} is equivalent to $e^{-\xi_t} = \mathcal{E}(U)_t$, where $\mathcal{E}(U)_t$ is the stochastic exponential of $U$, which is defined as the unique c\`adl\`ag solution $S$ of $S_t = 1 + \int_{(0,t]} S_{s-} dU_s$ (see e.g. \cite[Thm.~II.37]{protter}). Equation \eqref{SDEGOU} has a solution for any bivariate L\'evy process $(U,L)^T$ and any starting random variable $V_0$ independent of $(U,L)^T$, which in the case $\nu_U(\{-1\}) = 0$ is given by \begin{equation} \label{GOUdef2} V_t = \mathcal{E}(U)_t \left( \int_{(0,t]} \mathcal{E}(U)_{s-}^{-1} \, d\eta_s + V_0\right), \end{equation} where $\eta_t = L_t - \sum_{0<s\leq t} (1+\Delta U_s)^{-1} \Delta U_s \Delta L_s - t\sigma_{U,L}$, see \cite[Thm. 2.1]{behmelindnermaller11}. The processes of the form \eqref{GOUdef2} hence constitute a slightly larger class of stochastic processes than the GOU processes defined by \eqref{GOUdef} since they allow $U$ also to have jumps smaller than $-1$. Obviously, the GOU process defined in \eqref{GOUdef} as well as the process defined in \eqref{GOUdef2} are time homogeneous Markov processes \cite[Lem. 3.3]{behmelindnermaller11}.
In \cite{lindnermaller05} necessary and sufficient conditions for the existence of causal, strictly stationary solutions of the generalized Ornstein-Uhlenbeck process \eqref{GOUdef} are given. In particular it is shown (\cite[Thm. 2.1]{lindnermaller05}) that if $(V_t)_{t\geq 0}$ is strictly stationary and causal, then $\int_{(0,t]} e^{-\xi_{s-}}dL_s$ with $L$ as defined in \eqref{eq-def-UL} converges a.s. to a finite random variable as $t\to \infty$ and the stationary law $\mu$ is given by $\mu=\mathcal{L}(V_\infty)$ for $V_\infty = \int_{(0,\infty)}e^{-\xi_{s-}}dL_s.$ Observe that $L=\eta$ if $\xi$ and $\eta$ are independent.
The space of continuous functions $\mathbb{R}^d\to \mathbb{R}$ is denoted by $C(\mathbb{R}^d)$. The subspaces of bounded functions, functions vanishing at infinity and functions with compact support are written as $C_b(\mathbb{R}^d)$, $C_0(\mathbb{R}^d)$ and $C_c(\mathbb{R}^d)$, resp. For $n\in \mathbb{N}$ we write $C^n(\mathbb{R}^d)$ for the space of functions which are $n$-times continuously differentiable. Functions in $C^n_b(\mathbb{R}^d)$ are $n$-times continuously differentiable and the first $n$ derivatives are bounded.
$C^n_0(\mathbb{R}^d)$ and $C^n_c(\mathbb{R}^d)$ are defined likewise. For any bounded function $f$ we let $\|f\|=\|f\|_\infty$ denote its supremum norm. We write ``$\stackrel{d}{=}$'' to denote equality in distribution of random
variables, ``$\stackrel{d}{\to}$'' to denote convergence in
distribution of random variables,
{\it i.i.d.} for ``independent and identically distributed'', and $\log^+ (x) = \log (\max\{x,1\})$ for $x\in \mathbb{R}$. Throughout, the
characteristic function of a random variable $X$ is denoted by $\phi_X(u) = E e^{iuX}$, $u\in
\mathbb{R}$, and the Fourier transform of a finite measure $\mu$ on
$(\mathbb{R},\mathcal{B}_1)$ by $\widehat{\mu}(u) = \int_{\mathbb{R}} e^{iux} \,
\mu(dx)$. Here, $\mathcal{B}_1$ denotes the Borel-$\sigma$-algebra in
$\mathbb{R}$.
\section{Feller property and the infinitesimal generator of the GOU process}\setcounter{equation}{0} \label{sec-markov}
Let $(X_t)_{t\geq 0}$ be a time homogeneous Markov process on $\mathbb{R}^d$ with semigroup $T_t$, i.e. $$T_tf(x)=\int_{\mathbb{R}^d} f(y)\mu_t(x,dy)=E^x[f(X_t)]$$
where $\mu_t(x,dy)=P(X_t\in dy|X_0=x)$ are the transition probabilities of $X$ and $f\in C_0(\mathbb{R}^d)$. Then $X$ is a {\it Feller process} in $\mathbb{R}^d$ if its semigroup fulfills the {\it Feller properties} \begin{eqnarray*}
&\mbox{(F1)}& \quad T_t C_0(\mathbb{R}^d) \subset C_0(\mathbb{R}^d)\\ &\mbox{(F2)}& \quad T_tf \to f \mbox{ as } t\to 0 \quad \forall f\in C_0(\mathbb{R}^d), \end{eqnarray*}
where the convergence under (F2) is meant to hold in the Banach space $(C_0(\mathbb{R}^d), \|\cdot\|_\infty)$.
The {\it infinitesimal generator} $A^X$ of a Feller process $X$ is defined by $$A^Xf=\lim_{t\to 0} \frac{T_tf-f}{t}$$ for all functions $f$ in the domain of $A^X$, i.e. all $f$ in $$D(A^X)=\left\{f\in C_0(\mathbb{R}^d), \lim_{t\to 0} \frac{T_tf-f}{t}
\mbox{ exists in } \|\cdot \|_\infty \right\}.$$ A subspace $D$ of $D(A^X)$ is said to be a {\it core} for the generator $A^X$, if the closure of the restriction of $A^X$ to $D$ is equal to $A^X$.
Every L\'evy process $X$ is a Feller process. If the L\'evy process $X$ is real-valued its generator $A^X$ is given by (e.g. \cite[Thm. 31.5]{sato}) \begin{equation} \label{generatorlevy}
A^X f(x)= \frac{1}{2}\sigma_X^2 f''(x) + \gamma_X f'(x) +
\int_{\mathbb{R}} (f(x+y)-f(x)- yf'(x)\mathds{1}_{|y|\leq 1} )\nu_X(dy) \end{equation} and it holds $C_0^2(\mathbb{R})\subset D(A^X)$.
The generator of the OU process is well known in the literature, unlike the generator of the GOU process, which is presented in the next theorem. For L\'evy processes with finite second moment this generator is also given in \cite[Thm. 4.6.1]{kolokoltsov} and the formula for the generator can also be found in
\cite[Exercise V.7]{protter} (containing a typo). The fact that GOU processes are Feller processes and the
determination of the cores seems to be new.\\ Note that the equation $dV_t^x = x + \int_{(0,t]} V_{s-}^x \, dU_s + dL_t$ can be written as \begin{equation} \label{eq-Schilling} dV_t^x = x + \int_{(0,t]} g(V_{s-}^x) \, d(U_s,L_s)^T \end{equation} with $g(u) = (u,1) \in \mathbb{R}^{1\times 2}$. Solutions of \eqref{eq-Schilling} with {\it bounded} and locally Lipschitz $g$ are well known to constitute Feller process (e.g. \cite[Cor. 3.3]{schillingschnurr}), but the function $u \mapsto (u,1)$ is not bounded so that this theory cannot be applied. Further, in \cite[Rem. 3.4]{schillingschnurr} an example is given when $g$ is not bounded and the corresponding solution fails to be a Feller process.
\begin{theorem} \label{thm-generator-GOU}
Let $(Z_t)_{t\geq 0}=((U_t,L_t)^T)_{t\geq 0}$ be a bivariate L\'evy process with characteristic triplet $(\gamma_Z, A_Z, \nu_Z)$ where $\gamma_Z=(\gamma_U,\gamma_L)^T$, $A_Z=\begin{pmatrix} \sigma_U^2 & \sigma_{U,L} \\ \sigma_{U,L} & \sigma_L^2 \end{pmatrix}$ and $\nu_Z((dz_1,dz_2)^T)$ such that $\nu_Z((-1,dz_2)^T)=0$. Then the process $(V_t^x)_{t\geq 0}$ defined by \begin{equation} \label{GOUSDE-matrix}
V^x_t=x+ \int_{(0,t]} V_{s-}^x\, dU_s +L_t = x+\int_{(0,t]} g(V_{s-}^x)dZ_s,
\quad t\geq 0, \end{equation} for $g(u)=(u,1)$ is a Feller process whose generator $A^V$ has a domain containing
$$S(\mathbb{R}):=\left\{f\in C_0^2(\mathbb{R}) : \lim_{|x|\to\infty} \left(|x f'(x)|+ |x^2 f''(x)|\right)=0 \right\}.$$ In particular $C_c^\infty (\mathbb{R}) \subset C_c^2(\mathbb{R})\subset D(A^V)$. For any $f\in S(\mathbb{R})$ the generator can be written as \begin{eqnarray}
A^Vf(x)&=& f'(x) g(x) \gamma_Z + \frac{1}{2} f''(x)\left(g(x) A_Z g(x)^T\right) \label{generatorGOU-UL-matrix} \nonumber \\
&& + \int_{\mathbb{R}^2} \left( f(x+g(x)z)-f(x)-f'(x)g(x)z\mathds{1}_{|z|\leq 1}\right) \nu_Z(dz)\nonumber\\ &=& f'(x)(x\gamma_U+\gamma_L) + \frac{1}{2} f''(x)(x^2\sigma_U^2 + 2x\sigma_{U,L}+\sigma_L^2) \label{generatorGOU-UL}\\ && + \int_{\mathbb{R}^2} (f(x+xz_1+z_2) - f(x) - f'(x)(xz_1
+z_2)\mathds{1}_{|z|\leq 1} ) \nu_{U,L}(dz_1,dz_2). \nonumber \end{eqnarray} The spaces $S(\mathbb{R})$, $C_c^2(\mathbb{R})$ and $C_c^\infty(\mathbb{R})$ are cores for $A^V$. \end{theorem}
\begin{proof} $\,$\\ (i) Let us first establish the Feller property. It is well known that $V_t^x$ is a time homogenous Markov process (e.g. \cite[Lem. 3.3]{behmelindnermaller11}). By \eqref{GOUdef2}, $V_t^x$ is given by $V_t^x = \mathcal{E}(U)_t \left( x + \int_{(0,t]} \mathcal{E}(U)_{s-}^{-1} \, d\eta_s\right)$. Since $\mathcal{E}(U)_t \neq 0$ as a consequence of
$\nu_U(\{-1\}) = 0$, we have $\lim_{|x|\to \infty} |V_t^x| = \infty$
and hence $\lim_{|x|\to \infty} f(V_t^x) = 0$ for any $f\in C_0(\mathbb{R})$. By Lebesgue's dominated convergence theorem, this implies
$T_t f(x) = E [f(V_t^x)] \to 0$ as $|x|\to\infty$. The fact that for bounded and continuous $f$ the mapping $x\mapsto E[f(V_t^x)]$ is continuous is obvious using dominated convergence. Thus $T_t$ maps $C_0(\mathbb{R})$ into $C_0(\mathbb{R})$ and (F1) is shown. (F2) follows from (F1) and \cite[Thm. 3.15]{liggett}, observing that for each $x\in \mathbb{R}$, $V^x$ satisfies $P(V_0^x=x)=1$ and $(V_t^x)_{t\geq 0}$ is adapted to the smallest filtration satisfying the usual hypotheses induced by $((U_t,L_t)^T)_{t\geq 0}$, which is right continuous.
(ii) Before we prove \eqref{generatorGOU-UL}, we give a bound for the integrand appearing in \eqref{generatorGOU-UL} which will be used throughout. Let $f \in S(\mathbb{R})$ and set \begin{eqnarray}
K_1(f) & := & \sup_{y\in \mathbb{R}} \left\{ |f'(y)| (1+|y|) + |f''(y)|
(1+|y|)^2\right\} < \infty \quad \mbox{and} \label{eq-K1}\\ K_2 & := & \frac12 \, \sup_{y\in \mathbb{R}} \; \; \sup_{\zeta \in \mathbb{R}:
|\zeta|\leq (1+|y|)/2} \frac{ (1+|y|)^2}{(1+|y+\zeta|)^2}< \infty. \nonumber \end{eqnarray} We claim that \begin{eqnarray} \label{eq-majorant1}
\lefteqn{\left| f(x + xz_1 + z_2) - f(x) - f'(x) (x z_1 + z_2)
\mathds{1}_{|z|\leq 1} \right|} \\
&\leq & K_1(f) K_2 |z|^2 \mathds{1}_{|z|\leq 1/2} + K_1(f) |z|
\mathds{1}_{1/2 < |z| \leq 1} + 2 \|f\| \mathds{1}_{|z| > 1/2} \; \; \forall \; z = (z_1,z_2)^T \in \mathbb{R}^2, \; x \in \mathbb{R}. \nonumber \end{eqnarray}
Indeed, this is obvious for $|z|> 1/2$ since $|xz_1+z_2| \leq
\sqrt{1+x^2} |z| \leq (1+|x|) |z|$. For $|z|\leq 1/2$, by Taylor's theorem there is $\zeta\in \mathbb{R}$ with $0\leq |\zeta| \leq |xz_1+z_2|
\leq (1+|x|) |z| \leq (1+|x|)/2$ such that \begin{eqnarray*}
\lefteqn{\left| f(x+xz_1 + z_2) - f(x) - f'(x) (xz_1 + z_2)\right|} \\
& = & 2^{-1} |f''(x+\zeta)| (xz_1+z_2)^2 \\
& \leq & 2^{-1} \left|f''(x+\zeta) (1+|x+\zeta|)^2\right| \,
\frac{(1+|x|)^2}{(1+|x+\zeta|)^2} |z|^2 \\
& \leq & K_1(f) K_2 |z|^2, \end{eqnarray*}
which shows \eqref{eq-majorant1} also for $|z|\leq 1/2$. In particular, the right hand side of \eqref{generatorGOU-UL} is in $C_0(\mathbb{R})$ for $f\in S(\mathbb{R})$ by Lebesgue's dominated convergence theorem.
(iii) Let us show \eqref{generatorGOU-UL}. Let $f\in S(\mathbb{R})$, then by It\^o's formula (e.g. \cite[Thm. II.32]{protter}) we have \begin{eqnarray*}
\lefteqn{f(V^x_t)-f(V^x_0)}\\ &=& \int_{(0,t]} f'(V^x_{s-}) dV^x_s + \frac{1}{2}\int_{(0,t]} f''(V^x_{s-}) d[V^x,V^x]^c_s + \sum_{0< s\leq t}\left(f(V^x_s)-f(V^x_{s-})-f'(V^x_s)\Delta V^x_s\right) \end{eqnarray*} and hence \begin{eqnarray}
T_tf(x)-f(x)&=& E \left[f(V^x_t)-f(V^x_0)\right] \nonumber\\ &=& E \left[\int_{(0,t]} f'(V^x_{s-}) dV^x_s + \sum_{0< s\leq t}\left(f(V^x_s)-f(V^x_{s-})-f'(V^x_{s-})\Delta V^x_s\right)\right] \nonumber \\ && + \frac{1}{2}E \left[\int_{(0,t]} f''(V^x_{s-}) d[V^x,V^x]^c_s \right]\nonumber\\ &=:& \mbox{I}_t + \mbox{II}_t, \quad \mbox{say.} \label{eq-ito-numbers} \end{eqnarray} Observe that $dV_s^x = g(V_{s-}^x) d Z_s$ and $\Delta V^x_s= g(V^x_{s-})\Delta Z_s$. Since $Z$ is a L\'evy process, by the L\'evy-It\^o decomposition (e.g. \cite[Thm. 2.4.16]{applebaum}) we can write $Z_t = \gamma_Z t + M_t + \sum_{0<s\leq t} \Delta Z_s
\mathds{1}_{|\Delta Z_s| > 1}$, where $(M_t)_{t\geq 0}$ is a square integrable martingale with expectation 0. Hence we obtain for the first term \begin{eqnarray*}
\mbox{I}_t &=& E \left[\int_{(0,t]} f'(V^x_{s-})g(V^x_{s-}) \gamma_Z ds\right] + E \left[\int_{(0,t]} f'(V^x_{s-}) g(V^x_{s-}) dM_s\right]\\
&& + E \left[\sum_{0<s\leq t} f'(V^x_{s-}) g(V^x_{s-}) \Delta Z_s \mathds{1}_{|\Delta Z_s| > 1}+ \sum_{0< s\leq t}\left(f(V^x_s)-f(V^x_{s-})-f'(V^x_{s-})g(V^x_{s-})\Delta Z_s\right) \right].\\ \end{eqnarray*} Since $M$ is a square integrable martingale with expectation 0 and since $s\mapsto f'(V_{s-}^x) g(V_{s-}^x)$ is bounded because of $f\in S(\mathbb{R})$, the process $t\mapsto \int_{(0,t]} f'(V^x_{s-}) g(V^x_{s-}) dM_s$ is a martingale with expectation 0 (e.g. \cite[Prop. 2.24]{medvegyev}). Hence we conclude \begin{eqnarray*}
\mbox{I}_t &=& \int_{(0,t]} E \left[ f'(V^x_{s-})g(V^x_{s-})\right] \gamma_Z \, ds\\ && + E \left[ \sum_{0< s\leq t} \left( f(V^x_{s-} + g(V_{s-}^x) \Delta Z_s)
-f(V^x_{s-})-f'(V^x_{s-})g(V^x_{s-})\Delta Z_s \mathds{1}_{|\Delta Z_s| \leq 1}\right) \right]\\ &=& \int_{(0,t]} E \left[ f'(V^x_{s-})g(V^x_{s-})\right] \gamma_Z ds\\
&& + E \left[\int_{(0,t]} \int_{\mathbb{R}^2} \left( f\left(V^x_{s-}+ g(V^x_{s-})z
\right)-f(V^x_{s-})-f'(V^x_{s-})g(V^x_{s-})z\mathds{1}_{| z|\leq 1} \right)\nu_Z(dz) ds\right], \end{eqnarray*} where we used the compensation formula (e.g. \cite[Thm. 4.4]{kyprianou}), which may be applied since
$E\int_{(0,t]}\int_{\mathbb{R}^2} \left|f\left(V^x_{s-}+ g(V^x_{s-})z\right)
-f(V^x_{s-})-f'(V^x_{s-})g(V^x_{s-})z\mathds{1}_{| z|\leq 1}
\right|\nu_Z(dz) ds$ is finite by \eqref{eq-majorant1}. Using the continuity of $s\mapsto V_s^x$ at $s=0$ and again the bound from \eqref{eq-majorant1}, it follows from Lebesgue's dominated convergence theorem that $$\lim_{t\to 0} t^{-1} {\rm{I}}_t = f'(x) g(x) \gamma_Z + \int_{\mathbb{R}^2} \left(
f(x+g(x) z) - f(x) - f'(x) g(x) z \mathds{1}_{|z|\leq 1} \right)\, \nu_Z(dz).$$
For the second term in Equation \eqref{eq-ito-numbers} observe that by \cite[Eq. (4)]{karandikar} \begin{eqnarray*}
[V^x,V^x]^c_s&=& \left[ x + \int_{(0,\cdot]} g(V^x_{u-})dZ_u ,x + \int_{(0,\cdot]} g(V^x_{u-})dZ_u\right]_s^c\\ &=& \int_{(0,s]} g(V^x_{u-}) \,d[Z, Z^T]_u^c \,g(V^x_{u-})^T , \end{eqnarray*} and since $[Z,Z^T]^c_u = A_Z u$ it follows $$\mbox{II}_t=\frac{1}{2} E \left[\int_{(0,t]} f''(V^x_{s-}) g(V^x_{u-})\, A_Z \,g(V^x_{u-})^T \, du\right].$$ Together with the obtained formula for $\mbox{I}_t$, and inserting the definition of $g$ and $Z$, this shows that $\lim_{t\to 0} t^{-1} ({\rm{I}}_t + {\rm{II}}_t)$ is given by the right hand side of \eqref{generatorGOU-UL}. Since $V$ is a Feller process, and since the right hand side of \eqref{generatorGOU-UL} is in $C_0(\mathbb{R})$ for $f\in S(\mathbb{R})$ by \eqref{eq-majorant1}, this pointwise limit is actually uniform in $x$ (e.g. \cite[Lem. 31.7]{sato}), so that $S(\mathbb{R})$ is contained in the domain of the generator of $V$ and that $A^V f$ is given by \eqref{generatorGOU-UL} for all $f\in S(\mathbb{R})$.
(iv) We now show that $S(\mathbb{R})$ is a core for $A^V$ under the extra assumption that $E U_1^2 < \infty$ and $E L_1^2 < \infty$. Denote $A_t = \mathcal{E}(U)_t$ and $B_t = \mathcal{E}(U)_t \int_{(0,t]} \mathcal{E}(U)_s^{-1} \, d\eta_s$. Then $B_t \stackrel{d}{=} \int_{(0,t]} \mathcal{E}(U)_{s-} \, dL_s$ by \cite[Lem. 3.1]{behmelindnermaller11}. Then $E A_t^2 < \infty$ and $E B_t^2 < \infty$ as a consequence of Proposition 3.1 and Lemma 6.1 in \cite{behme2011} together with \cite[Thm. 25.18]{sato}. We conclude that $\diffp[2]{}{x} T_t f(x)$ exists for $f\in S(\mathbb{R})$ and that \begin{eqnarray*} \diffp{}{x} T_tf(x) & = & \diffp{}{x} E[f(A_t x+B_t)] = E[A_t f'(A_t x+B_t)] \quad \mbox{and}\\ \diffp[2]{}{x} T_t f(x) & = & \diffp[2]{}{x} E [f(A_tx + B_t)] = E [A_t^2 f''(A_tx + B_t)]. \end{eqnarray*} Since $E A_t^2 < \infty$, the mapping $x\mapsto \diffp[2]{}{x} T_t f(x)$ is obviously continuous, so that $T_t S(\mathbb{R}) \subset C^2(\mathbb{R})
\cap C_0(\mathbb{R})$. Using that $E |B_t| < \infty$ and $\lim_{|y|\to
\infty} |y f'(y)| = 0$ for $f\in S(\mathbb{R})$, we further see by dominated convergence that \begin{eqnarray*}
\left| x \diffp{}{x} T_t f(x)\right|
&\leq& E\left[\left|A_t x f'(A_t x + B_t)\right|\right] \\
&\leq& E\left[\left| (A_t x + B_t) f'(A_t x + B_t)\right|\right] + E\left[\left|B_t f'(A_t x + B_t)\right|\right]\\
&\to& 0,\quad \mbox{as }|x|\to\infty . \end{eqnarray*}
In the same way one can check that $\left|x^2 \diffp[2]{}{x}
T_tf(x)\right|\to 0$ as $|x|\to \infty$ such that $T_tS(\mathbb{R}) \subset S(\mathbb{R})$. By \cite[Prop. 1.3.3]{ethierkurtz} we thus obtain that $S(\mathbb{R})$ is a core for $A^V$, provided that $E U_t^2 < \infty$ and $E L_t^2 < \infty$.
(v) Now we drop the assumption that $E U_1^2 + E L_1^2 < \infty$ and show that $S(\mathbb{R})$ is a core for $A^V$. Similarly to the proof of Theorem 3.1 in Sato and Yamazato \cite{satoyamazato}, for $f \in S(\mathbb{R})$ denote the right hand side of \eqref{generatorGOU-UL} by $G f(x)$ and define \begin{eqnarray*} G_0 f(x) & := & f'(x)(x\gamma_U+\gamma_L) + \frac{1}{2} f''(x)(x^2\sigma_U^2 + 2x\sigma_{U,L}+\sigma_L^2) \\
&& + \int_{\{ z \in \mathbb{R}^2: |z|\leq 1\}} \left(f(x+xz_1+z_2) - f(x) -
f'(x)(xz_1 +z_2)\mathds{1}_{|z|\leq 1} \right) \nu_{U,L}(dz_1,dz_2). \end{eqnarray*} For $f\in C_0(\mathbb{R})$, denote further
$$W f(x) := \int_{\{ z \in \mathbb{R}^2 : |z|> 1\}} \left( f(x+ x z_1 + z_2) - f(x) \right) \, \nu_{U,L}(dz_1,dz_2).$$ Then $W: C_0(\mathbb{R}) \to C_0(\mathbb{R})$ is a bounded linear operator, and from (iii) we know that $A^V f = G f = G_0 f + W f$ for $f\in S(\mathbb{R})$. Consider the process $V_{(0)}$ defined by ${V}_{(0),t}^x = x + \int_{(0,t]} {V}_{(0),s-}^x \, d\tilde{U_s} + \tilde{L}_t$, where
$(\tilde{U},\tilde{L})^T$ is a L\'evy process with characteristic triplet $(\gamma_Z, A_Z, \mathds{1}_{|z|\leq 1} \nu_Z(dz))$. Again by (iii), $G_0 f = A^{V_{(0)}} f$ for $f\in S(\mathbb{R})$, and from (iv) we know that $S(\mathbb{R})$ is a core for $A^{V_{(0)}}$, so that the closure $\overline{G_0}$ of ${G_0}$ is $A^{{V_{(0)}}}$, in particular $D (\overline{{G_0}}) = D(A^{{V_{(0)}}})$. Since $G f = {G_0} f + W f$ for $f\in S(\mathbb{R})$ and $W$ is bounded, it follows that the closure $\overline{G}$ of $G$ satisfies $\overline{G} = \overline{{G_0}} + W$, in particular $D(A^{{V_{(0)}}}) = D (\overline{{G_0}}) = D(\overline{G})$. Since $A^V$ is a closed operator, we further know that $D(\overline{G}) \subset D(A^V)$ and that $A^V$ is a closed extension of $\overline{G}$. From the Hille-Yosida theorem (e.g. \cite[Thm. 1.2.6]{ethierkurtz}) it follows that for every $\lambda > 0$, $\lambda \,\mbox{Id} - \overline{{G_0}}: D(A^{{V_{(0)}}}) = D(\overline{{G_0}}) \to C_0(\mathbb{R})$, $f\mapsto \lambda f - \overline{{G_0}} f$
is a bijection with bounded inverse (the resolvent) satisfying $\| (\lambda\,
\mbox{Id} - \overline{{G_0}})^{-1}\| \leq \lambda^{-1}$. For $\lambda_0>
\|W\|$, it then follows from a perturbation result for closed linear operators (e.g. \cite[Thm. IV.1.16]{kato}), that also $\lambda_0\, \mbox{Id} - \overline{G} = \lambda_0\, \mbox{Id} - \overline{{G_0}} - W: D(\overline{G}) = D(\overline{{G_0}}) \to C_0(\mathbb{R})$ is a bijection with bounded inverse. Since $A^V$ is a closed extension of $\overline{G}$ and also $\lambda_0 \, \mbox{Id} - A^V: D(A^V) \to C_0(\mathbb{R})$ is a bijection (e.g. \cite[Prop. 1.2.1]{ethierkurtz}), we must have $D(\overline{G}) = D(A^V)$ and hence $\overline{G} = A^V$. This shows that $S(\mathbb{R})$ is a core for $A^V$.
(vi) Finally, we show that $C_c^2(\mathbb{R})$ and $C_c^\infty(\mathbb{R})$ are cores for $A^V$. Let $h$ be a function in $C^\infty_c$ with $h(x)=1$
if $|x|\leq 1$ and $h(x)=0$ if $|x|\geq 2$. Define $h_n(x)=h(x/n)$ and for any $f\in S(\mathbb{R})$ set $f_n(x)=f(x)h_n(x)$. Then $f_n\in C_c^2(\mathbb{R})$ and we obtain that $f_n\to f$, $f_n'\to f'$, $f_n''\to f''$, $xf_n'(x)\to xf'(x)$, $xf_n''(x)\to xf''(x)$ and $x^2f_n''(x)\to x^2f''(x)$ uniformly in $x$ as $n\to \infty$. In particular, with $K_1(\cdot)$ as defined in \eqref{eq-K1}, we see that $K_1(f_n)$ is bounded in $n$ and hence we conclude with \eqref{eq-majorant1} that $A^V f_n\to A^V f$ uniformly as $n\to \infty$. This shows that $C_c^2(\mathbb{R})$ is a core for $A^V$. Finally, for $f\in C_c^2(\mathbb{R})$ there are functions $g_n\in C_c^\infty(\mathbb{R})$ with uniformly bounded supports such that $g_n\to f$, $g_n'\to f'$ and $g_n'' \to f''$ uniformly as $n\to\infty$, hence also $x g_n'(x) \to x f'(x)$, $x g_n''(x) \to x f''(x)$ and $x^2 g_n''(x) \to x^2 f''(x)$ uniformly in $x$ as $n\to \infty$. Again, this gives $A^V g_n \to A^V f$ uniformly as $n\to\infty$ so that $C_c^\infty$ is a core for $A^V$. \end{proof} The following corollary is immediate from Theorem \ref{thm-generator-GOU}.
\begin{corollary} In the setting of Theorem \ref{thm-generator-GOU}, if $U$ and $L$ are additionally independent, Equation \eqref{generatorGOU-UL} simplifies to \begin{eqnarray}
A^Vf(x)&=& f'(x)(x\gamma_U+\gamma_L) + \frac{1}{2} f''(x)(x^2\sigma_U^2 +\sigma_L^2)\nonumber \\
&& + \int_{\mathbb{R}} (f(x+xy) - f(x) - f'(x)xy \mathds{1}_{|y|\leq 1}) \nu_U(dy) \nonumber\\
&& + \int_\mathbb{R} (f(x+y) - f(x) - f'(x)y\mathds{1}_{|y|\leq 1}) \nu_{L}(dy)\nonumber \\ &=& A^Lf(x) + f'(x) x\gamma_U + \frac{1}{2} f''(x) x^2\sigma_U^2 \label{generatorGOU-UL-ind}\\
&& + \int_{\mathbb{R}} (f(x+xy) - f(x) - f'(x)xy \mathds{1}_{|y|\leq 1}) \nu_U(dy) .\nonumber \end{eqnarray} \end{corollary}
\begin{corollary}\label{coro-generatorGOUxieta} Let $(\xi_t)_{t\geq 0}$ and $(\eta_t)_{t\geq 0}$ be two independent L\'evy processes and let $(V^x_t)_{t\geq 0}$ be the generalized Ornstein-Uhlenbeck process driven by $(\xi, \eta)^T$ with starting point $x$ as defined in \eqref{GOUdef}. Then $(V^x_t)_{t\geq 0}$ is a Feller process whose generator has a domain containing $S(\mathbb{R})$, and $S(\mathbb{R})$, $C_c^2(\mathbb{R})$ and $C_c^\infty(\mathbb{R})$ are cores for $A^V$. For any $f\in S(\mathbb{R})$ the generator can be written as \begin{eqnarray}
A^Vf(x)&=& A^{\eta}f(x) - f'(x) x\gamma_\xi + \frac{1}{2} (f''(x) x^2 + f'(x)x) \sigma_\xi^2 \label{generatorGOU-xieta-ind}\\
&& + \int_{\mathbb{R}} (f(xe^{-y}) - f(x) + f'(x)xy \mathds{1}_{|y|\leq 1}) \nu_\xi (dy).\nonumber \end{eqnarray} If $f\in S(\mathbb{R})$ and $f(0) = 0$, define $\tilde{f}(x)=f(e^x)$ and $\tilde{\tilde{f}}(x)=f(-e^x)$. Then $\tilde{f}, \tilde{\tilde{f}} \in C_0^2(\mathbb{R}) \subset D (A^{-\xi})$, and for such $f$ Equation \eqref{generatorGOU-xieta-ind} can be rewritten as \begin{equation} \label{generatorGOU-xieta-ind-short} A^V f(x) = A^{\eta}f(x)+ A^{-\xi} \tilde{f}(\log x)\mathds{1}_{x>0}
+ A^{-\xi} \tilde{\tilde{f}}(\log |x|)\mathds{1}_{x<0}. \end{equation} \end{corollary}
\begin{proof} Since $(V^x_t)_{t\geq 0}$ fulfills \eqref{GOUSDE-matrix} for $(U,L)^T$ defined in \eqref{eq-def-UL} the Feller property as well as the statements on the domain and cores of the generator follow directly from Theorem \ref{thm-generator-GOU}. Also observe from \eqref{eq-def-UL} that in the independent case we have $\eta_t=L_t$ and thus $A^{\eta}=A^L$ whereas the relation between $\xi$ and $U$ yields $\nu_U((-\infty, -1])=0$. In \cite[Lem. 3.4]{behmelindnermaller11} we have computed the characteristic triplet of $\xi$ in terms of the characteristic triplet of $U$ (the $\hat{U}$ used there is equal to $\xi$ whenever $\nu_U((-\infty,-1])=0$). Using these relations one obtains \eqref{generatorGOU-xieta-ind} from \eqref{generatorGOU-UL-ind} by standard computations. \\ Finally the fact that $\tilde{f}, \tilde{\tilde{f}} \in C_0^2(\mathbb{R})$ if $f\in S(\mathbb{R})$ such that $f(0) = 0$ and the validity of \eqref{generatorGOU-xieta-ind-short} may be checked directly from \eqref{generatorlevy} using the definitions of $\tilde{f}$ and $\tilde{\tilde{f}}$. \end{proof}
\begin{remark} In \cite{savovetal} the exponential functional for independent processes $\xi$ and $\eta$ is studied. Under the condition of finite first moments of $\xi$ and $\eta$, the authors prove that for suitable functions $f$ with support on the positive half line the generator of the GOU process can be written as \begin{equation} A^{V}f(x) = A^{-\xi}\tilde{f}(\log x) + A^{\eta} f(x) \label{eq-savov-et-al} \end{equation} where $\tilde{f}(x)=f(e^x)$ and $A^{-\xi}$ and $A^{\eta}$ are the generators of $-\xi$ and $\eta$ respectively. Remark that the $\xi$ used by the authors corresponds to $-\xi$ in our notation. The formula \eqref{eq-savov-et-al} for positive $x$ is also obtained in \cite[Proof of Thm. 1]{carmona-unpublished}. \end{remark}
\section{Relations between the exponential functional and the driving L\'evy processes}\setcounter{equation}{0} \label{sec-relations}
It is basic knowledge in the theory of Markov processes (see e.g. \cite[Prop. 4.9.2]{ethierkurtz}), that if $\mu$ is an invariant measure for the Markov process $X$ with strongly continuous contraction semigroup $T_t$ and generator $A$, i.e. if $\mu(B)=\int \mu_t(x,B) \mu(dx)$ for all Borel sets $B$, then \begin{equation} \label{generatorstat}
\int_{\mathbb{R}^d} Af(y)\mu(dy)= 0 \quad \forall f\in D(A). \end{equation} Conversely, if \eqref{generatorstat} holds, $\mu$ is an invariant measure under some additional conditions. In the special case of Feller processes Equation \eqref{generatorstat} holds if and only if $\mu$ is an invariant measure of the corresponding process $X$ \cite[Thm. 3.37]{liggett}.
In \cite{carmonapetityor} and \cite{carmonapetityorLP} the authors make use of Equation \eqref{generatorstat} to obtain the density of a specific stationary generalized Ornstein-Uhlenbeck process. More precisely they obtain the density of the exponential functional in the special case that $\xi$ is a Brownian motion with drift and $\eta$ is deterministic.
Let $(V_t)_{t\geq 0}$ be a GOU process as defined in \eqref{GOUdef} or even a process as defined in \eqref{GOUdef2}, fulfilling the SDE \eqref{SDEGOU}, with $\nu_U( \{-1\}) = 0$. Assume that $U$ and $L$ are independent, i.e. the generator of $(V_t)_{t\geq 0}$ is given by \eqref{generatorGOU-UL-ind} for $f\in S(\mathbb{R})$. Let $\mu=\mathcal{L}(V_\infty)$ be the invariant measure of $(V_t)_{t\geq 0}$, assuming its existence. Then by \eqref{generatorstat} we obtain for any $f \in S(\mathbb{R}) \subset D(A^V)$ \begin{eqnarray} \label{generatorinteq}
0 &=& \int_{\mathbb{R}} A^V f(x) \mu (dx) \nonumber\\ &=& \int_{\mathbb{R}}A^{L} f(x) \mu(dx)+ \gamma_U \int_{\mathbb{R}} f'(x) x \,\mu (dx) + \frac{\sigma_U^2}{2} \int_{\mathbb{R}} f''(x) x^2 \,\mu(dx) \label{eq-intgenerator} \\
&& + \int_{\mathbb{R}} \int_{\mathbb{R}\backslash\{-1\}} (f(x+xy) - f(x) - f'(x)xy \mathds{1}_{|y|\leq 1}) \nu_U(dy) \mu(dx).\nonumber \end{eqnarray}
This and the previous results allow to establish relationships between the characteristic functions of $V_\infty$, $U$ and $L$, as done in the following. Recall that $\psi_X(u)=\log E[e^{iuX_1}]$ is the characteristic exponent of the L\'evy process $X$.
\begin{theorem}\label{thm-generatorlimit} Let $(U_t)_{t\geq 0}$ and $(L_t)_{t\geq 0}$ be two independent L\'evy processes with $\nu_U(\{-1\}) = 0$ and such that $V_\infty=\int_0^\infty \mathcal{E}(U)_{s-}dL_s$ converges to a finite random variable. Then $\mu=\mathcal{L}(V_\infty)$ is the invariant law of the process $(V_t)_{t\geq 0}$ defined by \eqref{GOUdef2}.\\
Let $h\in C^\infty_c(\mathbb{R})$ be such that $h(x)=1$ for $|x|\leq 1$ and
$h(x)=0$ for $|x|\geq 2$ and set $h_n(x):=h(\frac{x}{n})$ and $f(x)=e^{iux}$, $f_n(x)=f(x)h_n(x)$ for $u\in \mathbb{R}$. Then \begin{eqnarray} \psi_L(u)\phi_{V_{\infty}}(u)&=&\lim_{n\to \infty} \left( -\gamma_U \int_{\mathbb{R}} xf_n'(x) \,\mu (dx) - \frac{\sigma_U^2}{2} \int_{\mathbb{R}} x^2f_n''(x) \,\mu(dx) \right. \label{eq-relation1}\\ && \left. - \int_{\mathbb{R}} \int_{\mathbb{R}} (f_n(x+xy) - f_n(x) - xyf_n'(x)
\mathds{1}_{|y|\leq 1}) \nu_U(dy) \mu(dx)\right). \nonumber \end{eqnarray} If additionally $E [V_\infty^2] = \int_{\mathbb{R}} x^2 \, \mu(dx) < \infty$, then \begin{eqnarray} \psi_L(u) \phi_{V_\infty} (u) &=& -iu\gamma_U E\left[V_\infty e^{iuV_\infty} \right] + \frac{\sigma_U^2 u^2}{2} E\left[V_\infty^2 e^{iuV_\infty} \right] \label{eq-relation2}\\
&&- \int_{\mathbb{R}} \left(\phi_{V_{\infty}}(u(1+y)) - \phi_{V_{\infty}}(u) - iuE\left[V_\infty e^{iuV_\infty} \right] y \mathds{1}_{|y|\leq 1}\right) \nu_U(dy) \nonumber \\
&=& - u \gamma_U \phi_{V_\infty}'(u) - \frac{\sigma_U^2 u^2}{2} \phi_{V_\infty}''(u) \label{eq-relation4}\\
&& - \int_{\mathbb{R}} \left(\phi_{V_{\infty}}(u(1+y)) - \phi_{V_{\infty}}(u) - u \phi_{V_\infty}'(u) y \mathds{1}_{|y|\leq 1}\right) \nu_U(dy) \nonumber\\ &=& - E \left[ e^{i u V_\infty} \psi_U ( u V_\infty) \right] \label{eq-relation3} \end{eqnarray} \end{theorem} Equation \eqref{eq-relation3} can also be written in the compact form $$E \left[ (\psi_U(u V_\infty)+ \psi_L(u) ) e^{iu V_\infty} \right] = 0 \quad \forall\; u \in \mathbb{R}.$$
For the proof we need the following lemma. We use the notation $S(\mathbb{R};\mathbb{C})$ to denote the class of complex valued functions $f:\mathbb{R}\to\mathbb{C}$ such that $\Re(f) \in S(\mathbb{R})$ and $\Im(f) \in S(\mathbb{R})$. Spaces like $C_c^\infty(\mathbb{R};\mathbb{C})$ are defined similarly. For a generator $A$ we also write $$D(A;\mathbb{C}) := \{ f\in C_0(\mathbb{R};\mathbb{C}) : \Re(f), \Im(f) \in D(A) \}.$$ It is clear that \eqref{generatorGOU-UL-ind} and hence \eqref{eq-intgenerator} remain valid for complex valued functions $f\in S(\mathbb{R};\mathbb{C})$.
\begin{lemma} \label{lemma-generatorlimit} Let $(L_t)_{t\geq 0}$ be a L\'evy process in $\mathbb{R}$ with generator $A^L$, $\mu$ a fixed finite measure on $(\mathbb{R},\mathcal{B}_1)$ and define $h,h_n,f$ and $f_n$ as in Theorem \ref{thm-generatorlimit}. Then $f_n\in C_c^\infty(\mathbb{R};\mathbb{C})\subset D(A^L; \mathbb{C})$ and $$\lim_{n\to\infty} \int_\mathbb{R} A^Lf_n(x) \mu(dx) = \psi_L(u) \int_\mathbb{R} e^{iux} \mu(dx) = \psi_L(u) \widehat{\mu} (u).$$ \end{lemma} \begin{proof} It is clear that $f_n\in C_c^\infty(\mathbb{R};\mathbb{C})$. From \eqref{generatorlevy} we obtain \begin{eqnarray*}
\int_\mathbb{R} A^Lf_n(x) \mu(dx) &=& \gamma_L \int_{\mathbb{R}} f_n'(x) \mu (dx) + \frac{\sigma_L^2}{2} \int_{\mathbb{R}} f_n''(x) \,\mu(dx) \\ && + \int_{\mathbb{R}} \int_{\mathbb{R}}(f_n(x+y) - f_n(x) - f_n'(x)
y\mathds{1}_{|y|\leq 1}) \nu_L(dy)\,\mu(dx).
\end{eqnarray*} By Taylor's formula, there are $\zeta_1, \zeta_2\in [-|y|,|y|]$ such that \begin{eqnarray*}
\lefteqn{\left| f_n(x+y) - f_n(x) - f_n'(x) y \mathds{1}_{|y|\leq 1}
\right|} \nonumber \\
& \leq & \left| f_n(x+y) - f_n(x) \right| \mathds{1}_{|y| > 1} +
2^{-1} \left[ |(\Re f_n'') (x+\zeta_1)| + |(\Im f_n'') (x+\zeta_2)| \right] y^2 \mathds{1}_{|y|\leq 1} \nonumber \\
& \leq & 2 \| f_n\| \mathds{1}_{|y|> 1} + \|f_n''\| y^2
\mathds{1}_{|y|\leq 1} . \end{eqnarray*} Computing the first two derivatives of $f_n$ one easily sees that they are uniformly bounded in $n$. Since additionally $\lim_{n\to \infty} f_n'(x)=iue^{iux}$ and $\lim_{n\to \infty} f_n''(x)=-u^2e^{iux}$ we obtain via dominated convergence \begin{eqnarray*} \lim_{n\to\infty} \int_{\mathbb{R}} A^L f_n(x) \, \mu(dx) &=& \gamma_L \int_{\mathbb{R}} iue^{iux} \,\mu (dx) - \frac{\sigma_L^2}{2} \int_{\mathbb{R}} u^2 e^{iux}\,\mu(dx) \\ & & + \int_{\mathbb{R}} \int_{\mathbb{R}} \left( e^{iu(x+y)} -e^{iux} -
iue^{iux}y\mathds{1}_{|y|\leq 1} \right) \, \nu_L(dy) \, \mu(dx) ,\\ \end{eqnarray*} which gives the claim.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm-generatorlimit}] Since $\int_0^t \mathcal{E}(U)_{s-} \, dL_s$ converges almost surely to the finite random variable $V_\infty$ as $t\to\infty$, $\mu = \mathcal{L}(V_\infty)$ is the unique stationary marginal distribution and hence invariant law of $V$ by \cite[Thms. 2.1 and 3.6]{behmelindnermaller11}. Equation \eqref{eq-relation1} then follows directly from \eqref{generatorstat}, \eqref{generatorinteq} and Lemma \ref{lemma-generatorlimit}.\\
To show \eqref{eq-relation2}, observe that by Taylor's formula there are $\zeta_1, \zeta_2\in [-|xy|,|xy|]$ such that \begin{eqnarray*}
\lefteqn{\left|f_n(x+xy) - f_n(x) - xy f_n'(x) \mathds{1}_{|y|\leq 1} \right|} \nonumber \\
& \leq & \left| f_n(x+xy) - f_n(x)\right| \mathds{1}_{|y|>1} +
2^{-1} \left[ |(\Re f_n'') (x+\zeta_1)| + |(\Im f_n'') (x+\zeta_2)| \,\right] x^2 y^2 \mathds{1}_{|y|\leq 1} \nonumber \\
& \leq & 2 \|f_n\| \mathds{1}_{|y|>1} + \|f_n''\| x^2 y^2
\mathds{1}_{|y|\leq 1}. \end{eqnarray*} Equation \eqref{eq-relation2} then follows directly from \eqref{eq-relation1} by dominated convergence and Fubini's theorem, observing as in the proof of Lemma \ref{lemma-generatorlimit} that $f_n$ and its first two derivatives are uniformly bounded in $n$. Finally, Equations \eqref{eq-relation4} and \eqref{eq-relation3} are immediate consequences of \eqref{eq-relation2}. \end{proof}
For GOU processes driven by $(\xi,\eta)$, Theorem \ref{thm-generatorlimit} gives the following.
\begin{corollary} \label{cor-generatorlimit} Let $(\xi_t)_{t\geq 0}$ and $(\eta_t)_{t\geq 0}$ be two independent L\'evy processes such that $V_\infty=\int_0^\infty e^{-\xi_{s-}}d\eta_s$ converges to a finite random variable. Then $\mu=\mathcal{L}(V_\infty)$ is the invariant law of the GOU process $(V_t)_{t\geq 0}$ driven by $(\xi, \eta)^T$ as defined in \eqref{GOUdef}.\\
Let $h\in C^\infty_c(\mathbb{R})$ such that $h(x)=1$ for $|x|\leq 1$ and
$h(x)=0$ for $|x|\geq 2$ and set $h_n(x):=h(\frac{x}{n})$ and $f(x)=e^{iux}$, $f_n(x)=f(x)h_n(x)$ for $u\in \mathbb{R}$. Then \begin{eqnarray} \psi_\eta(u)\phi_{V_{\infty}}(u)&=&\lim_{n\to \infty} \left( \gamma_\xi \int_{\mathbb{R}} xf_n'(x) \,\mu (dx) - \frac{\sigma_\xi^2}{2} \int_{\mathbb{R}} (x^2f_n''(x) + xf_n'(x) )\,\mu(dx) \right. \label{eq-relation5}\\ && \left. - \int_\mathbb{R} \int_{\mathbb{R}} (f_n(xe^{-y}) - f_n(x) + xyf_n'(x)
\mathds{1}_{|y|\leq 1}) \nu_\xi(dy) \mu(dx)\right). \nonumber \end{eqnarray} If additionally $E [V_\infty^2] < \infty$, then \begin{eqnarray} \psi_\eta(u) \phi_{V_\infty} (u) & = & \gamma_\xi u \phi_{V_\infty}'(u) - \frac{\sigma_\xi^2}{2} \left( u^2 \phi_{V_\infty}''(u) + u \phi_{V_\infty}'(u)\right) \label{eq-relation6} \\ & & - \int_{\mathbb{R}} \left( \phi_{V_\infty} (u e^{-y}) -
\phi_{V_\infty}(u) + u y \phi_{V_\infty}'(u) \mathds{1}_{|y|\leq 1} \right) \, \nu_\xi(dy). \nonumber \end{eqnarray} \end{corollary}
\begin{proof}
This follows directly from Theorem \ref{thm-generatorlimit} and the relations between $(U,L)$ and
$(\xi,\eta)$ as given in \eqref{eq-def-UL} and \cite[Lem.
3.4]{behmelindnermaller11}, or alternatively using
\eqref{generatorGOU-xieta-ind} and arguments as in the proof of
Theorem \ref{thm-generatorlimit}. \end{proof}
Observe that for $\xi$ being a compound Poisson process Equation \eqref{eq-relation6} immediately gives \eqref{eq-motivation}.
\begin{remark} Carmona \cite[Thm. 2]{carmona-unpublished} obtains a formula related to \eqref{eq-relation6} under certain, more restrictive assumptions. In particular, it is assumed in \cite{carmona-unpublished} that $e^{\xi_t}$ admits a strictly positive density on some interval $(0,r_t)$ for some $r_t > 0$. In the special case that $\eta$ is a compound Poisson process without negative jumps and $\xi$ is a Brownian motion with drift, formula \eqref{eq-relation6} has already been obtained by Nilsen and Paulsen \cite[Prop. 2]{NilsenPaulsen}, stated for Laplace transforms. \end{remark}
\begin{remark} Let $\eta$ be a subordinator, $\xi$ a L\'evy process independent of $\eta$, and suppose that $V_\infty := \int_0^\infty e^{-\xi_{s-}} \, d\eta_s$ is almost surely finite. Then $V_\infty \geq 0$, and we can also use Laplace transforms in the above derivation. More precisely, let $(U,L)^T$ be given by \eqref{eq-def-UL}, so that $L=\eta$ by independence and $e^{-\xi_t} = \mathcal{E}(U)_t$, where $\nu_U((-\infty,-1]) = 0$. Denote the Laplace transforms of $\eta=L$ and $V_\infty$ for $u\geq 0$ by $\mathbb{L}_\eta(u) = \mathbb{L}_L(u) = E [e^{-u \eta_1}] = \phi_\eta(iu)$ and $\mathbb{L}_{V_\infty} (u) = E [e^{-u V_\infty}]$, respectively. Let $f$ be a function in $S(\mathbb{R})$ with $f(x)=e^{-ux}, x\geq 0$, then $f$ is in $D(A^V)$ for $u>0$ and a direct computation starting from \eqref{eq-intgenerator} yields the following analogues of \eqref{eq-relation2} and \eqref{eq-relation6} without any further moment restrictions on the distribution of $V_\infty$.
\begin{eqnarray*} \lefteqn{\log \mathbb{L}_L(u)= \log \mathbb{L}_\eta(u)}\\ &=& u\gamma_U \frac{E\left[V_\infty e^{-uV_\infty} \right]}{\mathbb{L}_{V_\infty}(u)} -\frac{\sigma_U^2 u^2}{2} \frac{E\left[V_\infty^2 e^{-uV_\infty} \right]}{\mathbb{L}_{V_\infty}(u)}\\ && - \int_{(-1,\infty)} \left(\frac{\mathbb{L}_{V_\infty}(u(1+y))}{\mathbb{L}_{V_\infty}(u)}
- 1+ u\frac{E\left[V_\infty e^{-uV_\infty} \right]}{\mathbb{L}_{V_\infty}(u)} y \mathds{1}_{|y|\leq 1}\right) \nu_U(dy)\\ &=& -u\gamma_\xi \frac{E\left[V_\infty e^{-uV_\infty} \right]}{\mathbb{L}_{V_\infty}(u)} -\frac{\sigma_\xi^2 }{2} \left( \frac{E\left[V_\infty^2 e^{-uV_\infty} \right]}{\mathbb{L}_{V_\infty}(u)}u^2 - \frac{E\left[V_\infty e^{-uV_\infty} \right]}{\mathbb{L}_{V_\infty}(u)} u \right)\\ && -\int_{\mathbb{R}} \left(\frac{\mathbb{L}_{V_\infty}(ue^{-y})}{\mathbb{L}_{V_\infty}(u)}
- 1- u\frac{E\left[V_\infty e^{-uV_\infty} \right]}{\mathbb{L}_{V_\infty}(u)} y \mathds{1}_{|y|\leq 1}\right) \nu_\xi(dy), \quad u\geq 0. \end{eqnarray*} \end{remark}
The formula given in Corollary \ref{cor-generatorlimit} will be useful in determining $\mathcal{L}(\eta_1)$ from $\mathcal{L}(V_\infty)$ and $\mathcal{L}(\xi_1)$ as observed in Theorem \ref{thm-injectivity} below. For the determination of $\mathcal{L}(\xi_1)$ from $\mathcal{L}(V_\infty)$ and $\mathcal{L}(\eta_1)$, the following relation between the characteristic triplets of $\xi$, $L$ and the characteristic function of $\log
|V_\infty|$ will be helpful.
\begin{theorem}\label{thm-generatorlimit-2}
Let $(\xi_t)_{t\geq 0}$ and $(\eta_t)_{t\geq 0}$ be two independent L\'evy processes such that $V_\infty=\int_0^\infty e^{-\xi_{s-}}d\eta_s$ converges to a finite random variable and such that $\eta$ is not the zero process. Then $\mu=\mathcal{L}(V_\infty)$ is the invariant law of the GOU process $(V_t)_{t\geq 0}$ driven by $(\xi, \eta)^T$.\\
Let $h\in C^\infty_c(\mathbb{R})$ such that $h(x)=1$ for $|x|\leq 1$ and
$h(x)=0$ for $|x|\geq 2$ and set $h_n(x):=h(\frac{x}{n})$ and for
$x\neq 0$ and $u\in \mathbb{R}$ define $f(x)=e^{iu\log |x| }$ and
$f_n(x)=e^{iu\log |x|}h_n(\log |x|)$ with $f_n(0) = 0$. Then \begin{eqnarray}
\psi_{-\xi}(u)\phi_{\log |V_{\infty}|}(u)&=&- \lim_{n\to \infty} \int_{\mathbb{R}} A^\eta f_n(x) \mu(dx)\label{eq-relation7}\\ &=&\lim_{n\to \infty} \left( -\gamma_\eta \int_{\mathbb{R}} f_n'(x) \mu (dx) - \frac{\sigma_\eta^2}{2} \int_\mathbb{R} f_n''(x) \mu(dx) \right. \label{eq-relation8}\\ &&\left. - \int_\mathbb{R} \int_{\mathbb{R}} (f_n(x+y) -
f_n(x)-f_n'(x)y\mathds{1}_{|y|\leq 1} \nu_\eta(dy) \mu(dx) \right). \nonumber \end{eqnarray} If additionally $E [V_\infty^{-2}] < \infty$, then \begin{eqnarray}
\lefteqn{\psi_{-\xi}(u)\phi_{\log |V_{\infty}|}(u)} \label{eq-relation9} \\ &=& - iu \gamma_\eta E \left[ V_\infty^{-1}
e^{i u \log |V_\infty|} \right] + \frac{\sigma_\eta^2}{2} (iu + u^2) E \left[ V_\infty^{-2} e^{iu \log
|V_\infty|} \right] \nonumber \\
& & - \int_{\mathbb{R}} \left( E\left[ e^{iu \log |V_\infty + y|} \right] -
E \left[ e^{iu \log |V_\infty|} \right] - iuy E \left[ V_\infty^{-1}
e^{iu \log |V_\infty|} \right] \mathds{1}_{|y|\leq 1} \right) \, \nu_\eta(dy). \nonumber \end{eqnarray} \end{theorem} \begin{proof}
Observe that obviously $f_n\in C_c^\infty(\mathbb{R};\mathbb{C})$ and thus $f_n\in D(A^V;\mathbb{C})\cap D(A^\eta;\mathbb{C})$. On the other hand we obtain for $\tilde{f}(x)=f(e^x)$ and $\tilde{f}_n(x)=f_n(e^x)$ that $\tilde{f}(x)= e^{iux}$ and $\tilde{f}_n(x)=\tilde{f}(x)h_n(x)$ and hence $\tilde{f}_n\in C_c^\infty(\mathbb{R};\mathbb{C})\subset D(A^{-\xi};\mathbb{C})$. Similarly for $\tilde{\tilde{f}}(x)=f(-e^x)$ and $\tilde{\tilde{f}}_n(x)=f_n(-e^x)$ we have $\tilde{\tilde{f}}(x)= e^{iux}$ and $\tilde{\tilde{f}}_n(x)=\tilde{\tilde{f}}(x)h_n(x)$ and also $\tilde{\tilde{f}}_n\in C_c^\infty(\mathbb{R};\mathbb{C})\subset D(A^{-\xi};\mathbb{C})$. \\ Since $\mu(\{0\})=0$ by \cite[Thm. 2.2]{bertoinlindnermaller08}, we obtain from \eqref{generatorGOU-xieta-ind-short} and \eqref{generatorstat} \begin{eqnarray*}
0&=& \int_\mathbb{R} A^Vf_n(x)\mu(dx)\\ &=& \int_{\mathbb{R}} A^\eta f_n(x)\mu(dx)+ \int_{(0,\infty)} A^{-\xi} \tilde{f}_n(\log x)\mu(dx)
+ \int_{(-\infty,0)} A^{-\xi} \tilde{\tilde{f}}_n(\log |x|)\mu(dx). \end{eqnarray*} Setting $S_1: (0,\infty)\to \mathbb{R}, x\mapsto \log x$, and $S_2: (-\infty,0)\to \mathbb{R}, x\mapsto \log (-x)$, we compute using Lemma \ref{lemma-generatorlimit} \begin{eqnarray*}
\lefteqn{\lim_{n\to\infty}\left(\int_{(0,\infty)} A^{-\xi} \tilde{f}_n(\log x)\mu(dx)+ \int_{(-\infty,0)} A^{-\xi} \tilde{\tilde{f}}_n(\log |x|)\mu(dx) \right)}\\
&=& \lim_{n\to\infty} \left( \int_{\mathbb{R}} A^{-\xi} \tilde{f}_n(y) dS_1(\mu_{|(0,\infty)})(y)+ \int_{\mathbb{R}} A^{-\xi} \tilde{\tilde{f}}_n(y)dS_2(\mu_{|(-\infty,0)})(y) \right)\\
&=& \psi_{-\xi}(u) \left(\int_{\mathbb{R}} e^{iuy} dS_1(\mu_{|(0,\infty)})(y)+ \int_{\mathbb{R}} e^{iuy} dS_2(\mu_{|(-\infty,0)})(y)\right)\\
&=& \psi_{-\xi}(u) \left(\int_{(0,\infty)} e^{iu\log x} \mu(dx)+ \int_{(-\infty,0)} e^{iu\log |x|} \mu(dx)\right)\\
&=& \psi_{-\xi}(u) \phi_{\log |V_{\infty}|}(u) \end{eqnarray*} which yields \eqref{eq-relation7} and \eqref{eq-relation8} via \eqref{generatorlevy}.
Now assume that $E [V_\infty^{-2}] < \infty$. We have $\tilde{f}_n(x) = e^{iux} h_n(x)$ and $f_n(x) = \tilde{f}_n(\log
|x|)$ for all $x\in \mathbb{R}$. In particular, $f_n'(x) = x^{-1}
\tilde{f}_n'( \log |x|)$ and $f_n''(x) = x^{-2} ( \tilde{f}_n''(\log
|x|) - \tilde{f}_n' (\log |x|))$ for $x\neq 0$. For $|y|>1$, we further have $|f_n(x+y) - f_n(x)| \leq 2 \|h\| < \infty$, and for
$|y|\leq 1$ such that $xy (x+y) \neq 0$ there are $\zeta_1,\ldots, \zeta_4\in \mathbb{R}$ by Taylor's theorem such that \begin{eqnarray*}
\lefteqn{ \left| f_n(x+y) - f_n(x) - f_n'(x) y \right|} \\
& = & \left| \tilde{f}_n ( \log |x+y|) - \tilde{f}_n (\log |x|) -
\tilde{f}_n'( \log |x|) yx^{-1} \right| \\
& \leq & \left| \tilde{f}_n ((\log |x|) + yx^{-1}) - \tilde{f}_n (
\log |x|) - \tilde{f}_n' (\log |x|) yx^{-1} \right| \\
& & + \left| \tilde{f}_n ( \log |x| + \log |1+ yx^{-1}|) -
\tilde{f}_n ((\log|x|) + yx^{-1}) \right| \\
& = & 2^{-1} \left| (\Re \tilde{f}_n'')((\log |x|) + \zeta_1) y^2 x^{-2}\right| + \left| (\Re \tilde{f}_n')((\log |x|) + yx^{-1} +
\zeta_2) \left( \log |1 + yx^{-1}| - yx^{-1} \right) \right| \\
& & + 2^{-1} \left| (\Im \tilde{f}_n'')((\log |x|) + \zeta_3) y^2 x^{-2}\right| + \left| (\Im \tilde{f}_n')((\log |x|) + yx^{-1} +
\zeta_4) \left( \log |1 + yx^{-1}| - yx^{-1} \right) \right| \\
& \leq & \| \tilde{f}_n''\| y^2 x^{-2} + \| \tilde{f}_n'\| C y^2 x^{-2} \end{eqnarray*}
for some universal constant $C$. Since $\|\tilde{f}_n\|$,
$\|\tilde{f}_n'\|$ and $\| \tilde{f}_n''\|$ are uniformly bounded in $n$, since $\mu$ is continuous (cf. \cite[Thm. 2]{bertoinlindnermaller08}) so that $(\nu \otimes \mu) (\{(x,y)^T\in \mathbb{R}^2 : xy (x+y) = 0\} = 0$, since $\int_{\mathbb{R}} x^{-2} \mu(dx) < \infty$ by assumption and since $f_n$, $f_n'$ and $f_n''$ converge on $\mathbb{R} \setminus \{0\}$ to $f$, $f'$ and $f''$, respectively, by dominated convergence the right hand side of \eqref{eq-relation8} is equal to $$\int_{\mathbb{R}} \left( - \gamma_\eta f'(x) - \frac{\sigma_\eta^2}{2} f''(x) - \int_{\mathbb{R}} \left(f(x+y) - f(x) -
f'(x) y \mathds{1}_{|y|\leq 1}\right) \, \nu_\eta(dy) \right) \mu(dx),$$ which gives \eqref{eq-relation9}. \end{proof}
\section{Injectivity} \label{sec-inj} \setcounter{equation}{0}
Let $\xi=(\xi_t)_{t\geq 0}$ and $(\eta_t)_{t\geq 0}$ be two independent L\'evy processes such that $V_\infty := \int_0^\infty e^{-\xi_{s-}} \, d\eta_s$ converges almost surely. By \cite[Thm. 2]{ericksonmaller05}, this implies that $\xi$ drifts to $+\infty$. As in the introduction, for a L\'evy process $(\xi_t)_{t\geq 0}$ such that $\xi_t \to +\infty$ a.s. as $t\to\infty$ denote $$D_\xi := \{ \mathcal{L} (\eta_1) : \eta \;\mbox{ L\'evy process, independent of }\xi, \;\mbox{such that } \int_0^\infty e^{-\xi_{s-}} \, d\eta_s \;\mbox{converges a.s.}\}$$ and consider the mapping $$ \Phi_\xi : D_\xi \to \mathcal{P}(\mathbb{R}) ,\quad \mathcal{L}(\eta_1) \mapsto \mathcal{L} \left( \int_0^\infty e^{-\xi_{s-}} \, d\eta_s \right), \quad \mbox{where $\eta$ and $\xi$ are independent.} $$ Here $\mathcal{P}(\mathbb{R})$ denotes the set of probability distributions on $(\mathbb{R},\mathcal{B}_1)$. For a L\'evy process $(\eta_t)_{t \geq 0}$ denote further $$\tilde{D}_\eta := \{ \mathcal{L} (\xi_1) : \xi \;\mbox{ L\'evy process, independent of }\eta, \;\mbox{such that } \int_0^\infty e^{-\xi_{s-}} \, d\eta_s \;\mbox{converges a.s.}\}$$ and define $\tilde{\Phi}_\eta$ by $$\tilde{\Phi}_\eta: \tilde{D}_\eta \to \mathcal{P}(\mathbb{R}), \quad \mathcal{L}(\xi_1) \mapsto \mathcal{L} \left( \int_0^\infty e^{-\xi_{s-}} \, d\eta_s \right), \quad \mbox{where $\eta$ and $\xi$ are independent.}$$
We are interested in injectivity of the mappings $\Phi_\xi$ and
$\tilde{\Phi}_\eta$, or at least in injectivity of these mappings when restricted to certain subsets. A key result for these investigations will be the following theorem, which follows immediately from \eqref{eq-relation5} and \eqref{eq-relation8}, by dividing by $\phi_{V_\infty}(u)$ and $\phi_{\log |V_\infty|}(u)$ when different from zero, which is always the case for $u$ in a neighborhood of zero.
\begin{theorem} \label{thm-injectivity} Let $(\xi_t)_{t\geq 0}$ and $(\eta_t)_{t\geq 0}$ be two independent L\'evy processes such that $V_\infty := \int_0^\infty e^{-\xi_{s-}} \, d\eta_s$ converges almost surely. If $\phi_{V_\infty} (u) \neq 0$ for $u$ from a dense subset of $\mathbb{R}$, or if $\mathcal{L}(\eta_1)$ is uniquely determined by the values of its characteristic function in a neighborhood of the origin, then $\mathcal{L}(\eta_1)$ is uniquely determined by $\mathcal{L}(V_\infty)$ and $\mathcal{L}(\xi_1)$. Similarly, if
$\eta$ is not the zero process and $\phi_{\log |V_\infty|}(u) \neq 0$ for $u$ from a dense subset of $\mathbb{R}$, or if $\mathcal{L}(\xi_1)$ is uniquely determined by the values of its characteristic function in a neighborhood of the origin, then $\mathcal{L}(\xi_1)$ is uniquely determined by $\mathcal{L}(V_\infty)$ and $\mathcal{L}(\eta_1)$. \end{theorem}
It is well known (e.g. \cite{loeve}) that not every distribution is characterized by the values of its characteristic function in a neighborhood of the origin. This remains true for infinitely divisible distributions. To see this take two different distributions $\mu_1$ and $\mu_2$ without atoms at $0$ whose characteristic functions coincide in a neighborhood of the origin and consider the corresponding compound Poisson distributions with L\'evy measures $\mu_1$ and $\mu_2$. These are both infinitely divisible and their characteristic functions $\exp(\hat{\mu}_i(u)-1)$ coincide in a neighborhood of the origin.
We do not know if the characteristic function of the stationary distribution of a GOU process cannot vanish on a non-empty open interval. As shown by Il'inskii \cite[Cor.~1]{Ilinskii}, a set $A \subset \mathbb{R}$ is the zero set of some characteristic function if and only if $A$ is closed, does not contain 0 and is symmetric with respect to the origin. Hence, a priori there is no reason why $\phi_{V_\infty}$ appearing in Theorem \ref{thm-injectivity} should not vanish identically on some interval.
Still, it is possible to give some sufficient conditions. We start with the following lemma, which is a minor reformulation of results in Kawata \cite{kawata1972} and Lucasz \cite{lucasz1970}.
\begin{lemma} \label{lem-zeros} Let $X$ be a random variable with law $\mu$ and assume that there is some $\varepsilon > 0$ such that $E e^{\varepsilon X} < \infty$ or $E e^{-\varepsilon X} < \infty$. Then the characteristic function $\phi_X=\widehat{\mu}$ cannot be identically zero on non-empty open intervals. Furthermore, if $Y$ is another random variable whose characteristic function coincides with that of $X$ in a neighborhood of 0, then $\mathcal{L}(Y) = \mathcal{L}(X) = \mu$. \end{lemma} \begin{proof} Without loss of generality assume that $E e^{-\varepsilon X} < \infty$. Then $g(z) := E e^{i z X}$ can be defined for all $z\in \mathbb{C}$ such that $0 \leq \Im z < \varepsilon$, it is continuous there and analytic in $0 < \Im z < \varepsilon$. That $\phi_X$ cannot be identically zero on non-empty open intervals then follows from \cite[Cor. 1.14.1]{kawata1972}. Let $Y$ be another random variable such that $\phi_Y(u) = \phi_X(u)$ for all $u \in (-a,a)$ with some $a>0$. Since $\phi_Y(u) = \lim_{y\downarrow 0} g(u + iy)$ for $u\in (-a,a)$, it follows from \cite[Thm. 11.1.1]{lucasz1970} and its proof that $ E e^{- \varepsilon Y} < \infty$. That $\mathcal{L}(Y) = \mathcal{L}(X)$ then follows from \cite[Thm. 9.6.2]{kawata1972}. \end{proof}
Define $ID^{\rm sym}$ to be the set of all infinitely divisible distributions $\mathcal{L}(\eta_1)$ which are symmetric, and $ID^{\rm exp}$ to be the set of all infinitely divisible distributions whose L\'evy measure $\nu_\eta$ has some one-sided exponential moment, i.e. for which there is $\varepsilon > 0$ such that $$\int_{1}^\infty e^{\varepsilon x} \, \nu_\eta(dx) < \infty \quad \mbox{or} \quad \int_{-\infty}^{-1} e^{- \varepsilon x} \, \nu_\eta(dx) < \infty.$$ Denote $$D_\xi^{\rm sym, exp} := D_\xi \cap (ID^{\rm sym} \cup ID^{\rm exp}), \quad D_\xi^{\rm sym} := D_\xi \cap ID^{\rm sym} \quad \mbox{and} \quad D_\xi^{\rm exp} := D_\xi \cap ID^{\rm exp}.$$ With these notions, we get the following result:
\begin{theorem} \label{cor-phi-xi-injective} Let $(\xi_t)_{t\geq 0}$ be a L\'evy process such that $\xi_t$ converges almost surely to $\infty$ as $t\to \infty$. Then
$(\Phi_\xi)|_{D_{\xi}^{\rm sym, exp}}$ is injective and $$\Phi_\xi (D_\xi^{\rm sym, exp}) \cap \Phi_\xi ( D_\xi \setminus D_\xi^{\rm sym, exp}) = \emptyset.$$ If additionally $\xi$ is spectrally negative, or $\xi = q N$ for some constant $q>0$ and a Poisson process $N$, then $\Phi_\xi$ is injective on $D_\xi$. \end{theorem}
In the special case when $\xi_t = t$, we have a spectrally negative $\xi$, and we recover the well known result (e.g. \cite[Prop. 3.6.10]{jurekmason}) that $\Phi_{\xi_t = t}$ is injective.
\begin{proof}[Proof of Theorem \ref{cor-phi-xi-injective}] If $\xi$ is spectrally negative, then $V_\infty=\int_0^\infty e^{-\xi_{t-}} \, d\eta_t$ is self-decomposable by Remark (ii) to Theorem 2.2 in \cite{bertoinlindnermaller08}, hence infinitely divisible so that $\phi_{V_\infty}(u) \neq 0$ for all $u\in \mathbb{R}$. Injectivity of $\Phi_\xi$ then follows from Theorem \ref{thm-injectivity}. If $\xi = q N_t$ for $q>0$ and a Poisson process $N$, then by Example \ref{example-start} we can write $V_\infty = \sum_{i=0}^\infty e^{-qi} (\eta_{T_{i+1}}-\eta_{T_i}),$ where $(\eta_{T_{i+1}} - \eta_{T_i})_{i=0,1,2,\ldots}$ is i.i.d. and infinitely divisible by \cite[Thm. 30.1]{sato}. Hence $V_\infty$ is infinitely divisible, and injectivity of $\Phi_\xi$ follows from Theorem \ref{thm-injectivity}.
Now let $\xi$ be an arbitrary L\'evy process drifting to infinity. If $\mathcal{L}(\eta_1) \in D_\xi \cap ID^{\rm exp}$, then there is $\varepsilon > 0$ such that $E e^{\varepsilon \eta_1} < \infty$ or
$E e^{-\varepsilon \eta_1} < \infty$ (cf. \cite[Thm. 25.17]{sato}), and Theorem~\ref{thm-injectivity} and Lemma \ref{lem-zeros} show that $(\Phi_\xi)|_{D_\xi^{\rm exp}}$ is injective and $\Phi_\xi (D_\xi^{\rm exp}) \cap \Phi_\xi (D_\xi \setminus D_\xi^{\rm exp}) = \emptyset$.
Finally, let $\xi$ be an arbitrary L\'evy process drifting to infinity and $\mathcal{L}(\eta_1) \in D_\xi^{\rm sym}$. Conditioning on $\xi$, for $f$ in the Skorokhod space $D([0,\infty), \mathbb{R})$ of c\`adl\`ag functions, we have
$$\left( V_\infty | \xi = f \right) = \int_0^\infty e^{-f(t)} \, d\eta_t,$$ which converges for
$P_\xi$-almost every $f$. For such $f$, $\int_0^\infty e^{-f(t)} \, d\eta_t$ is infinitely divisible (e.g. Sato~\cite{sato2007}), and hence $E (e^{iuV_\infty}|\xi = f) \neq 0$ for all $u\in \mathbb{R}$. Since $\int_0^\infty e^{-f(t)} \, d\eta_t$ is also symmetric, $E
(e^{iuV_\infty}|\xi = f)$ is real valued and continuous in $u$ and hence strictly positive for all $u\in \mathbb{R}$. It follows that $$\phi_{V_\infty} (u) =
\int_{D([0,\infty),\mathbb{R})} E \left[ e^{iu V_\infty} |\xi= f \right] \, P_\xi (df) > 0 \quad \forall u \in \mathbb{R}.$$ Theorem
\ref{thm-injectivity} then shows that $(\Phi_\xi)|_{D_\xi^{\rm sym}}$ is injective and $\Phi_\xi (D_\xi^{\rm sym}) \cap \Phi_\xi (D_\xi \setminus D_\xi^{\rm sym}) = \emptyset$. This finishes the proof. \end{proof}
\begin{remark} Theorem \ref{cor-phi-xi-injective} shows in particular that if $\xi$ is arbitrary (but drifting to $+\infty$), and $\eta$ is spectrally positive or negative (which applies in particular if $\eta$ is a subordinator or the negative of a subordinator), then the distribution of $\eta_1$ is uniquely determined by $\mathcal{L}(V_\infty)$ and $\mathcal{L}(\xi_1)$. \end{remark}
Let us now turn to injectivity properties of $\tilde{\Phi}_\eta$. We start with the following lemma, which is immediate from Lemma \ref{lem-zeros}.
\begin{lemma} \label{lem-zeros2}
Let $X$ be a random variable which has no atom at 0 and assume that there is $\varepsilon > 0$ such that $E |X|^\varepsilon < \infty$ or
$E |X|^{-\varepsilon} < \infty$. Then the characteristic function
$\phi_{\log |X|}$ of $\log |X|$ cannot be identically zero on non-empty open intervals. \end{lemma}
Examples of random variables $X$ with finite negative fractional moment $E |X|^{-\varepsilon}<\infty$ are given by random variables which have a density $f$ in a neighborhood of zero such that $f(x) =
O(x^{\alpha})$ as $|x|\to 0$ for some $\alpha > \varepsilon -1 $. In particular, if $\mathcal{L}(X)$ is a self-decomposable non-degenerate distribution, then $X$ has a density satisfying this condition for some $\varepsilon > 0$, which follows from Theorems 28.4, 53.6 and 53.8 in \cite{sato}; observe that this is trivial if $X$ has a non-zero Gaussian component.
Hence, whenever $X\not\equiv 0$ is self-decomposable, then $\phi_{\log |X|}$ cannot be identically zero on non-empty open intervals.
Other examples are given in the next lemma, which shows that $\int_0^\infty e^{-\xi_{s-}} \, d\eta_s$ will always have certain negative fractional moments if $\eta$ is a subordinator with strictly positive drift, or if $\eta$ has a non-trivial Brownian motion component. This complements \cite[Lem. 2.1]{MaulikZwart} and \cite[Lem. 3.3]{savovetal} who assume $\xi$ to have finite mean.
\begin{lemma} \label{lem-fractional-moment} Let $\xi$ and $\eta$ be two independent L\'evy processes such that $V_\infty = \int_0^\infty e^{-\xi_{s-}} \, d\eta_s$ converges almost surely. Suppose that $\eta$ is a subordinator with strictly positive drift, or that the Brownian motion part of $\eta$ is non-trivial
(i.e. $\sigma_\eta^2 > 0$). Then $E |V_\infty|^{-\varepsilon} < \infty$ for every $\varepsilon \in [0,1)$. In the latter case (i.e. when $\sigma_\eta^2 > 0)$, $V_\infty$ has a bounded density on $\mathbb{R}$. \end{lemma}
\begin{proof}
Suppose first that $\eta$ is a subordinator with strictly positive drift $\gamma_\eta^0$. Let $\varepsilon \in (0,1)$. Define the L\'evy process $\xi^\flat$ by $\xi_t^\flat = \xi_t - \sum_{0<s\leq t, |\Delta \xi_s| > 1} \Delta \xi_s$. Let $\tau$ be the time of the first jump of $\xi$ whose size is greater than $1$ in magnitude. Then $$V_\infty \geq \gamma_\eta^0 \int_0^{1\wedge \tau} e^{-\xi_{s-}} \, ds \geq \gamma_\eta^0 (1\wedge \tau) \exp \left( - \sup_{0\leq s
\leq 1} |\xi_s^\flat|\right).$$ Since $\tau$ and $\xi^\flat$ are independent and $\tau$ is exponentially distributed (or $\tau \equiv \infty$), it follows $E (1\wedge \tau)^{-\varepsilon} < \infty$ and $E \exp \left(\varepsilon \sup_{0\leq s \leq 1}
|\xi_s^\flat|\right) < \infty$ (cf. \cite[Thms. 25.17, 25.18]{sato}), so that $E V_\infty^{-\varepsilon} < \infty$ if $\eta$ is a subordinator with strictly positive drift.
Now suppose that $\eta$ is a L\'evy process such that $\sigma_\eta^2 > 0$. Denote the Brownian motion component of $\eta$ by $B$, so that $B$ and $\eta-B$ are independent. Then the conditional distribution of $V_\infty$ given $\xi=f$ is given by $\int_0^\infty e^{-f(t-)} \, dB_t + \int_0^\infty e^{-f(t-)} \, d(\eta_t - B_t)$. But $\int_0^\infty e^{-f(t-)} \, dB_t$ is $N(0, \sigma_\eta^2
\int_0^\infty e^{-2f(s)} \, ds)$-distributed, hence its density is bounded by $(2\pi \sigma_\eta^2\int_0^\infty e^{-2f(s)}\, ds)^{-1/2}$. Hence also $(V_\infty|\xi = f)$ has a density, $g_f$ say, which is bounded by $(2\pi \sigma_\eta^2\int_0^\infty e^{-2f(s)}\, ds)^{-1/2}$. It follows that $V_\infty$ has a density given by $x\mapsto \int_{D([0,\infty))} g_f(x) \, P_\xi(df)$, and since
$$\int_{D([0,\infty))} \left(2\pi \sigma_\eta^2 \int_0^\infty e^{-2f(s)} ds \right)^{-1/2} \, P_\xi(df) = (2\pi\sigma_\eta^2)^{-1/2} E \left[ \left( \int_0^\infty e^{-2\xi_{s-}} \, ds\right)^{-1/2}\right] < \infty$$ by the part just proved, this density is bounded on $\mathbb{R}$. This then also shows that $E |V_\infty|^{-\varepsilon} < \infty$ for all $\varepsilon \in [0,1)$. \end{proof}
Recall that $ID^{\rm exp}$ denotes the set of all infinitely divisible distributions whose L\'evy measure has some one-sided exponential moment. Denote $$\tilde{D}_\eta^{\rm exp} := \tilde{D}_\eta \cap ID^{\rm \exp}.$$ We can now prove the following injectivity result regarding $\tilde{\Phi}_\eta$:
\begin{theorem} \label{cor-eta-injective1} Let $\eta=(\eta_t)_{t\geq 0}$ be a non-zero L\'evy process. Then
$(\tilde{\Phi}_\eta)|_{\tilde{D}_\eta^{\rm exp}}$ is injective and \begin{equation} \label{eq-injectivity1} \tilde{\Phi}_\eta (\tilde{D}_\eta^{\rm exp}) \cap \tilde{\Phi}_\eta (\tilde{D}_\eta \setminus \tilde{D}_\eta^{\rm exp}) = \emptyset. \end{equation} If additionally $\eta$ is a subordinator with strictly positive drift, or if the Brownian motion part of $\eta$ is non-trivial (i.e. $\sigma_\eta^2 > 0$), or if $\eta$ is a compound Poisson process without drift such that $\nu_\eta((-\infty,0)) = 0$ and $\int_0^1 x^{-\varepsilon} \, \nu_\eta(dx) < \infty$ for some $\varepsilon > 0$, then $\tilde{\Phi}_\eta$ is injective on $\tilde{D}_\eta$. \end{theorem}
Observe that $\tilde{D}_\eta^{\rm exp}$ contains all $\mathcal{L}(\xi_1) \in \tilde{D}_\eta$ such that $\xi$ is spectrally negative or spectrally positive. In particular, subordinators are uniquely determined by $\mathcal{L}(V_\infty)$ and $\mathcal{L}(\eta_1)$.
\begin{proof}[Proof of Theorem \ref{cor-eta-injective1}] The injectivity of $\tilde{\Phi}_\eta$ on $\tilde{D}_\eta^{\rm exp}$ as well as \eqref{eq-injectivity1} are clear from Theorem~\ref{thm-injectivity} and Lemma \ref{lem-zeros}. Similarly, injectivity of $\tilde{\Phi}_\eta$ on $\tilde{D}_\eta$ follows from Lemmas \ref{lem-zeros2}, \ref{lem-fractional-moment} and Theorem~\ref{thm-injectivity} if $\eta$ is a subordinator with strictly positive drift or if $\sigma_\eta^2 > 0$.
Finally, let us prove injectivity of $\tilde{\Phi}_\eta$ when $\eta$ is a compound Poisson process with $\nu_\eta((-\infty,0)) = 0$ and $\int_0^1 x^{-\varepsilon} \, \nu_\eta(dx) < \infty$ for some $\varepsilon > 0$.
Denote by $T$ the time of the first jump of $\eta$. Then $$V_\infty = \int_0^\infty e^{-\xi_{s-}} \, d\eta_s = e^{-\xi_{T-}} \Delta \eta_T + e^{-\xi_T} \int_T^\infty e^{-(\xi_{s-} - \xi_T)} \, d\eta_s = e^{-\xi_T} (\Delta \eta_T + V_\infty') \quad \mbox{a.s.},$$ since $\xi$ and $\eta$ almost surely do not jump together. The random variable $V_\infty'$ has the same distribution as $V_\infty$ and is independent of $(e^{-\xi_T}, \Delta \eta_T)$. Observe further that also $\xi_T$ and $\Delta \eta_T$ are independent. It follows that \begin{equation*} \phi_{\log V_\infty}(u) = \phi_{-\xi_T}(u) \, \phi_{\log (\Delta \eta_T + V_\infty')} (u), \quad u\in \mathbb{R}.\end{equation*} Since $$E (\Delta \eta_T + V_\infty')^{-\varepsilon} \leq E (\Delta \eta_T)^{-\varepsilon} < \infty$$ as a consequence of $V_\infty' \geq 0$ and $\int_0^1 x^{-\varepsilon} \, \nu_\eta(dx) < \infty$, it follows from Lemma~\ref{lem-zeros2} that $\phi_{\log (\Delta \eta_T + V_\infty')}$ cannot vanish identically on non-empty open intervals. Since $\phi_{-\xi_T}(u) \neq 0$ for all $u\in \mathbb{R}$ as $\xi_T$ is infinitely divisible, it follows that $\phi_{\log V_\infty}$ cannot vanish identically on non-empty open intervals. Injectivity of $\tilde{\Phi}_\eta$ then follows from Theorem \ref{thm-injectivity}. \end{proof}
We do not know if $\Phi_\xi$ and $\tilde{\Phi}_\eta$ will always be injective, but as we have seen in Theorems \ref{cor-phi-xi-injective} and \ref{cor-eta-injective1}, the mappings $\Phi_\xi$ and $\tilde{\Phi}_\eta$ are injective in many cases. However, if we drop the condition of independence of $\xi$ and $\eta$, an injectivity result does not hold, as shown in the following. Therefore, additionally to the definitions at the beginning of this section, for a L\'evy process $\xi$, let $$D_\xi^{\rm dep} := \{ \mathcal{L} (\chi_1,\eta_1) : (\chi,\eta) \; \mbox{biv. LP such that $\int_0^\infty e^{-\chi_{s-}} \, d\eta_s$ converges a.s. and}\; \mathcal{L}(\chi_1) = \mathcal{L}(\xi_1)\}$$ and define the mapping \begin{eqnarray*} \Phi_\xi^{\rm dep} : D_\xi^{\rm dep} \to \mathcal{P}(\mathbb{R}) ,\quad \mathcal{L}(\chi_1,\eta_1) \mapsto \mathcal{L} \left( \int_0^\infty e^{-\chi_{s-}} \, d\eta_s \right). \end{eqnarray*} Then we obtain the following counterexample of injectivity.
\begin{example} \label{ex-1} Let $\xi= N$ be a Poisson process. Then $\Phi_\xi^{\rm dep}$ is not injective. \end{example}
\begin{proof} Let $(\chi,\eta)$ be a bivariate L\'evy process such that $\mathcal{L}( \chi_1,\eta_1) \in D_\xi^{\rm dep}$. By \cite[Thm. 2]{ericksonmaller05}, this means $\mathcal{L}(\chi_1) = \mathcal{L}(\xi_1)$ and
$E\log^+| \eta_1| < \infty$. Denote the time of the first jump of $\chi$ by $T = T(\chi)$. Then \begin{equation} \label{Al-12} \int_0^\infty e^{-\chi_{t-}} \, d\eta_t = \eta_{T} + e^{-1} \int_{T}^\infty e^{-(\chi_{t-} - \chi_{T})} \, d\eta_t. \end{equation} Since $\int_{T}^\infty e^{-(\chi_{t-} - \chi_{T})} \, d\eta_t$ has the same distribution as $\int_0^\infty e^{-\chi_{t-}} \, d\eta_t =: W$, it follows that the characteristic function $\phi_W$ of $W$ satisfies $$\phi_W(x) = \prod_{k=0}^\infty \phi_{\eta_{T}} (e^{-k} x), \quad x \in \mathbb{R}$$ as shown in \cite{behme-et-al}. Thus, $\mathcal{L}(W)$ is determined by $\rho_{\chi,\eta} := \mathcal{L} (\eta_{T})$ (not necessarily vice versa!). Now let $(\chi^{(1)},\eta^{(1)}) \in D_\xi^{\rm dep}$ be such that $\eta^{(1)}$ is independent of $\chi^{(1)}$, $\eta^{(1)}$ is not the zero process and $E \log^+
|\eta^{(1)}_{T(\chi^{(1)})}| < \infty$, and let $(\chi^{(2)},\eta^{(2)})$ be a bivariate compound Poisson process without drift and L\'evy measure $$\nu_{\chi^{(2)},\eta^{(2)}} (dx,dy) = \delta_1(dx) \rho_{\chi^{(1)}, \eta^{(1)}} (dy).$$ Then $(\chi^{(2)}, \eta^{(2)}) \in D_\xi^{\rm dep}$ and $$\rho_{\chi^{(2)},\eta^{(2)}}= \mathcal{L}( \eta^{(2)}_{T(\chi^{(2)})})= \rho_{\chi^{(1)}, \eta^{(1)}}.$$ It follows that both $(\chi^{(1)},\eta^{(1)})$ and $(\chi^{(2)},\eta^{(2)})$ lead to the same distribution, giving an example that injectivity is violated. \end{proof}
\section{Ranges}\label{sec-ranges} \setcounter{equation}{0}
The results of the previous section may now be used to determine information on the ranges of the mappings $\Phi_\xi$ and $\tilde{\Phi}_\eta$ as defined in Section \ref{sec-inj}. We start with an elementary conclusion, which also follows from \cite[Thm. 2.2]{bertoinlindnermaller08} or \cite[Lem. 3.1]{behme2011}.
\begin{proposition}
Let $\xi$ be non-deterministic, then $\Phi_\xi (D_\xi \setminus \{ \mathcal{L}(0)\})$ is a subset of the continuous distributions. Analoguously, if $\eta$ is non-deterministic, then the range of $\tilde{\Phi}_\eta$ is a subset of the continuous distributions. \end{proposition} \begin{proof}
It follows directly from \cite[Thm. 1.3]{alsmeyeretal} that the distribution of the treated exponential functional fulfills a pure type theorem, in particular it is either continuous, or a Dirac measure. Suppose that $L_1=\eta_1\not\equiv 0 $. Inserting the characteristic function $\phi(u)=e^{iuk}$, $k\in\mathbb{R}$, of a Dirac measure in \eqref{eq-relation3}, one immediately obtains $\psi_L(u)=-\psi_U(ku)$ which can only hold for deterministic processes $L_t=-kU_t=\gamma_L t$ with $k\neq 0$ and hence deterministic $\eta$ and $\xi$. \end{proof}
Recall the definition of $\Phi_\xi^{\rm dep}$ from the previous section. Also recall that a distribution $\mu$ on $(\mathbb{R},\mathcal{B}_1)$ is called {\it $b$-decomposable}, where $b\in (0,1)$, if there exists a probability measure $\rho$ on $(\mathbb{R},\mathcal{B}_1)$ such that $\widehat{\mu}(z) = \widehat{\mu}(bz) \widehat{\rho}(z)$ for all $z\in \mathbb{R}$.
\begin{proposition} \label{prop-range1} Let $\xi=N$ be a Poisson process. Then the range of $\Phi_{\xi}^{\rm dep}$ is the class of all $e^{-1}$-decomposable distributions. \end{proposition}
\begin{proof} That all distributions in the range of $\Phi_{\xi}^{\rm dep}$ are $e^{-1}$-decomposable is clear from \eqref{Al-12}. Conversely, let $\mathcal{L}(W)$ be an $e^{-1}$-decomposable distribution. Then there exists an i.i.d. noise sequence $(Z_n)_{n\in \mathbb{N}_0}$ such that \begin{equation} \label{eq-range1} \sum_{k=0}^n e^{-k} Z_k \stackrel{d}{\to} W, \quad n\to\infty, \end{equation} which follows by iterating the defining equation $W \stackrel{d}{=} e^{-1} W' + Z$ with $W'$ independent of $Z$ for $e^{-1}$-decomposability. Hence $\sum_{k=0}^n e^{-k} Z_k$ converges in distribution and hence almost surely as $n\to\infty$ and the Borel-Cantelli-lemma implies that $Z_0$ must have finite $\mbox{log}^+$-moment. Now define the compound Poisson process $(\chi,\eta)$ without drift and L\'evy measure $$\nu_{\chi,\eta} (dx,dy) = \delta_1 (dx) \mathcal{L}(Z_0)(dy).$$ Then $\mathcal{L}(\chi,\eta) \in D_\xi^{\rm dep}$ (due to the finite log$^+$ moment of $Z_0$), and with the notations of Example \ref{ex-1} it follows that $\mathcal{L}(\eta_{T(\chi)}) = \mathcal{L}(Z_0)$. Hence $\Phi_\xi^{\rm dep} (\mathcal{L}(\chi,\eta)) = \mathcal{L}(W)$. \end{proof}
\begin{proposition}
Let $\xi=N$ be a Poisson process. Then the range of $\Phi_{\xi}$ is a subset of the class of infinitely divisible $e^{-1}$-decomposable distributions
without Gaussian part. \end{proposition} \begin{proof}
By Proposition \ref{prop-range1} it remains to show that $W=\int_{(0,\infty)} e^{-N_{s-}}d\eta_s$ is infinitely divisible and has zero Gaussian part. Therefore denote the time of the first jump of $N$ by $T$, then $Z_0:=\eta_T$ is infinitely divisible without Gaussian part as a consequence of \cite[Thm. 30.1]{sato}. Hence by \eqref{eq-range1} also $W$ is infinitely divisible and the Gaussian part of $W$ is zero. \end{proof}
It is well known that the OU process is a Gaussian process whose stationary distribution is normally distributed. In particular $\int_{(0,\infty)} e^{-t\sigma^2/2}d(\sigma W_t)$ for $W_t$ a standard Brownian motion (Wiener process) is standard normally distributed. The following theorem shows that this is the only possible choice of $(\xi,\eta)^T$ which leads to a centered normal distribution.
\begin{theorem} \label{thm-nonormal} Let $\xi$ and $\eta$ be two independent L\'evy processes such that $\int_0^\infty e^{-\xi_{s-}} \, d\eta_s$ converges almost surely. Let $v> 0$. Then $\mathcal{L}(V_\infty) = N(0,v^2)$ if and only if there is $\gamma_\xi > 0$ such that $\xi_t = \gamma_\xi t$ and $\eta_t = (2\gamma_\xi)^{1/2} v W_t$, where $(W_t)_{t\geq 0}$ is a standard Brownian motion. \end{theorem}
\begin{proof} That $\mathcal{L}(V_\infty) = N(0,v^2)$ if $\xi$ and $\eta$ are as described is well known and follows as discussed above. Let us show the converse and assume that $V_\infty$ is $N(0,v^2)$-distributed. By replacing $\eta$ by $v^{-1}\eta$ we may assume that $v=1$. Inserting $\phi_{V_\infty}(u) = e^{-u^2/2}$ in \eqref{eq-relation6}, we obtain for $u\in \mathbb{R}$ \begin{equation} \label{eq-relation10} \psi_\eta(u) = -\gamma_\xi u^2 - \sigma_\xi^2 (u^4-2u^2)/2 - \int_{\mathbb{R}} \left( e^{-u^2 (e^{-2y} -1)/2} -1 - u^2 y
\mathds{1}_{|y|\leq 1} \right)\, \nu_\xi(dy). \end{equation} For given $u\in \mathbb{R}$ denote $$f_u(y) := e^{-u^2 (e^{-2y} -1)/2} -1 -
u^2 y \mathds{1}_{|y|\leq 1}, \quad y \in \mathbb{R}\setminus \{0\}.$$
We shall first investigate the limit behavior of \eqref{eq-relation10} as $u\to \infty$ when divided by appropriate powers of $u$ and from that obtain information about the characteristic triplet of $\xi$. To do so, observe first that there are constants $C_1, C_2, C_3>0$ such that \begin{eqnarray*}
|e^{-x} - 1 + x - x^2/2| & \leq & C_1 x^2 \quad \forall\; x > 0,\\
|e^{-2y }- 1 + 2y| & \leq & C_2 y^2 \quad \forall\; y \in [-1,1], \quad \mbox{and} \\ (e^{-2y} - 1)^2 & \leq & C_3 y^2 \quad \forall\; y \in [-1,1]. \end{eqnarray*}
Let $y_0 \in [-1,0)$. Then $|f_u(y)| \leq 1+ u^2$ for $y< y_0$, and for $y \in [y_0,0)$ we can estimate \begin{eqnarray*}
|f_u(y)| & \leq & \left| -u^2 (e^{-2y}-1)/2 + u^4 (e^{-2y} -1)^2/8
- u^2 y\right| + C_1 u^4 (e^{-2y} -1)^2 /4 \\ & \leq & u^2 C_2 y^2/2 + u^4 C_3 y^2/8 + C_1 C_3 u^4y^2/4. \end{eqnarray*} Using dominated convergence, this gives
$$\limsup_{u\to \infty} u^{-4} \int_{-\infty}^0 |f_u(y)| \, \nu_\xi(dy) \leq (C_3/8 + C_1 C_3/4) \int_{[y_0, 0)} y^2\, \nu_\xi(dy),$$ and letting $y_0 \uparrow 0$ we see that \begin{equation} \label{eq-relation11}
\lim_{u\to\infty} u^{-4} \int_{-\infty}^0 |f_u(y)| \, \nu_\xi(dy) = 0. \end{equation} Now let $y>0$. Then $$f_u(y) \geq \left( u^2 (1-e^{-2y})/2 - u^2 y\right) \mathds{1}_{(0,1]}(y) \geq -u^2 (C_2/2) y^2 \mathds{1}_{(0,1]}(y).$$ Since also $\lim_{u\to\infty} u^{-5} f_u(y) = +\infty$ for $y>0$ and $\lim_{u\to \infty} \int_{(0,1]} u^{-3} y^2 \, \nu_\xi(dy) = 0$, we obtain from Fatou's lemma \begin{eqnarray*} \lefteqn{\liminf_{u\to\infty} u^{-5} \int_{(0,\infty)} f_u(y) \, \nu_\xi(dy) }\\ & = & \liminf_{u\to\infty} \int_{(0,\infty)} u^{-5} \left( f_u(y) + u^2 (C_2/2) y^2 \mathds{1}_{(0,1]} (y) \right) \, \nu_\xi(dy) \\ & \geq & \int_{(0,\infty)} \liminf_{u\to\infty} \left( u^{-5} f_u(y) + u^{-3} (C_2/2) y^2 \mathds{1}_{(0,1]} (y) \right) \, \nu_\xi(dy) \\ & = & \int_{(0,\infty)} \infty \, \nu_\xi(dy) = \infty \, \nu_\xi((0,\infty)).
\end{eqnarray*} Dividing \eqref{eq-relation10} by $u^5$ and observing that $\lim_{u\to\infty} u^{-2} \psi_\eta(u) = -\sigma_\eta^2 /2 < \infty$ (cf. \cite[Lem. 43.11]{sato}) and hence $\lim_{u\to\infty} u^{-5} \psi_\eta(u) = 0$, this together with \eqref{eq-relation11} gives $\nu_\xi((0,\infty)) = 0$. Similarly, dividing \eqref{eq-relation10} by $u^4$, we obtain $\sigma_\xi^2 = 0$ by \eqref{eq-relation11}.
It remains to show that $\nu_\xi((-\infty,0)) = 0$. In doing so, we shall first establish that $\xi$ must be of finite variation. Recall that \begin{eqnarray*} e^{-x} - 1 + x & \geq & 0 \quad \forall\; x \geq 0 \quad \mbox{and}\\ e^{-x} - 1 + x & \geq & x/2 \quad \forall\; x \geq 4. \end{eqnarray*} Let $y <0$. Then $f_u(y) \geq -1$ for $y< -1$, and for $y\in [-1,0)$ we estimate \begin{eqnarray*} f_u(y) & = & e^{-u^2 (e^{-2y}-1)/2} - 1 + u^2 (e^{-2y}-1)/2 - u^2 (e^{-2y}-1) / 2 - u^2 y \\ & \geq & u^2 (e^{-2y}-1)/4 \, \mathds{1}_{ \{u^2 (e^{-2y}-1)/2 \geq 4\} } - C_2 u^2 y^2/2. \end{eqnarray*} An application of Fatou's lemma then shows $$\liminf_{u\to \infty} u^{-2} \int_{(-\infty, 0)} f_u(y) \, \nu_\xi(dy) \geq -C_2 / 2 \, \int_{[-1,0)} y^2 \, \nu_\xi(dy) + \int_{[-1,0)} (e^{-2y}-1)/4 \, \nu_\xi(dy).$$ But since
$\lim_{u\to\infty} u^{-2} |\psi_\eta (u)| < \infty$, dividing \eqref{eq-relation10} by $u^2$ and letting $u\to\infty$ gives $\int_{[-1,0)} (e^{-2y}-1) \, \nu_\xi(dy) < \infty$, hence
$\int_{[-1,0)} |y| \, \nu_\xi (dy) < \infty$, so that $\xi$ is of finite variation. Equation \eqref{eq-relation10} can now be rewritten as \begin{equation}\label{eq-relation12} \psi_\eta(u) + \int_{(-\infty,0)} \left( e^{-u^2 (e^{-2y}-1)/2} -1\right) \, \nu_\xi(dy) = -\gamma_\xi^0 u^2, \end{equation} where $\gamma_\xi^0$ is the drift of $\xi$. Since $\xi_t\to \infty$ as $t\to\infty$ and $\xi$ is spectrally negative, we must have $\gamma_\xi^0 > 0$.
Let $\rho$ denote the standard normal distribution and define the mapping $T$ by $$T: \mathbb{R} \times (-\infty,0) \to \mathbb{R}, \quad (x,y) \mapsto x \sqrt{e^{-2y}-1}.$$ Then for any $\varepsilon > 0$, \begin{eqnarray*} \lefteqn{\int_{(-\infty,-\varepsilon]} \left( e^{-u^2 (e^{-2y}-1)/2} - 1 \right) \, \nu_\xi(dy)} \\
& = & \int_{\mathbb{R}} \int_{\mathbb{R}} \left(e^{iu x \sqrt{e^{-2y}-1} }- 1\right) \, \rho(dx) \,
{\nu_\xi}|_{(-\infty,-\varepsilon]}(dy) \\ & = & \int_{\mathbb{R}} (e^{iuz} - 1) \, T(\rho \otimes
{\nu_\xi}|_{(-\infty,-\varepsilon]}) (dz). \end{eqnarray*} With $$\overline{\psi}_\varepsilon (u) := \psi_\eta(u) + \int_{\mathbb{R}} (e^{iuz} - 1) \, T(\rho \otimes
{\nu_\xi}|_{(-\infty,-\varepsilon]}) (dz)$$ it follows from \eqref{eq-relation12} that $\lim_{\varepsilon \downarrow 0} \overline{\psi}_\varepsilon (u) = -\gamma_\xi^0 u^2$. But since $\overline{\psi}_\varepsilon$ is the L\'evy-Khintchine exponent of an infinitely divisible distribution with L\'evy measure $\nu_\eta +
T(\rho \otimes {\nu_\xi}|_{(-\infty,-\varepsilon]})$, since $T(\rho
\otimes {\nu_\xi}|_{(-\infty,-\varepsilon]})$ is increasing as $\varepsilon \downarrow 0$, and since $u\mapsto -\gamma_\xi^0 u^2$ is the L\'evy-Khintchine exponent of a Gaussian random variable, it follows from \cite[Thm. 8.7]{sato} that $T(\rho \otimes
{\nu_\xi}|_{(-\infty,-\varepsilon]}) = 0$ for any $\varepsilon > 0$, hence $\nu_\xi ((-\infty,0)) = 0$.
We have shown that $\xi_t = \gamma_\xi^0 t$. Injectivity of the mapping $\Phi_\xi$ (cf. Theorem~\ref{cor-phi-xi-injective}) together with the sufficiency part show that necessarily $\eta_t = (2\gamma_\xi)^{1/2} v W_t$, completing the proof. \end{proof}
\section{Continuity}\setcounter{equation}{0} \label{sec-cont}
Another natural question about the mappings $\Phi_\xi$ and $\tilde{\Phi}_\eta$ as defined in Section \ref{sec-inj} is, whether they are continuous. Hereby we say, that $\Phi_\xi$ is continuous, if for each sequence of L\'evy processes $(\eta^{(n)})_{n\in\mathbb{N}}$
such that $\eta_1^{(n)}\overset{d}\to \eta_1$ as $n\to\infty$ and $\mathcal{L}(\eta_1^{(n)})\in D_\xi$, $\mathcal{L}(\eta_1)\in D_\xi$, the sequence $\Phi_\xi(\mathcal{L}(\eta_1^{(n)}))$ converges weakly to $\Phi_\xi(\mathcal{L}(\eta_1))$ as $n\to\infty$, denoted as $\Phi_\xi(\mathcal{L}(\eta_1^{(n)})) \overset{w}\to \Phi_\xi(\mathcal{L}(\eta_1))$ in the following. Continuity of $\tilde{\Phi}_\eta$ is defined similarly. \\ In general $\Phi_\xi$ is not continuous as proven by the following counterexample. We expect that failure of continuity of $\Phi_{\xi_t=t}$ is known as it is a very well studied mapping, but since we were unable to find a ready reference we give a short proof.
\begin{example} \label{ex-2} Let $(\xi_t=t)_{t\geq 0}$ be deterministic. Then $\Phi_\xi$ is not continuous. \end{example} \begin{proof} In the given setting we have that $D_\xi$ is $ID_{\log}$, the set of infinitely divisible distributions with finite log$^+$-moment. Now let $(Y_i^{(n)})_{i\in\mathbb{N}}$ be sequences of i.i.d. random variables such that $$\nu^{(n)}:=\mathcal{L}(Y_1^{(n)})=(1-\frac 1 n )\left( \frac 1 2 \delta_1 + \frac 1 2 \delta_{-1}\right) + \frac 1 n \left( \frac 1 2 \delta_{n^n} + \frac 1 2 \delta_{-n^n}\right)$$ and define the sequence $(Y_i^{(0)})_{i\in\mathbb{N}}$ of i.i.d. random variables with $$\nu^{(0)}:=\mathcal{L}(Y_1^{(0)})=\left( \frac 1 2 \delta_1 + \frac 1 2 \delta_{-1}\right). $$ Then obviously we have $Y_i^{(n)}\overset{d}\to Y_i^{(0)}$ as $n\to \infty$. Now for all $n\in\mathbb{N}_0$ define the compound Poisson process $\eta_t^{(n)}:=\sum_{i=1}^{N_t} Y_i^{(n)}$ where $N$ is a Poisson process with rate $1$, independent of $(Y_i^{(n)})_{i\in \mathbb{N}}$. Then $\mu^{(n)}:=\mathcal{L}(\eta_1^{(n)})\in D_\xi$ for all $n\in \mathbb{N}_0$ and in particular for $n\geq 1$ and $z\in\mathbb{R}$ we have that \begin{eqnarray*} \widehat{\mu^{(n)}}(z)&=&\exp\left(\int_{\mathbb{R}} (e^{izx}-1)\,\nu^{(n)}(dx)\right)= \exp(\widehat{\nu^{(n)}}(z)-1) \\&\overset{n\to\infty}\to& \exp(\widehat{\nu^{(0)}}(z)-1) = \widehat{\mu^{(0)}}(z) \end{eqnarray*} such that $\mu^{(n)}\to \mu^{(0)}$ as $n\to \infty$. But $\phi_\xi(\mu^{(n)})$ does not converge to $\phi_\xi(\mu^{(0)})$ as will be shown in the following. Herefore observe that by \cite[Eq. (17.14)]{sato} the L\'evy measure $\tilde{\nu}^{(n)}$ of $\phi_\xi(\mu^{(n)})$ fulfills for all $n\geq 0$ \begin{equation*} \tilde{\nu}^{(n)}([1,\infty)) = \int_\mathbb{R} \int_0^\infty \mathds{1}_{[1,\infty)}(e^{-s}y) ds \,\nu^{(n)}(dy) = \int_{(0,\infty)} \log y \,\nu^{(n)}(dy) \end{equation*} such that for all $n\geq 1$ $$\tilde{\nu}^{(n)}([1,\infty))= \frac{1}{2} \log n \to \infty \mbox{ as } n\to \infty,$$ whereas $\tilde{\nu}^{(0)}([1,\infty))= 0$. Using \cite[Thm. 8.7]{sato} this shows that $\Phi_\xi(\mu^{(n)}) \not\overset{w}\to \Phi_\xi(\mu)$ as $n\to\infty$, so that $\Phi_\xi$ is not continuous. \end{proof}
Continuity of stationary solutions of random recurrence equations has been studied by Brandt \cite{brandt}. The following is a special case of his result for i.i.d. sequences, but does not assume that $E [\log |B_0^{(n)}|]$, $E [|\log B_0|]$ are finite and that
$E [\log |B_0^{(n)}|] \to E [|\log |B_0|]$ as $n\to\infty$. That these conditions can be omitted follows readily by an inspection of Brandt's proof \cite[Thm. 2]{brandt}.
\begin{proposition} \label{prop-brandt}
Let the sequences $(A_i ,B_i)_{i\in\mathbb{N}_0}$, $(A_i^{(1)} ,B_i^{(1)})_{i\in\mathbb{N}_0}$ $(A_i^{(2)} ,B_i^{(2)})_{i\in\mathbb{N}_0}$, $\ldots$ be i.i.d. such that
$E [\log^+|A_0^{(n)}|] < \infty$, $E [\log^+|B_0^{(n)}|]<\infty$ for all $n$, $E[\log^+|A_0|] < \infty$ and $E [\log^+|B_0|] < \infty$. Assume further that $$-\infty< E[\log |A_0^{(n)}|]<0 \quad \mbox{for all $n$,} \quad \quad -\infty< E[\log |A_0|]<0$$ and that for $n\to \infty$ \begin{eqnarray*}
(A_0^{(n)} ,B_0^{(n)}) &\overset{d}\to& (A_0 ,B_0),\\
E[\log^+ |A_0^{(n)}|] &\to& E[\log^+ |A_0|],\\
E[\log^+ |B_0^{(n)}|] &\to& E[\log^+ |B_0|] \\
\mbox{and }E[\log |A_0^{(n)}|] &\to& E[\log |A_0|]. \end{eqnarray*} Let $Y^{(n)}_\infty$ be the unique stationary marginal distribution of the random recurrence equation $Y^{(n)}_{i+1}=A^{(n)}_i Y_i^{(n)}+ B_i^{(n)}$, $i\in\mathbb{N}_0$, and define $Y_\infty$ analoguously. Then $$(A_0^{(n)} ,B_0^{(n)}, Y^{(n)}_\infty) \overset{d}\to (A_0 ,B_0, Y_\infty)\quad \mbox{as}\quad n\to \infty$$ such that in particular $Y^{(n)}_\infty \overset{d}\to Y_\infty$ as $n\to\infty$. \end{proposition}
Due to the fact that generalized Ornstein-Uhlenbeck processes are the conti\-nuous-time analogon of the solutions to random recurrence equations with i.i.d. coefficients, we can use the above proposition in our setting to obtain the following.
\begin{theorem} \label{thm-continuity}
Let $(\xi^{(n)}, \eta^{(n)})$, $n\in\mathbb{N}$, and $(\xi,\eta)$ be bivariate L\'evy processes such that $$(\xi_1^{(n)}, \eta_1^{(n)}) \stackrel{d}{\to} (\xi_1,\eta_1), \quad n\to\infty.$$ Suppose there exists $\delta>0$ such that \begin{eqnarray} \label{contcond6} \sup_{n\in \mathbb{N}} \int_{\mathbb{R} \setminus [-1,1]} (\log^+
|x|)^{1+\delta} \, \nu_{\eta^{(n)}}(dx) &<& \infty\\
\label{contcond1} \mbox{and}\quad \sup_{n\in \mathbb{N}}
E[|\xi_1^{(n)}|^{1+\delta}]&<&\infty. \end{eqnarray}
Then $E \log^+ |\eta_1| < \infty$ and $E |\xi_1| < \infty$. Assume further that \begin{equation} \label{contcond3}
E \xi_1>0 \quad \mbox{and}\quad E \xi_1^{(n)}>0, \; n\in \mathbb{N}. \end{equation} Then $\int_0^\infty e^{-\xi_{s-}^{(n)}} \, d\eta_s^{(n)}$ converges almost surely absolutely for each $n\in \mathbb{N}$, as does $\int_0^\infty e^{-\xi_{s-}} \, d\eta_s$, and \begin{equation} \label{eq-8.16} \int_0^\infty e^{-\xi_{s-}^{(n)}} \, d\eta_s^{(n)} \stackrel{d}{\to} \int_0^\infty e^{-\xi_{s-}} \, d\eta_s, \quad n\to\infty. \end{equation} \end{theorem}
For the proof of Theorem \ref{thm-continuity} we need the following Lemma, which is of its own interest.
\begin{lemma} \label{lem-8.5} Let $L = (L_t)_{t\geq 0}$ be a L\'evy process in $\mathbb{R}$ with characteristic triplet $(\gamma_L, \sigma_L^2, \nu_L)$. Let $b>0$. Then there exist universal constants $C_1, C_2, C_3 \in (0,\infty)$, depending only on $b$, such that for every adapted c\`adl\`ag process $H$ satisfying
$$E \left( \log^+ \sup_{0\leq s \leq 1} |H_s| \right)^b < \infty$$ the following estimate holds: \begin{eqnarray}
\lefteqn{ E \left( \log^+ \sup_{0<s\leq 1} \left| \int_0^s H_{u-} \, dL_u \right| \right)^b } \nonumber \\
& \leq & C_1 \left( 1 + \sigma^2_L + \int_{|x|\leq 1} x^2 \, \nu_L(dx) +
\log^+ |\gamma_L| + \exp \left\{ C_2 \int_{|x|>1} (\log^+ |x|)^b \, \nu_L(dx) \right\} \right) \nonumber \\
& & + C_3 \, E\left( \log^+ \sup_{0\leq s\leq 1} |H_s| \right)^b . \label{eq-8.12} \end{eqnarray} \end{lemma}
\begin{proof} Write $L_t = L_t^\sharp + L_t^\flat$, where $L^\sharp = (L_t^\sharp)_{t\geq 0}$ has characteristic triplet
$$(\gamma_L^\sharp := 0, (\sigma_L^\sharp)^2 := \sigma^2_L, \nu_L^\sharp := {\nu_L}|_{[-1,1]})$$ and $L^\flat = (L^\flat_t)_{t\geq 0}$ has characteristic triplet
$$(\gamma_L^\flat := \gamma_L, (\sigma_L^\flat)^2 := 0 , \nu_L^\flat := {\nu_L|}_{ \mathbb{R} \setminus [-1,1]}).$$ Then $L^\sharp$ has expectation zero (e.g.~\cite[Ex. 25.12]{sato}) and is a square integrable martingale, and $L^\flat$ is a compound Poisson process together with drift $\gamma_L$. Observe that for proving \eqref{eq-8.12} it is obviously sufficient to prove it for $L^\sharp$ and $L^\flat$ separately, which we shall do.
For the estimate for $L^\sharp$, let $x>0$. Then \begin{eqnarray}
\lefteqn{ P \left( \left( \log^+ \sup_{0<s\leq 1} \left| \int_0^s H_{u-} \, dL_u^\sharp \right| \right)^b > x \right) } \nonumber \\
& = & P \left( \sup_{0<s\leq 1} \left| \int_0^s H_{u-} \, dL_u^\sharp \right| > \exp (x^{1/b}) \right) \nonumber \\
& \leq & P \left( \sup_{0<s\leq 1} \left| \int_0^s H_{u-} \, dL_u^\sharp \right| > \exp (x^{1/b}), \, \sup_{0\leq s \leq 1} |H_s| \leq \exp (x^{1/b}/2) \right) \nonumber \\
& & + P \left( \sup_{0\leq s \leq 1} |H_s| > \exp (x^{1/b} / 2) \right). \label{eq-8.13} \end{eqnarray} Denote $H_s^{(x)} := H_s \wedge \exp (x^{1/b}/2)$. Then on
$\{\sup_{0\leq s\leq 1} |H_s| \leq \exp (x^{1/b}/2)\}$, $\int_0^s H_{u-} \, dL_u^\sharp = \int_0^s H_{u-}^{(x)} \, dL_u^\sharp$ for all $0\leq s \leq 1$, so that by Markov's inequality and Doob's maximal quadratic inequality, we obtain \begin{eqnarray}
\lefteqn{P \left( \sup_{0<s\leq 1} \left| \int_0^s H_{u-} \, dL_u^\sharp \right| > \exp (x^{1/b}), \, \sup_{0\leq s \leq 1} |H_s| \leq \exp (x^{1/b}/2) \right)} \nonumber \\
& \leq & P \left( \sup_{0 \leq s \leq 1} \left| \int_0^1 H_{u-}^{(x)} \, dL_u^\sharp\right| > \exp (x^{1/b}) \right) \nonumber \\
& \leq & \exp (- 2 x^{1/b}) \, E \sup_{0\leq s \leq 1} \left|
\int_0^1 H_{u-}^{(x)} \, dL_u^\sharp \right|^2 \nonumber \\
& \leq & 4 \exp (- 2 x^{1/b}) \, E \left| \int_0^1 H_{u-}^{(x)} \, dL_u^\sharp \right|^2 \nonumber \\
& = & 4 \exp (- 2 x^{1/b}) \, \int_0^1 E |H_{u-}^{(x)}|^2 \, dL_u^\sharp \; \mbox{\rm Var}(L_1^\sharp) \nonumber \\ & \leq & 4 \exp (- 2 x^{1/b}) \exp (x^{1/b})\; \mbox{\rm Var}(L_1^\sharp) \nonumber \\
& = & 4 \exp (- x^{1/b}) \,(\sigma^2_L + \int_{|y|\leq 1} y^2\, \nu_L(dy)), \nonumber \end{eqnarray} where we used \cite[Ex. 25.12]{sato} to express the variance $\mbox{\rm Var}(L_1^\sharp)$ in terms of the characteristic triplet. Combining this with \eqref{eq-8.13}, we obtain \begin{eqnarray*}
\lefteqn{ E \left( \log^+ \sup_{0<s\leq 1} \left| \int_0^s H_{u-} \, dL_u^\sharp \right|\right)^b } \\
& \leq & 4 \left(\sigma^2_L + \int_{|y|\leq 1} y^2 \, \nu_L(dy)\right) \int_0^\infty \exp (-x^{1/b}) \, dx + \int_0^\infty P \left( \left( \log^+
\sup_{0\leq s \leq 1} |H_s|\right)^b > x 2^{-b} \right) \, dx \\
& = & 4 \left(\sigma^2_L + \int_{|y|\leq 1} y^2 \, \nu_L(dy)\right) \int_0^\infty \exp (-x^{1/b}) \, dx + 2^b \, E \left( \log^+
\sup_{0\leq s \leq 1} |H_s| \right)^b, \end{eqnarray*} establishing \eqref{eq-8.12} for $L^\sharp$.
In order to obtain \eqref{eq-8.12} for $L^\flat$, denote
$$R_t := |\gamma_L| t + \sum_{0<s \leq t} |\Delta L_s^\flat|.$$ Then $R=(R_t)_{t\geq 0}$ is a subordinator and \begin{eqnarray}
\lefteqn{ \left( \log^+ \sup_{0\leq s \leq 1} \left| \int_0^s H_{u-}
\, dL_u^\flat \right| \right)^b} \nonumber \\
& \leq & \left( \log^+ \left( R_1 \sup_{0\leq s \leq 1} |H_s| \right) \right)^b \nonumber \\
& \leq & \left( \log^+ R_1 + \log^+ \sup_{0<s\leq 1} |H_s|\right)^b \nonumber \\ & \leq & (2^{b-1} \vee 1) (\log^+ R_1)^b + (2^{b-1} \vee 1) \left(
\log^+ \sup_{0\leq s \leq 1} |H_s| \right)^b . \label{eq-8.14} \end{eqnarray} Since the function $x\mapsto (\log (x\vee e))^b$ is submultiplicative (cf. Sato~\cite[Prop. 25.4]{sato}), it follows from the proof of Theorem~25.3 in Sato~\cite{sato} that there is a constant $C_2 = C_2(b)$, depending only on $b$, such that \begin{equation*}
E \left( \log \left( e \vee \sum_{0<s\leq 1} |\Delta L_s^\flat|
\right)\right)^b \leq \exp \left\{ C_2 \int_{|x|>1} (\log^+ |x|)^b \, \nu_L(dx) \right\}. \end{equation*} Hence, there is a constant $C_4 = C_4(b) \in (0,\infty)$ such that \begin{eqnarray} \lefteqn{ E (\log^+ R_1)^b } \nonumber \\ & \leq & 1 + E (\log (e \vee R_1))^b \nonumber \\
& \leq & C_4 \left( 1 + (\log^+ |\gamma_L|)^b + \exp \left\{ C_2
\int_{|x|>1} (\log^+ |x|)^b \, \nu_L(dx) \right\} \right). \nonumber \end{eqnarray} Together with \eqref{eq-8.14} this gives \eqref{eq-8.12} for $L^\flat$. \end{proof}
\begin{proof}[Proof of Theorem \ref{thm-continuity}] Recall that for any real numbers $a$ and $b$ and $\delta>0$ it holds
$$|a+b|^{1+\delta}\leq C_\delta (|a|^{1+\delta}+|b|^{1+\delta})$$ for some constant $C_\delta$. Using this together with Doob's martingale inequality (c.f. \cite[Eq. (25.16)]{sato}) and Jensen's inequality we obtain \begin{eqnarray*}
E[\sup_{0<s\leq 1} |\xi_s^{(n)}|^{1+\delta} ]
&\leq& C_\delta \left( E[\sup_{0<s\leq 1} |\xi_s^{(n)}-sE[\xi_1^{(n)}] |^{1+\delta}] + |E[\xi_1^{(n)}]|^{1+\delta}\right)\\
&\leq& C_\delta \left( 8 E[|\xi_1^{(n)}-E[\xi_1^{(n)}] |^{1+\delta}] + |E[\xi_1^{(n)}]|^{1+\delta}\right)\\
&\leq& 8 C_\delta^2 E[|\xi_1^{(n)}|^{1+\delta}] + (8C_\delta^2+C_\delta) |E[\xi_1^{(n)}]|^{1+\delta}\\
&\leq& (16C_\delta^2+C_\delta) E[|\xi_1^{(n)}|^{1+\delta}]. \end{eqnarray*} Hence from \eqref{contcond1} we conclude \begin{equation} \label{contcond5b}
\sup_{n\in\mathbb{N}} E [(\sup_{0<s\leq 1} |\xi_s^{(n)}|)^{1+\delta} ]<\infty \end{equation} and therefore also \begin{equation} \label{contcond5}
\sup_{n\in\mathbb{N}} E [(\sup_{0<s\leq 1} |\xi_s^{(n)}\vee 0|)^{1+\delta} ]<\infty . \end{equation} Denote by $(\gamma_{\eta^{(n)}}, \sigma^2_{\eta^{(n)}}, \nu_{\eta^{(n)}})$ and $(\gamma_\eta,\sigma_\eta^2,\nu_\eta)$ the characteristic triplets of $\eta^{(n)}$ and $\eta$, respectively. Denote by $h$ the continuous truncation function $h(x) = x
\mathbf{1}_{|x|\leq 1} + (2-|x|) \mbox{sgn}(x) \, \mathbf{1}_{|x|\in (1,2]}$. Set $$\beta_{\eta^{(n)}} := \gamma_{\eta^{(n)}} + \int_{[-2,2]} (h(x) - x
\mathbf{1}_{|x| \leq 1}) \, \nu_{\eta^{(n)}} (dx) = \gamma_{\eta^{(n)}} + \int_{[-2,2]} x\left(\frac{h(x)}{x} -
\mathbf{1}_{|x| \leq 1}\right) \, \nu_{\eta^{(n)}} (dx),$$ i.e. the constant term in the L\'evy-Khintchine triplet of $\eta^{(n)}$ with respect to the truncation function $h$ (c.f. \cite[Eqs. (8.5), (8.6)]{sato}). Define $\beta_\eta$ similarly. Since $\eta^{(n)}_1 \stackrel{d}{\to} \eta_1$, it follows from \cite[Thm. VII.2.9, p.396]{JacodShiryaev} that $\beta_{\eta^{(n)}} \to \beta_\eta$, \begin{eqnarray*}
\lefteqn{\sigma^2_{\eta^{(n)}} + \int_{|x|\leq 1} x^2 \nu_{\eta^{(n)}} (dx)
+ \int_{1< |x| \leq 2} (2-|x|)^2 \, \nu_{\eta^{(n)}} (dx) }\\
&\to& \sigma^2_{\eta} + \int_{|x|\leq 1} x^2 \nu_{\eta} ( dx) + \int_{1<
|x| \leq 2} (2-|x|)^2 \, \nu_{\eta} (dx) \end{eqnarray*} and $\int_{\mathbb{R}} f(x) \, \nu_{\xi^{(n)}} (dx) \to \int_{\mathbb{R}} f(x) \, \nu_\xi (dx)$ as $n\to\infty$ for every continuous bounded function $f$ vanishing in a neighbourhood of zero. In particular, \begin{equation} \label{eq-8.17} \sup_{n\in \mathbb{N}} \sigma^2_{\eta^{(n)}} < \infty, \quad \sup_{n\in \mathbb{N}} \int_{[-1,1]} x^2 \, \nu_{\eta^{(n)}} (dx) <
\infty \quad \mbox{and} \quad \sup_{n\in \mathbb{N}} |\gamma_{\eta^{(n)}}| < \infty. \end{equation} Applying Lemma~\ref{lem-8.5} with $b=1+\delta$ and $H_s = 1$ and using \eqref{contcond6} then shows that $\sup_{n\in \mathbb{N}} E (\log^+
|\eta_1^{(n)}|)^{1+\delta}<\infty$, and hence that $E (\log^+
|\eta_1|)^{1+\delta} < \infty$ by Fatou's lemma for weak convergence
(cf. Kallenberg~\cite[Lem.~4.11]{kallenberg}), and similarly we obtain $E |\xi_1|^{1+\delta} < \infty$ from \eqref{contcond5b}.
Since $E \log^+|\eta_1| < \infty$, $E \xi_1 > 0$, $E \log^+
|\eta_1^{(n)}| < \infty$ and $E \xi_1^{(n)} > 0$, the integrals $\int_0^\infty e^{-\xi_{s-}} \, d\eta_s$ and
$\int_0^\infty e^{-\xi_{s-}^{(n)}} \, d\eta_s^{(n)}$ converge almost surely absolutely (cf. \cite[Thm. 2]{ericksonmaller05}). Writing $$\int_0^\infty e^{-\xi_{s-}^{(n)}} \, d\eta_s^{(n)} = \int_0^1 e^{-\xi_{s-}^{(n)}} \, d\eta_s^{(n)} + e^{-\xi_1^{(n)}} \int_1^\infty e^{-(\xi_{s-}^{(n)} - \xi_1^{(n)})} \, d(\eta_s^{(n)} - \eta_1^{(n)}),$$ we have $$\int_0^\infty e^{-\xi_{s-}^{(n)}} \, d\eta_s^{(n)} \stackrel{d}{=} \sum_{k=0}^\infty \left( \prod_{i=0}^{k-1} A_i^{(n)} \right) B_k^{(n)}$$ with some i.i.d. sequences $(A_k^{(n)}, B_k^{(n)})_{k\in \mathbb{N}_0}$ such that $$(A_0^{(n)}, B_0^{(n)}) {=} (e^{-\xi_1^{(n)}}, \int_0^1 e^{-\xi_{s-}^{(n)}} \, d\eta_s^{(n)}),$$ and a similar statement holds for $\int_0^\infty e^{-\xi_{s-}} \, d\eta_s$ with $(A_k,B_k)_{k\in \mathbb{N}_0}$ i.i.d. such that $(A_0,B_0) = (e^{-\xi_1}, \int_0^1 e^{-\xi_{s-}} \, d\eta_s)$.
Now, to apply Proposition \ref{prop-brandt}, we have to check its conditions on the sequences $(A_0^{(n)})_{n\in\mathbb{N}}$ and $(B_0^{(n)})_{n\in\mathbb{N}}$ which we shall do in the following.
Since $(\xi_1^{(n)}, \eta_1^{(n)}) \stackrel{d}{\to} (\xi_1,\eta_1)$, $n\to\infty,$ it follows from \cite[Cor. VII.3.6, p. 415]{JacodShiryaev} that $(\xi^{(n)},\eta^{(n)}) \stackrel{\mathcal{L}}{\to} (\xi,\eta)$, where ``$\stackrel{\mathcal{L}}{\to}$'' denotes convergence in the Skorokhod topology. Additionally, the sequences $(\xi^{(n)}), n\in\mathbb{N}$, and $(\eta^{(n)}), n\in \mathbb{N}$, satisfy the P-UT condition (cf. \cite[Def.~VI.6.1, p.~377]{JacodShiryaev}). To see this, let $h:\mathbb{R}^2 \to \mathbb{R}^2$ be a continuous bounded function satisfying $h(x) = x$ in a neighbourhood of 0. Then if $(\gamma_h,A,\nu)_h$ is the L\'evy-Khintchine triplet of $(\xi,\eta)$ with respect to $h$ (we use the notations here as in \cite[Eq. II.4.21, p.~107]{JacodShiryaev}), then $(\gamma_h t, At, dt \, \nu(dx))$, $t\geq 0$, is the semimartingale characteristic of $(\xi,\eta)$ with respect to $h$, cf. \cite[Cor.~II.4.19, p. 107]{JacodShiryaev}. A similar statement holds for $(\xi^{(n)},\eta^{(n)})$. Since $(\xi^{(n)}, \eta^{(n)}) \stackrel{\mathcal{L}}{\to} (\xi,\eta)$ as $n\to\infty$, the sequence $(\xi^{(n)}, \eta^{(n)})$, $n\in\mathbb{N}$, is tight. Furthermore, since again by \cite[Cor. VII.3.6, p. 415]{JacodShiryaev}, $\gamma^{(n)}_h \to \gamma_h$ as $n\to\infty$, and since the total variation of $s\mapsto \gamma^{(n)}_h s$ on
$[0,t]$ is $|\gamma^{(n)}_h| t$, condition (iii) of \cite[Thm. VI.6.15, p. 380]{JacodShiryaev} is satisfied, and it follows from
\cite[Thm. VI.6.21, p. 382]{JacodShiryaev} that $(\xi^{(n)},\eta^{(n)})$, $n\in \mathbb{N}$, is P-UT. Then also $(\eta^{(n)})_{n\in \mathbb{N}}$ is P-UT (cf. \cite[Eq. VI.6.3, p. 377]{JacodShiryaev}).\\ From \cite[Thm.~VI.6.22, p. 383]{JacodShiryaev} it now follows that \begin{equation} \label{eq-8.8} (\xi^{(n)}, \eta^{(n)}, \int_0^{\cdot} e^{-\xi_{s-}^{(n)}} \, d\eta_s^{(n)}) \stackrel{\mathcal{L}}{\to} (\xi,\eta,\int_0^{\cdot} e^{-\xi_{s-}} \, d\eta_s), \quad n\to\infty, \end{equation} in the Skorokhod topology. Since none of the components has a discontinuity at fixed $t\geq 0$ with positive probability, this implies \begin{equation}\label{brandtcond1}
(A_0^{(n)}, B_0^{(n)})\overset{d}\to (A_0,B_0), \quad n\to \infty. \end{equation}
By assumption we have $\log|A_0^{(n)}| = -\xi_1^{(n)}
\stackrel{d}\to -\xi_1 = \log|A_0|$. Since additionally the sequence
$(\log|A_0^{(n)}|)_{n\in\mathbb{N}}$ is uniformly integrable by \eqref{contcond1} (see e.g. \cite[Condition (3.18)]{billingsley}), this yields by \cite[Thm. 3.5]{billingsley} \begin{equation}\label{brandtcond2}
E[\log |A_0^{(n)}|] \to E[\log|A_0|], \quad n\to \infty. \end{equation}
Since \eqref{contcond5} implies $\sup_n E[|\log^+
|A_0^{(n)}||^{1+\delta}]<\infty$ we obtain similarly \begin{equation}\label{brandtcond3}
E[\log^+ |A_0^{(n)}|] \to E[\log^+|A_0|], \quad n\to \infty. \end{equation} Also, it is obvious that \eqref{contcond3} and \eqref{contcond1} yield \begin{equation}\label{brandtcond4}
-\infty<E[\log |A_0|]<0 \quad \mbox{and} \quad -\infty<E[\log
|A_0^{(n)}|]<0. \end{equation} Finally, observe that \eqref{contcond5} implies \begin{equation*} \label{eq-cont1}
\sup_{n\in \mathbb{N}} E[\log^+ \sup_{0<s\leq 1} |e^{-\xi_s^{(n)}}|]^{1+\delta}<\infty \end{equation*} which, together with \eqref{contcond6} and \eqref{eq-8.17}, yields by Lemma \ref{lem-8.5}
$$\sup_{n\in \mathbb{N}} E [\log^+ |B_0^{(n)}|]^{1+\delta} < \infty.$$
Again, this gives $E [\log^+ |B_0|]^{1+\delta} < \infty$ and \begin{equation}\label{brandtcond5}
E[\log^+ |B_0^{(n)}|] \to E[\log^+|B_0|]<\infty, \quad n\to \infty. \end{equation} Now, by Proposition \ref{prop-brandt} we obtain the stated result from \eqref{brandtcond1}, \eqref{brandtcond2}, \eqref{brandtcond3}, \eqref{brandtcond4} and \eqref{brandtcond5}. \end{proof}
From the above theorem we immediately obtain the following corollary on injectivity of $\Phi_\xi$ and $\tilde{\Phi}_\eta$. Observe that the conditions in part $(i)$ have been violated in Example \ref{ex-2}.
\begin{corollary}\begin{enumerate}
\item Let $(\xi_t)_{t\geq 0}$ be a L\'evy process such that $E[\xi_1]>0$ and $E[|\xi_1|^{1+\delta}]<\infty$ for some $\delta>0$. Let $(\eta^{(n)})_{n\in\mathbb{N}}$ be a sequence of L\'evy processes such that $\eta_1^{(n)}\overset{d}\to \eta_1$ as $n\to\infty$, $\mathcal{L}(\eta^{(n)}_1)\in D_\xi$, $\mathcal{L}(\eta_1)\in D_\xi$ and
$$\sup_{n\in\mathbb{N}} \int_{|x|>1} (\log^+|x|)^{1+\delta} \nu_{\eta^{(n)}}(dx)<\infty.$$ Then $\Phi_\xi(\mathcal{L}(\eta^{(n)}_1)) \overset{w}\to \Phi_\xi(\mathcal{L}(\eta_1))$ as $n\to\infty$.
\item Let $(\eta_t)_{t\geq 0}$ be a L\'evy process such that $E[\log^+|\eta_1|^{1+\delta}]<\infty$ for some $\delta>0$. Let $(\xi^{(n)})_{n\in\mathbb{N}}$ be a sequence of L\'evy processes such that $\xi_1^{(n)}\overset{d}\to \xi_1$ as $n\to\infty$, $\mathcal{L}(\xi^{(n)}_1)\in \tilde{D}_\eta$, $\mathcal{L}(\xi_1)\in \tilde{D}_\eta$, $E[\xi_1^{(n)}]>0$, $E[\xi_1]>0$ and
$$\sup_{n\in\mathbb{N}} E[|\xi_1^{(n)}|^{1+\delta}]<\infty.$$ Then $\tilde{\Phi}_\eta(\mathcal{L}(\xi^{(n)}_1)) \overset{w}\to \tilde{\Phi}_\eta(\mathcal{L}(\xi_1))$ as $n\to\infty$. \end{enumerate} \end{corollary}
\end{document} | arXiv |
Infinite-order square tiling
In geometry, the infinite-order square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,∞}. All vertices are ideal, located at "infinity", seen on the boundary of the Poincaré hyperbolic disk projection.
Infinite-order square tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration4∞
Schläfli symbol{4,∞}
Wythoff symbol∞ | 4 2
Coxeter diagram
Symmetry group[∞,4], (*∞42)
DualOrder-4 apeirogonal tiling
PropertiesVertex-transitive, edge-transitive, face-transitive
Uniform colorings
There is a half symmetry form, , seen with alternating colors:
Symmetry
This tiling represents the mirror lines of *∞∞∞∞ symmetry. The dual to this tiling defines the fundamental domains of (*2∞) orbifold symmetry.
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).
*n42 symmetry mutation of regular tilings: {4,n}
Spherical Euclidean Compact hyperbolic Paracompact
{4,3}
{4,4}
{4,5}
{4,6}
{4,7}
{4,8}...
{4,∞}
Paracompact uniform tilings in [∞,4] family
{∞,4} t{∞,4} r{∞,4} 2t{∞,4}=t{4,∞} 2r{∞,4}={4,∞} rr{∞,4} tr{∞,4}
Dual figures
V∞4 V4.∞.∞ V(4.∞)2 V8.8.∞ V4∞ V43.∞ V4.8.∞
Alternations
[1+,∞,4]
(*44∞)
[∞+,4]
(∞*2)
[∞,1+,4]
(*2∞2∞)
[∞,4+]
(4*∞)
[∞,4,1+]
(*∞∞2)
[(∞,4,2+)]
(2*2∞)
[∞,4]+
(∞42)
=
=
h{∞,4} s{∞,4} hr{∞,4} s{4,∞} h{4,∞} hrr{∞,4} s{∞,4}
Alternation duals
V(∞.4)4 V3.(3.∞)2 V(4.∞.4)2 V3.∞.(3.4)2 V∞∞ V∞.44 V3.3.4.3.∞
See also
Wikimedia Commons has media related to Infinite-order square tiling.
• Square tiling
• Uniform tilings in hyperbolic plane
• List of regular polytopes
References
• John H. Conway; Heidi Burgiel; Chaim Goodman-Strass (2008). "Chapter 19, The Hyperbolic Archimedean Tessellations". The Symmetries of Things. ISBN 978-1-56881-220-5.
• H. S. M. Coxeter (1999). "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. ISBN 0-486-40919-8. LCCN 99035678.
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
What Is EBITDAR?
Formula and Calculation
What Does EBITDAR Tell You?
EBITDAR vs. Other Metrics
EBITDAR FAQs
EDITDAR: Meaning, Formula & Calculations, Example, Pros/Cons
Julia Kagan
Julia Kagan is a financial/consumer journalist and senior editor, personal finance, of Investopedia.
Amilcar Chavarria
Reviewed by Amilcar Chavarria
Amilcar has 10 years of FinTech, blockchain, and crypto startup experience and advises financial institutions, governments, regulators, and startups.
Earnings before interest, taxes, depreciation, amortization, and restructuring or rent costs (EBITDAR) is a non-GAAP tool used to measure a company's financial performance. Although EBITDAR does not appear on a company's income statement, it can be calculated using information from the income statement.
EBITDAR is a profitability measure like EBIT or EBITDA that adjusts net income to be internally analyzed by removing certain costs.
It's better for casinos, restaurants, and other companies that have non-recurring or highly variable rent or restructuring costs as these expenses are taken out of net income.
EBITDAR gives analysts a view of a company's core operational performance apart from expenses unrelated to operations, such as taxes, rent, restructuring costs, and non-cash expenses.
Using EBITDAR allows for easier comparison of one firm to another by minimizing unique variables that don't relate directly to operations.
EBITDAR may unjustly remove controllable costs which may not hold management accountable for some costs incurred.
Formula and Calculation of EBITDAR
EBITDAR can be calculated in several different ways. Because EBITDA is a heavily used financial calculation, the most common way is to add restructuring and/or rental costs to EBITDA:
EBITDAR = EBITDA + Restructuring/Rental Costs where: EBITDA = Earnings before interest, taxes, depreciation, and amortization \begin{aligned} &\text{EBITDAR}=\text{EBITDA + Restructuring/Rental Costs}\\ &\textbf{where:}\\ &\text{EBITDA = Earnings before interest, taxes,}\\ &\text{depreciation, and amortization}\\ \end{aligned} EBITDAR=EBITDA + Restructuring/Rental Costswhere:EBITDA = Earnings before interest, taxes,depreciation, and amortization
Different approaches to calculating EBITDAR may start with different earnings or income calculations. In general, the earnings portion refers to net income. This is the all-inclusive, non-restrictive earnings that a company has made in a given period that is not yet adjusted for any items below.
Interest expense is the cost incurred for securing a debt or line of credit with an outstanding balance. A company may choose to eliminate this cost because it may not be controllable by management. In addition, it may be strategically advantageous to have opted to finance something using low-cost debt instead of relying on internal capital or higher-cost methods such as issuing equity shares.
Tax expense is the cost imposed on a company for local, state, or federal taxes. Because a company often does not have a say in its tax assessment, it may be removed for internal analysis. However, companies also have the discretion of forming favorable legal structures to help minimize its tax assessment. Some may argue that if a company fails to strategically plan its future tax liability, it should be held accountable for the taxes assessed when analyzing financial results internally.
Depreciation is the allocated cost of a tangible asset over its useful life. Though a company may outright purchase an asset, it will likely not receive the benefit of the asset all at one time but instead over a period of time. Although there are different depreciation rates and methods, a company may have much control over how depreciation impacts their net income calculation. In addition, companies may not care to see such non-cash transactions when analyzing results.
In a very similar manner as depreciation, amortization is the spreading of costs over the useful life of an asset. However, amortization occurs for intangible assets such as trademarks, patents, and goodwill. The benefit of these items is received over time; however, the worth theoretically deteriorates over time and they become less valuable as they are used or competitors make them obsolete. Just like depreciation, amortization is a non-cash, uncontrollable expense that management may not care to analyze.
Restructuring or Rental Costs
The element that makes EBITDAR different from other calculations is the elimination of restructuring costs or rental costs. These costs may not yield financial results comparable with other companies or comparable for a single company across a period of time. For certain industries and sectors, it may be more favorable to remove these costs when analyzing financial results for reasons discussed below.
EBITDAR is an internal analysis tool only. Though it may be discussed within the notes to a company's financial statements, companies are not required to publicly disclose their EBITDAR calculations.
EBITDAR is a metric used primarily to analyze the financial health and performance of companies that have gone through restructuring within the past year. It is also useful for businesses such as restaurants or casinos that have unique rent costs. It exists alongside earnings before interest and tax (EBIT) and earnings before interest, tax, depreciation, and amortization (EBITDA).
Using EBITDAR in analysis helps to reduce variability from one company's expenses to the next, in order to focus only on costs that are related to operations. This is helpful when comparing peer companies within the same industry.
EBITDAR doesn't take rent or restructuring into account because this metric seeks to measure a company's core operational performance. For example, imagine an investor comparing two restaurants, one in New York City with expensive rent and the other in Omaha with significantly lower rent. To compare those two businesses effectively, the investor excludes their rent costs, as well as interest, tax, depreciation, and amortization.
Similarly, an investor may exclude restructuring costs when a company has gone through a restructuring and has incurred costs from the plan. These costs, which are included on the income statement, are usually seen as nonrecurring and are excluded from EBITDAR to give a better idea of the company's ongoing operations.
EBITDAR is most often calculated for internal purposes only, as it is not a required financial reporting metric for public companies. A firm might calculate it each quarter to isolate and review operational expenses without having to consider fluctuating costs such as restructuring, or rent costs that may differ within various subsidiaries of the company or among the firm's competitors.
EBITDAR Example
Imagine Company XYZ earns $1 million in a year in revenue and incurs $400,000 in total operating expenses. Included in the firm's $400,000 operating expenses is depreciation of $15,000, amortization of $10,000, and rent of $50,000. The company also incurred $20,000 of interest expenses and $10,000 of tax expenses for the period.
Company XYZ can begin by calculating its net income. This is the total amount of revenue less the total amount of expenses.
Net Income = $1,000,000 Revenue - $400,000 Operating Expenses - $20,000 Interest - $10,000 Taxes = $570,000
Company XYZ can then back into EBIT by adding back interest and taxes.
EBIT = $570,000 Net Income + $20,000 Interest + $10,000 Taxes = $600,000.
Company XYZ can further back out additional costs to arrive at EBITDA.
EBITDA = $600,000 EBIT + $15,000 Depreciation + $10,000 Amortization = $625,000
Last, Company XYZ can reincorporate rental costs to arrive at EBITDAR.
EBITDAR = $625,000 EBITDA + $50,000 Rental Expenses = $675,000
EBITDAR can be calculated many different ways. For example, if you know EBITDA, you can simply add restructuring or rent costs. As another example, if you know EBIT, just add back depreciation, amortization, and restructuring/rent costs. The ultimate calculation across all different methods should be the same.
Advantages and Limitations of EBITDAR
Advantages of EBITDAR
EBITDAR is more useful than other financial calculations in several different situations:
EBITDAR removes one-time restructuring costs. As these expenses are often non-recurring, it is less useful to analyze earnings after these one-time costs.
EBITDAR makes certain companies more comparable. By removing rental costs, it becomes more reasonable to compare the operations of different companies without discrepancies arising based on whether the company owns its assets or not.
EBITDAR adjusts for geographical regions with higher costs. Some locations may have higher rent costs based on the nature of that area.
EBITDAR communicates a more controllable earnings calculation. Management can more strategically approach earnings calculations when less controllable elements have been removed.
Limitations of EBITDAR
However, there are several cases where EBITDAR is not as advantageous to use:
EBITDAR manipulates what may be a recurring reorganizational process. Larger companies may restructure their entity very frequently. As this may be an inherent cost of the company, some may argue it is unfair to eliminate this naturally-occurring cost.
EBITDAR may eliminate controllable costs. An organization must still be held responsible for inefficiency if it continually must undergo restructuring. Because EBITDAR "hides" the restructuring cost, management may not take full ownership of this semi-controllable aspect of operations when only looking at this calculation.
EBITDAR does not reflect potentially higher selling prices. The argument is to eliminate rent costs as some areas incur higher expenses; however, these areas may also be subject to geographical pricing and more likely to charge higher rates for their products and incur greater income (which is not adjusted for).
EBITDAR attempts to align reporting to cash activity but may be misleading. A company must still incur cash outlays for interest, taxes, restructurings, and rental costs. By removing these amounts, a company may be misled regarding how much cash it actually goes through in a period.
Strives to exclude non-recurring or one-time expenses
Disregards different capital structures and attempts to compare companies based on their operations only
Adjusts for how different regions may have different costs
Aims to include only the major expenses that management has the ability to control
Removes restructuring costs which may be recurring and part of the normal course of operations for a large company
May remove controllable costs that management should be held accountable for
Does not reduce income for higher cost areas although expenses are adjusted for
May mislead management regarding cash flow needs
EBITDAR vs. Other Financial Calculations
EBITDAR vs. EBITDA
The difference between EBITDA and EBITDAR is that the latter excludes restructuring or rent costs. However, both metrics are utilized to compare the financial performance of two companies without considering their taxes or non-cash expenses such as depreciation and amortization.
A company may choose EBITDAR over EBITDA if it has undergone a recent reorganization that will make it more difficult to analyze year-over-year results. In the year of the reorganization, expenses will likely be higher due to conversions, training, and temporary inefficiencies.
A majority of companies are able to stick with EBITDA because (1) they have not recently undergone a reorganization, (2) they wish to still include the cost of that reorganization as part of their earnings analysis since it may have been controllable, and (3) it is a much more widely accepted earnings calculation.
EBITDAR vs. EBIT
The difference between EBITDAR and EBIT is more substantial. EBIT adjusts earnings for interest and taxes, but it still includes the costs allocated to a good over its useful life. EBIT also includes restructuring and rental costs.
The argument for EBIT is that the cost of depreciable assets is still a controllable cost. Although management may not have full discretion on how long as asset is depreciated for or what its depreciable is, the company still decided to incur the cost of acquiring the asset to use as part of operations. For this reason, depreciation is included in EBIT.
The same concept applies to intangible assets that must be amortized. A company can argue it receives a financial benefit (i.e. greater brand awareness, better product recognition) from goodwill; therefore, because it is recognizing the financial benefit, it must also consider the financial cost (amortization).
Potentially the largest difference between EBITDAR and EBIT relates to cashflow. EBITDAR removes many more non-cash expenses and one-time expenses; therefore, EBITDAR may be a more accurate reflection on what a company will need in terms of cash on a recurring basis. On the other hand, EBIT is usually a greater reflection of what a company's accounting profit will be.
EBITDAR vs. Net Income
The greatest difference lies between EBITDAR and net income. Net income is the ultimate bottom line. It includes all company-wide expenses whether they require cash outlay or not. Net income does not distinguish between different types of costs; all expenses are included.
Net income is heavily dictated by accounting rules and non-cash transactions. Though the financial industry heavily relies on analyzing and comparing net income across companies, there are simply too many variables impact this single calculation to make it truly useful for analysis. This idea stemmed the calculations above; instead of relying on a single, broad number, analysists could choose the aspects of a company to look into by forming different metrics such as EBITDAR.
How Do You Calculate EBITDAR?
EBITDAR is calculated by subtracting interest, taxes, depreciation, amortization, and restructuring/rent costs from earnings. Because EBIT and EBITDA are commonly used measurements as well, a company can calculate EBITDAR by manipulating either of those two measurements. For example, a company can simply subtract depreciation, amortization, and restructuring/rent costs from EBIT.
What Is a Good EBITDAR Margin?
It is not uncommon to see an EBITDA ratio exceed 20%. The general rule of thumb is a strong EBITDA measurement is 10%; because EBITDAR may not be substantially different from EBITDA for many companies, a good EBITDAR margin will be at least double-digits.
What Companies Use EBITDAR?
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\begin{document}
\title[]{Experimental Snapshot Verification of non-Markovianity with Unknown System-Probe Coupling}
\author{Henri Lyyra}\email{[email protected]}
\affiliation{Department of Physics and Nanoscience Center, University of Jyv\"askyl\"a, FI-40014 University of Jyv\"askyl\"a, Finland}
\email{[email protected]}
\author{Olli Siltanen} \affiliation{Laboratory of Quantum Optics, Department of Physics and Astronomy, University of Turku, FI-20014, Turun yliopisto, Finland} \affiliation{Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014, Turun yliopisto, Finland}
\author{Jyrki Piilo} \affiliation{Laboratory of Quantum Optics, Department of Physics and Astronomy, University of Turku, FI-20014, Turun yliopisto, Finland} \affiliation{Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014, Turun yliopisto, Finland}
\author{Subhashish Banerjee} \affiliation{Indian Institute of Technology Jodhpur, Jodhpur 342011, India}
\author{Tom Kuusela} \affiliation{Laboratory of Quantum Optics, Department of Physics and Astronomy, University of Turku, FI-20014, Turun yliopisto, Finland} \affiliation{Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014, Turun yliopisto, Finland}
\begin{abstract} We apply the recently proposed quantum probing protocols with an unknown system-probe coupling to probe the convex coefficients in mixtures of commuting states. By using two reference states instead of one as originally suggested, we are able to probe both lower and upper bounds for the convex coefficient. We perform extensive analysis for the roles of the parameters characterizing the double peaked Gaussian frequency spectrum in the Markovian-to-non-Markovian transition of the polarization dynamics of a single photon. We apply the probing of the convex coefficient to the transition-inducing frequency parameter and show that the non-Markovianity of the polarization dynamics can be confirmed with a single snapshot measurement of the polarization qubit performed at unknown time and even with unknown coupling. We also show how the protocol can identify Markovian and non-Markovian time intervals in the dynamics. The results are validated with single photon experiments. \end{abstract}
\maketitle
\iffalse \section*{Action plan} \begin{enumerate}
\item Find the transition value (from Markovian to non-Markovian) $A_{crit} \in [0,1/2]$ of the amplitude parameter $A$. If the experimental probing data shows that $A < A_{crit}$, we know that the dynamics is Markovian. If the data shows that $ A_{crit} \le A \le 1 - A_{crit}$, the dynamics is non-Markovian. If $A \ge 1 - A_{crit}$, the dynamics is Markovian.~\color{green}\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\color{black} \item Do the analysis for the $\alpha$-fidelities~\color{green}\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\color{black} \item Do the analysis for the trace distance~\color{green}\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\color{black}
\item Find the other parameters such that we can get a definitive answer \em{the dynamics is Markovian} or \em{the dynamics is non-Markovian}, AND they can be experimentally implemented. \textbf{We showed that the Markovian case is impossible.}~\color{green}\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\color{black} \item Combine notes and experimental results. \item Finalize manuscript. \end{enumerate} \fi
\section{Introduction}
Due to its foundational role in realistic quantum systems and the implementability of quantum technologies, the study of open quantum systems has attracted a lot of attention \cite{petru,RHbook,banerjeebook}. Whenever a quantum system interacts with some other system, its environment, the system and environment form together a closed system whose dynamics is unitary. Due to this interaction, the dynamics of the reduced system state is not necessarily unitary and it is said to be an open system. When the open system is used to store or process information, the open system dynamics causes information flow from the system to the environment and correlations between the system and environment. This means that all of the information initially encoded in the open system cannot be retrieved just by measurements on the evolved open system.
Luckily, the loss of information is not always monotonic and in some cases the information has been shown to partially return to the open system as the interaction is prolonged. Multiple definitions of quantum non-Markovianity based on such revivals of information and also other dynamical properties have been proposed and there is no agreement on a single definition \cite{BLP,lorenzomeasure,BCM,hemeasure,RHPreview,BLPreview,vegareview,lihallreview,EPLN-MWhatIs}. In this paper we refer to non-Markovianity as revivals of the trace distance of the optimal pair of initial states, namely the BLP non-Markovianity, which directly quantifies the increases of distinguishability of the state pair during the open system's dynamics \cite{BLP}. Since for our specific dynamics BLP non-Markovianity is equivalent to multiple definitions of non-Markovianity, such as CP divisibility, our results apply also more generally \cite{nonMarkEquivalence,teittinen2018}.
In addition to fundamental interest of the nature of information flow, non-Markovianity has found many uses in making quantum protocols more efficient under noisy circumstances \cite{EPLN-MGoodFor}. Non-Markovianity has been experimentally shown to increase the success rate of Deutsch-Josza algorithm implemented in NV-center in diamond , and allow for perfect superdense coding and quantum teleportation with mixed two photon polarization states in noisy transmission \cite{panalgorithm,karlssonEPL,laine2014,liu2020}. Non-Markovianity has been shown to help in entanglement generation and distribution , and it has been shown to improve the secure key rate in quantum key distribution \cite{huelgarivasplenio2012,Xiang2014,utagi}.
Multiple different measures and indicators of non-Markovianity have been developed but directly experimentally confirming non-Markovianity requires comparison of the system state at two different times, such that the information measure in question is larger at the later time.
Even though the open system dynamics is generally harmful for the information carrier, the information flow into the combined system-environment state can be exploited in specific types of measurements. Since the initial state of the environment influences how the open system evolves in time, measurements on the evolved system can be used to deduce some unknown properties of the environment. Such measurements are useful in situations where direct measurements on the system could hinder the operation of a quantum device - or in the worst case - destroy its information carriers. In these so-called quantum probing measurements, the unitary coupling operator describing the total system-environment dynamics is commonly assumed as known. This assumption is used to form a mapping between the measured open system acting as the probe, and the unknown properties of the environment, which is the system of interest in the measurement \cite{probing2006,probingnonmark2013,probing2013,probing2016,probing2017,haikka}. While using such assumption has proven successful in specific cases, it has certain disadvantages: As the whole measurement strategy is based on one fixed unitary coupling, it has to be faithfully implemented in the experiment. If the coupling is changed, the protocol fails as the connection between the unknown properties and the evolved probe state changes also.
Recently, a new approach to quantum probing was introduced in \cite{tukiainen}. The approach is based on a generalized data processing inequality of the so-called $\alpha$-fidelities which was shown useful for multiple purposes. Among its applications a quantum probing protocol with unknown system-probe coupling was proposed. In the protocol, the generalized data processing inequality is applied to compare two probe states after they have interacted with the system prepared in an unknown state and some reference state. This comparison yields to information of the unknown state.
Later, such protocol was experimentally implemented in a single photon experiment \cite{lyyrasiltanen2020}. In that paper, the upper bound for the width of a Gaussian frequency spectrum was probed by measurements on a polarization probe qubit that had interacted with frequency in a combination of quartz plates rotated in randomly chosen angles, corresponding to an unknown system-probe coupling. These works showed that it is possible to construct and implement measurements whose action is arbitrarily chosen and completely unknown but still result to non-trivial and reliable information.
In this paper, we show how the aforementioned probing protocols based on the generalized data processing inequalities can be used to extract lower and upper bounds for the convex coefficients in mixtures of commuting states by using unknown system-probe coupling. We apply this result in detecting the global feature of non-Markovianity of the dynamics and identifying the Markovian and non-Markovian time intervals from snapshot measurements at unknown time - and even with completely unknown system-probe coupling. Thus we show that the probing protocol can be used to obtain qualitative information about the probe's important dynamical features in addition to the static properties of the system.
Previously, a method for determining the non-Markovianity in terms of the minimal deviation of a snapshot channel from the set of all dynamical semigroups and CP-divisible dynamical maps was developed \cite{WolfEisertCirac}. In contrast, our goal is to determine if the whole dynamical map is non-Markovian and to identify the Markovian and non-Markovian time intervals of the dynamics in terms of information back flow as revivals of trace distance - all from a single snapshot at unknown time and with unknown system-probe coupling. We note here, that a sequence of system state preparations can also be used to define and study non-Markovianity in another way by combining them with a sequence of control operations performed during the dynamics \cite{pollock2018}.
The paper is structured as follows: First, we discuss the necessary background of open quantum systems, quantum probing, and generalized data processing inequalities. Then, we develop the measurement strategy for probing the convex coefficients in mixtures of commuting states with unknown system-probe couplings and list some of its possible applications. After that, we concentrate on a specific quantum optical implementation, namely probing the convex coefficient of a mixture of two Gaussian frequency spectra. We analyze extensively the roles of the parameters of the double peaked spectrum in the non-Markovianity of the polarization dynamics and show that by using our probing measurements, it is possible to verify the non-Markovianity of the polarization dynamics with no assumptions on the system-probe coupling or any of the parameters of the dynamics-inducing frequency state. Finally, the protocol is experimentally realized in an all-optical setup and the results are discussed.
\section{Open Quantum Systems, Quantum Probing, and Generalized Data Processing Inequalities}\label{theory}
We say that a quantum system $A$ is open if it interacts with some other system, the environment $B$. Commonly, it is assumed that $A$ and $B$ are uncorrelated before the interaction begins. The dynamics of the total closed system $AB$ is described by a unitary coupling $U$ which makes $A$ and $B$ interact. The state of the system $A$ after the interaction becomes \begin{equation}\label{reduced} \Phi(\rho) = \text{tr}_B [ U ( \rho \otimes \xi ) U^\dagger ]\,, \end{equation} where $\rho$ and $\xi$ are the initial states of $A$ and $B$, respectively, and tr$_B [X]$ is the partial trace of $X$ over the Hilbert space of the environment $B$. As a concatenation of completely positive and trace preserving (CPTP) maps $\Phi$ is also CPTP, or in other words a \emph{channel}.
The effects of CPTP maps have been widely studied and it has been shown that, in terms of some information measures, information can never increase in channels. One of such commonly used measures is the \textit{trace distance} of states $\rho_1$ and $\rho_2$, defined as \begin{equation} D_{tr} (\rho_1,\rho_2) := \frac{1}{2} \text{tr} \left[ \sqrt{ (\rho_1-\rho_2)^\dagger (\rho_1-\rho_2) } \right]\,, \end{equation} where $\sqrt{X}$ is the unique positive operator satisfying $\sqrt{X}\sqrt{X} = X$ for $X \ge 0$. Trace distance gives directly the probability of distinguishing two equally likely states $\rho_1$ and $\rho_2$ in a single realization of an optimal measurement. Trace distance was shown to satisfy the following \textit{data processing inequality} \begin{equation} D_{tr} (\rho_1,\rho_2) \ge D_{tr} \big(\Phi( \rho_1 ), \Phi( \rho_2 ) \big)\,, \end{equation} for any channel $\Phi$, and any pair of initial states $\rho_1$ and $\rho_2$, meaning that the distinguishability of any pair of states can never increase in CPTP maps \cite{nielsenchuang}. Also the commonly used \textit{fidelity of quantum states}, defined as \begin{equation} F_{1/2} (\rho_1, \rho_2) := \tr \left[ \sqrt{ \sqrt{\rho_2} \, \rho_1 \, \sqrt{\rho_2} } \right]\,. \end{equation} satisfies the following data processing inequality \begin{equation} F_{1/2} (\rho_1, \rho_2) \le F_{1/2} \big( \Phi( \rho_1 ), \Phi( \rho_2 ) \big)\,, \end{equation} which means that the similarity of two states in terms of fidelity can never decrease in CPTP maps.
\iffalse \begin{figure}
\caption{\textbf{The common quantum probing approach.} The system $S$ and the probe $P$ interact under the unitary coupling $U$. After the interaction, measurements on $P$ in the evolved state $\Phi(\rho)$ are used to determine unknown properties of $S$. Here $U$ is known and the unknown parameters of $\xi$ are mapped to the state transformation $\rho \mapsto \Phi(\rho)$ of $P$.}
\label{env_in_dyn}
\end{figure} \fi
Even though the data processing inequalities say that information can be only lost from the open system $A$, it still remains in the state of the total closed system AB whose dynamics is unitary. This feature has been exploited in the so-called \textit{quantum probing} protocols. In quantum probing, the system $S$ has some unknown parameter $x$ and the goal is to evaluate the value of $x$ without making direct measurements on $S$. This can be the case for example when $S$ is a part of a device such as a quantum computer and we want to monitor its behavior while it is running, but direct measurements would disturb or even destroy it.
In quantum probing, direct measurements are avoided by preparing a disposable probe system $P$ in some known state $\rho$ and letting it interact with $S$ under some unitary $U$. Then, measurements on $P$ are used to extract information on $x$. In the above description of open quantum system, probe $P$ corresponds to the open system $A$, and system $S$ is its environment $B$. Equation \eqref{reduced} shows that the channel $\Phi$ that is induced on probe $P$ depends on the initial state $\xi$ of $S$. In the usual quantum probing protocols, the unitary coupling $U$ is known and a mapping between the unknown parameter $x$ in $\xi$ and the transformed probe state $\Phi(\rho)$ is used to evaluate $x$ from measurements on $\Phi(\rho)$. Such protocols depend on using a specific coupling $U$, and if it is not known or cannot be properly implemented, the unknown parameter of $S$ cannot be mapped to the evolved probe state $\Phi(\rho)$ and thus the protocol cannot be applied.
To see how quantum probing can be performed even if $U$ is unknown, let us consider the two cases in Fig.~\ref{env_in_dyn-12}. Once the coupling $U$ is fixed, the dynamics of the probe $P$ depends on the initial state $\xi$ of the system $S$. If $S$ is prepared in different states $\xi_1$ and $\xi_2$ and it is coupled to $P$, this can induce two different channels $\Phi_1$ and $\Phi_2$ on $P$ even if $U$ is the same for both initial states of $S$.
\begin{figure}
\caption{
\textbf{The unknown coupling quantum probing approach.} The system $S$ interacts with the probe $P$. The unitary $U$ is the same in panels a) and b). The analytical form of $\xi_1$ and $\xi_2$ is known, but some parameters in $\xi_1$ and $\xi_2$ are different. Thus, the induced probe channels $\Phi_1$ and $\Phi_2$ may be different. The system-probe coupling $U$ is unknown, and consequently so are the solutions for the channels $\Phi_1$ and $\Phi_2$. Nevertheless, comparing the measured probe states $\Phi_1(\rho_1)$ and $\Phi_2(\rho_2)$ can be used to gain reliable and non-trivial information on the unknown parameter.
}
\label{env_in_dyn-12}
\end{figure}
This observation was combined with the data processing inequality to form a mathematical tool for studying open quantum systems. This tool utilizes the comparisons between the initial system and probe states, and the evolved probe states \cite{tukiainen}. The {\it $\alpha$-fidelity of states} was defined for $\alpha \in (0,1)$ as \cite{tukiainen} \begin{eqnarray} &&F_{\alpha} \big(\rho_1, \rho_2\big) := \tr \left[ \left(\rho_2^{\frac{1-\alpha}{2\alpha}} \rho_1 \, \rho_2^{\frac{1-\alpha}{2\alpha}} \right)^\alpha \right]\,. \end{eqnarray} We note that in the special case $\alpha=1/2$, $F_{1/2}$ becomes the commonly used fidelity of states.
Now, let us consider the $\alpha$-fidelities in the context of Fig.~\ref{env_in_dyn-12}. The unitary coupling $U$ between $P$ and $S$ is fixed but in Fig.~\ref{env_in_dyn-12} a) and b) the initial states of $P$ and $S$ can be different. Thus, different choices of states $\xi_1$ and $\xi_2$ of $S$ induce channels $\Phi_1$ and $\Phi_2$ to $P$ in the interaction, respectively.
\iffalse By exploiting the properties of the quantum Rényi divergence, defined as \cite{renyi1,renyi2} \begin{align}
S_\alpha(\rho_1 || \rho_2) := \left\{ \begin{aligned} &\frac{1}{ \alpha - 1} \ln \left\{ \tr \left[ \left( \rho_2^{ \frac{ 1 - \alpha }{ 2\alpha } } \rho_1 \rho_2^{ \frac{ 1 - \alpha }{ 2\alpha } } \right)^{\alpha} \right] \right\}& &\text{when }\rho_1 \not\perp \rho_2 \\ &\infty& &\text{otherwise}& \end{aligned} \right. \,, \end{align} and the monotonicity of the logarithm function, we get \fi
In this open system picture, it was shown that the $\alpha$-fidelities satisfy the following \textit{generalized data processing inequality} \cite{tukiainen} \begin{eqnarray}\label{eq:mainineq2} F_{\alpha} \big(\rho_1, \rho_2\big) F_{\alpha} \big(\xi_1, \xi_2\big) \leq F_{\alpha} \big(\Phi_1(\rho_1), \Phi_2(\rho_2)\big) \,, \end{eqnarray}
$\forall\,\alpha\in[1/2,1)$ and for all unitary couplings $U$ and initial states $\rho_1$, $\rho_2$, $\xi_1$, and $\xi_2$.
Equation \eqref{eq:mainineq2} allows us to develop new kind of probing protocols which do not require any knowledge of the system-probe coupling. Let us assume that the analytical form of $F_{\alpha} \big(\xi_1, \xi_2\big)$ is calculated and we want to get bounds for some parameters characterizing the system state $\xi_1$ or $\xi_2$. In the protocol, the experimenter lets the system, prepared in states $\xi_1$ and $\xi_2$, interact with probes prepared in known states $\rho_1$ and $\rho_2$, respectively, as in Fig.~\ref{env_in_dyn-12}. Then, the evolved probe states $\Phi_1(\rho_1)$ and $\Phi_2(\rho_2)$ are determined with tomographical measurements. Solving for the unknown parameter in Eq.~\eqref{eq:mainineq2} and inserting the measured density operators in $F_{\alpha} \big(\Phi_1(\rho_1), \Phi_2(\rho_2)\big)$ yields non-trivial bounds for the unknown parameter.
\iffalse The trace distance of states $\rho_1$ and $\rho_2$, is defined as \begin{equation} D_{tr} (\rho_1,\rho_2) = \frac{1}{2} \text{tr} \left[ \sqrt{ (\rho_1-\rho_2)^\dagger (\rho_1-\rho_2) } \right]\,, \end{equation} where $\sqrt{X}$ is the unique operator satisfying $\sqrt{X}\sqrt{X} = X$ for the positive operator $X$. Trace distance gives directly the probability of distinguishing two equally likely states $\rho_1$ and $\rho_2$ in a single realization of an optimal measurement. \fi
By using the subadditivity w.r.t.~tensor products \cite{wilde}, unitary invariance \cite{nielsenchuang}, and the data processing inequality, trace distance can be shown to satisfy the following generalized data processing inequality in the open quantum system picture of Fig.~\ref{env_in_dyn-12} \cite{mikkothesis} \begin{equation}\label{tracedistanceineq} D_{tr} \left( \Phi_1(\rho_1),\Phi_2(\rho_2) \right) \leq D_{tr} \left( \rho_1, \rho_2 \right) + D_{tr} \left( \xi_1, \xi_2 \right)\,, \end{equation} for all unitary couplings $U$ and initial states $\rho_1$, $\rho_2$, $\xi_1$, and $\xi_2$. As described above for $\alpha$-fidelities, also the generalized data processing inequality of trace distance can be used to construct quantum probing protocols with unknown system-probe couplings.
The generalized data processing inequalities \eqref{eq:mainineq2} and \eqref{tracedistanceineq} are our main mathematical tool in the probing protocols. In the next section we use them to construct quantum probing protocols with unknown system-probe couplings to determine lower and upper bounds for convex coefficients in mixtures of commuting states which we apply later in snapshot verification of the probe's non-Markovianity.
\section{Probing of Convex Coefficients}
Let us consider a state of interest, given by $\xi_1 = p \xi_2 + (1 - p)\xi_3$, which is a convex combination of two known commuting states $\xi_2$ and $\xi_3$ with the unknown convex coefficient $p \in [0,1]$. We prepare three probe systems in the states $\rho_1$, $\rho_2$, $\rho_3$ and let them interact with the system in the state $\xi_1$ and the two reference states $\xi_2$ and $\xi_3$, respectively. By measuring the evolved probe states $\Phi_1(\rho_1)$, $\Phi_2(\rho_2)$, and $\Phi_3(\rho_3)$, we can perform a probing measurement to obtain bounds for the convex coefficient $p$.
The commuting states $\xi_2$ and $\xi_3$ can be written diagonal in the same basis $\{\ket{k}\}_{k = 1}^{d^S}$ as $\xi_2 = \sum_k \lambda_k \ket{k}\bra{k}$ and $\xi_3 = \sum_k \nu_k \ket{k}\bra{k}$. Thus we get \begin{align} F_{\alpha} ( \xi_1, \xi_2 )
&= \sum_{k} \left( [ p \lambda_k + (1 - p) \nu_k ] \right)^\alpha \lambda_k^{1-\alpha} \\ &\geq \sum_{k} \left( p \lambda_k \right)^\alpha \lambda_k^{1-\alpha}
= p^\alpha \\ &\Rightarrow F_{\alpha} ( \xi_1, \xi_2 ) \geq p^\alpha \,.\label{commstatefidelity} \end{align}
More specifically, if $\xi_2$ and $\xi_3$ are orthogonal, the $\alpha$-fidelity between the state of interest $\xi_1$ and the first reference state $\xi_2$ becomes \begin{equation}\label{statefidelity} F_\alpha (\xi_1, \xi_2) = p^\alpha\,. \end{equation}
By using Eq.~\eqref{commstatefidelity} or \eqref{statefidelity} in Eq.~\eqref{eq:mainineq2}, we get an upper bound for the convex coefficient as \begin{equation}\label{ainequality} p \leq \left( \frac{F_{\alpha} \big(\Phi_1(\rho_1), \Phi_2(\rho_2)\big)}{F_{\alpha} \big(\rho_1, \rho_2\big)} \right)^{1/\alpha}\,. \end{equation}
Similarly, by using the $\alpha$-fidelity between the state of interest $\xi_1$ and the second reference state $\xi_3$, we get \begin{align} F_{\alpha} ( \xi_1, \xi_3 ) &\geq (1- p)^\alpha \,,\label{commstatefidelity2} \end{align} for commuting $\xi_2$ and $\xi_3$, and \begin{equation}\label{statefidelity2} F_\alpha (\xi_1, \xi_3) = (1-p)^\alpha\,, \end{equation} when $\xi_2$ and $\xi_3$ are orthogonal. As a consequence, we get also a lower bound for the convex coefficient $p$ as \begin{equation}\label{ainequality2} p \geq 1 - \left( \frac{F_{\alpha} \big(\Phi_1(\rho_1), \Phi_3(\rho_3)\big)}{F_{\alpha} \big(\rho_1, \rho_3\big)} \right)^{1/\alpha}\,, \end{equation} whenever $\xi_1 = p \xi_2 + (1 - p)\xi_3$ where $\xi_2$ and $\xi_3$ are orthogonal or commute.
By combining Eqs.~\eqref{ainequality} and \eqref{ainequality2}, we get \begin{gather} 1 - \left( \frac{F_{\alpha_3} \big(\Phi_1(\rho_1), \Phi_3(\rho_3)\big)}{F_{\alpha_3} \big(\rho_1, \rho_3\big)} \right)^{1/\alpha_3} \nonumber \\ \leq ~ p ~ \leq \label{ainequalityfull} \\ \left( \frac{F_{\alpha_2} \big(\Phi_1(\rho_1), \Phi_2(\rho_2)\big)}{F_{\alpha_2} \big(\rho_1, \rho_2\big)} \right)^{1/\alpha_2} \,, \nonumber \end{gather} where $\alpha_2$ and $\alpha_3$ are independent parameters in the interval $[1/2,1)$.
For the trace distance between the state of interest $\xi_1$ and the commuting reference states $\xi_2$ and $\xi_3$ we get \begin{align} D_{tr} (\xi_1,\xi_2)
&= \frac{1}{2} \sum_k (1-p) |\lambda_k - \nu_k | \\ &\leq \frac{1}{2} \sum_k (1-p) (\lambda_k + \nu_k)
= 1-p \\ &\Rightarrow D_{tr} (\xi_1,\xi_2) \leq 1-p \,, \end{align} where we used the same spectral decompositions for $\xi_2$ and $\xi_3$ as above, and similarly \begin{align} D_{tr} (\xi_1,\xi_3) &\leq p\,. \end{align}
If $\xi_2$ and $\xi_3$ are orthogonal, we get \begin{align} D_{tr} (\xi_1,\xi_2) &= 1 - p\,,\\ D_{tr} (\xi_1,\xi_3) &= p\,. \end{align}
As a consequence of Eq.~\eqref{tracedistanceineq}, we get another set of bounds for $p$ as \begin{gather} D_{tr} ( \Phi_1( \rho_1 ), \Phi_3( \rho_3 ) ) - D_{tr} ( \rho_1,\rho_3 ) \nonumber \\ ~ \leq ~ p ~ \leq ~ \label{trainequality} \\ 1 - \left[ D_{tr} ( \Phi_1( \rho_1 ), \Phi_2( \rho_2 ) ) - D_{tr} ( \rho_1,\rho_2 ) \right]\,, \nonumber \end{gather} whenever $\xi_1 = p \xi_2 + (1 - p)\xi_3$ where $\xi_2$ and $\xi_3$ are orthogonal or commute.
\iffalse Finally, we write the tightest lower and upper bounds for the unknown convex coefficient $p$ as \begin{align}\label{boundstotal} \begin{aligned} &\max \left\{ 1 - \left( \frac{F_{\alpha_3} \big(\Phi_1(\rho_1), \Phi_3(\rho_3)\big)}{F_{\alpha_3} \big(\rho_1, \rho_3\big)} \right)^{1/\alpha_3} ,~ \left[ D_{tr} ( \Phi_1( \rho_1 ), \Phi_3( \rho_3 ) ) - D_{tr} ( \rho_1,\rho_3 ) \right] \right\} \\ &\leq p \leq \\ &\min \left\{ \left( \frac{F_{\alpha_2} \big(\Phi_1(\rho_1), \Phi_2(\rho_2)\big)}{F_{\alpha_2} \big(\rho_1, \rho_2\big)} \right)^{1/\alpha_2} ,~ 1 - \left[ D_{tr} ( \Phi_1( \rho_1 ), \Phi_2( \rho_2 ) ) - D_{tr} ( \rho_1,\rho_2 ) \right] \right\}\,. \end{aligned} \end{align} \fi Interestingly, this result does not depend on knowing anything about how the system and probe interact, as it is based on the same approach as studied in \cite{tukiainen,lyyrasiltanen2020}. This means that the protocol is not sensitive to imperfections in the implementation of the system-probe coupling $U$, and the same strategy can be used for multiple different couplings to obtain even tighter bounds. As Eq.~\eqref{ainequalityfull} has the freedom to choose $\alpha_2$ and $\alpha_3$, the bounds corresponding to each measurement of $\Phi_1( \rho_1 ),\,\Phi_2( \rho_2 )$, and $\Phi_3( \rho_3 )$ can be optimized w.r.t.~the $\alpha$ parameters.
The bounds of the convex coefficient $p$ can be used for different purposes. First of all, when $\xi_2 \perp \xi_3$, the purity \cite{teikobook} of the state $\xi_1$ can be given as $P(\xi_1):=\text{tr}[\rho^2] = P(\xi_2)p^2 + P(\xi_3)(1 - p)^2$, so when the orthogonal states $\xi_2$ and $\xi_3$ in the convex combination are fixed, the measured bounds of $p$ can be used to get bounds also for the purity of $\xi_1$.
Secondly, when $\xi_2 \perp \xi_3$, the von Neumann entropy \cite{nielsenchuang,teikobook} of $\xi_1$ becomes $S(\xi_1) = p S( \xi_2 ) + ( 1 - p )S( \xi_3 ) - [ p \ln (p) + (1-p) \ln (1-p)]$, so bounds of $p$ yield bounds for von Neumann entropy, as the states $\xi_2$ and $\xi_3$ are known. If $\xi_1$ represents a two-qubit state, where $\xi_2$ and $\xi_3$ are two different Bell states, the entanglement can be quantified with the concurrence measure \cite{concurrence,lyyraconcurrence} as $C(\xi_1) = |2p - 1|$. As in the case of purity and von Neumann entropy, also bounds of entanglement can be experimentally determined with our approach.
The same strategy can be used to probe upper bounds of the $N$ convex coefficients $p_i$ if the state $\xi_1$ is a mixture $\xi_1 = \sum_{i = 2}^{N+1} p_i \xi_i$ where $\xi_i$ commute. If the eigenbasis of $\xi_1$ is known, it can be written as $\xi_1 = \sum_{i = 1}^{d^S} \lambda_i \ket{\phi_i}\bra{\phi_i}$, where $\ket{\phi_i}$ is the eigenstate corresponding to the eigenvalue $\lambda_i$. As a consequence, our strategy can be used to obtain upper bounds for all the eigenvalues by performing the probing measurement with $d^S$ reference states where $d^S$ is the dimension of the Hilbert space of the system.
Next, we apply our probing protocol to a quantum optical system where the convex coefficient influences the non-Markovianity in the probe dynamics. We perform extensive analysis for the system and determine the critical values of the convex coefficient determining whether non-Markovianity appears in the probe dynamics for any combination of the other system parameters.
\section{Snapshot verification of non-Markovianity}
\iffalse \begin{figure}
\caption{
\textbf{Effect of the amplitude parameter $A$.} The unknown parameter $A$ controls the amplitude relation between two Gaussian frequency peaks. Here, we have denoted $G(\omega) = A G_1(\omega) + (1-A)G_2(\omega)$. The amplitude parameter controls also the qubit dynamics transition from Markovian to non-Markovian and back to Markovian. \textbf{CHANGE THE FIGURE: Three panels, $\xi_1$, $\xi_2$, and $\xi_3$.}
}
\label{gaussians}
\end{figure} \fi
\begin{figure}
\caption{
\textbf{Illustration of the frequency states $\xi_1,\,\xi_2$, and $\xi_3$.} The state of interest $\xi_1$ is a convex combination of the two reference states $\xi_2$ and $\xi_3$, and its spectrum is given by $G(\omega) = A G_1(\omega) + (1-A)G_2(\omega)$. The amplitude parameter $A$, the width of the Gaussians, and the distance of the central frequencies together control the transition of polarization probe's dynamics between Markovian and non-Markovian.
}
\label{gaussians}
\end{figure}
From now on we apply the above results in a specific optical system where the convex coefficient influences whether the dynamics of the probe is Markovian or non-Markovian. Our system of interest is the frequency of a single photon and our probe is the photon's polarization. Usually in the open system literature, the photon's polarization is referred to as the system and its frequency is the environment, but since we probe the frequency by measurements on the polarization, we use the names probe and system.
Our goal is to determine from a snapshot measurement whether the polarization probe's dynamics is Markovian or non-Markovian when the frequency has been prepared in the state $\xi_1$. The state corresponds to a double peaked Gaussian spectrum with an unknown relative peak amplitude parameter $A$, central peak frequencies $\mu_1$ and $\mu_2$, and peak widths $\sigma$. In order to measure the properties, we prepare the frequency of two other photons into reference states $\xi_2$ and $\xi_3$, characterized by single peaked Gaussian spectra. The frequency states can be written as \begin{align}\label{systemStates} \begin{aligned} &\xi_1 = A \xi_2 + (1 - A)\xi_3\,, ~~~~~ A \in [0,1]\,,\\ &\xi_2 = \int G_1( \omega ) \ket{\omega} \bra{\omega} d \omega \,,\\ &\xi_3 = \int G_2( \omega ) \ket{\omega} \bra{\omega} d \omega \,, \\ & \text{where}~ G_k(\omega) = \frac{1}{ \sqrt{2 \pi \sigma^2} } e^{- \frac{ (\omega - \mu_k )^2}{2 \sigma^2}}\,,~k\in \{1,2\} \,. \end{aligned} \end{align} Here, $\sigma$ is the standard deviation, and $\mu_k$ is the mean frequency of the Gaussian distribution $G_k(\omega)$, illustrated in Fig~\ref{gaussians}. Here, $\omega$ are the frequency values appearing with probability $G_k(\omega)$. Since the frequency states $\xi_2$ and $\xi_3$ commute, we can apply the quantum probing protocol to obtain lower and upper bounds for $A$ by using Eqs.~\eqref {ainequalityfull} and \eqref{trainequality}.
\iffalse We assume that the Gaussians $G_1$ and $G_2$ are so far apart that their overlap is neglible. The approximation is justified even if the real Gaussians would have some overlap, since \begin{align} F_{\alpha} ( \xi_1, \xi_2 ) &= \tr \left[ \left( \xi_2^{\frac{1-\alpha}{2\alpha}} \xi_1 \, \xi_2^{\frac{1-\alpha}{2\alpha}} \right)^\alpha \right] \\ &= \tr \left[ \xi_2^{\frac{1-\alpha}{2}} \xi_1^\alpha \, \xi_2^{\frac{1-\alpha}{2}} \right] \\ &= \tr \left[ \xi_1^\alpha \, \xi_2^{1-\alpha} \right] \\ &= \tr \left[ \left( \int [ A G_1( \omega ) + (1 - A) G_2( \omega ) ] \ket{\omega} \bra{\omega} d \omega \right)^\alpha \left( \int G_1( \omega' ) \ket{\omega'} \bra{\omega'} d \omega' \right)^{1-\alpha} \right] \\ &= \tr \left[ \int [ A G_1( \omega ) + (1 - A) G_2( \omega ) ]^\alpha G_1( \omega )^{1-\alpha} \ket{\omega} \bra{\omega} d \omega \right] \\ &= \int [ A G_1( \omega ) + (1 - A) G_2( \omega ) ]^\alpha G_1( \omega )^{1-\alpha} d \omega \\ &\geq \int [ A G_1( \omega ) ]^\alpha G_1( \omega )^{1-\alpha} d \omega \\ &= A^\alpha \,, \end{align} where we have exploited the fact that commuting states are diagonal in the same basis. Thus, we get $A^\alpha \leq F_\alpha ( \xi_1, \xi_2 ) \leq \frac{ F_\alpha ( \Phi_1 ( \rho_1 ), \Phi_2 ( \rho_2 ) ) }{ F_\alpha ( \rho_1, \rho_2) }$ \fi
\iffalse
By applying Eq.~\eqref{boundstotal}, we can use our probing approach to get estimates for the amplitude parameter A as \begin{equation} \max \left\{ 1 - F_{\alpha_3} \big( \Phi_1(\rho), \Phi_3(\rho) \big)^{1/\alpha_3} ,~ 2 D_{tr} \big( \Phi_1( \rho ), \Phi_3( \rho ) \big) \right\} \leq A \leq \min \left\{ F_{\alpha_2} \big( \Phi_1(\rho), \Phi_2(\rho) \big)^{1/\alpha_2} ,~ 1 - 2 D_{tr} \big( \Phi_1( \rho ), \Phi_2( \rho ) \big) \right\}\,. \end{equation}
\fi
When the polarization and frequency interact in a birefringent medium such as a combination of quartz plates with fast axes aligned, the reduced dynamics of the polarization qubit becomes \begin{equation}\label{dephasing} \rho(t) = \Phi^t \big( \rho(0) \big) = \begin{pmatrix} \rho_{HH} & \kappa(t)\rho_{HV}\\ \kappa^*(t)\rho_{VH} & \rho_{VV} \end{pmatrix}\,, \end{equation} where $\kappa(t) = \int f(\omega)e^{i 2 \pi \omega \Delta n t } d \omega$
is the decoherence function and $f(\omega)$ is the frequency spectrum \cite{setuphamiltonian}. Here $\Delta n = n_H - n_V$ is the birefringence of the medium, where $n_H$ and $n_V$ are the refractive indices in the horizontal (H) and vertical (V) directions, respectively. This polarization-frequency model has been recently popular in the studies of quantum information in open quantum systems \cite{liu2011,setuphamiltonian,karlssonLyyra,karlssonEPL,sina2017,liu2018,sina2020,olli,liu2020,lyyrasiltanen2020,olliarxiv}. If the fast axes of the quartz plates in the combination are not aligned, the dynamics becomes significantly more complicated.
We concentrate on the non-Markovianity given by the commonly used BLP measure of non-Markovianity \cite{BLP}, which is based on the trace distance. Trace distance quantifies the distinguishability of two states, which can be interpreted as the amount of information encoded into a sequence of information carriers prepared in the two given states. Thus, increases of trace distance mean increases in information.
If $D_{tr} (\Phi^t(\rho_1),\Phi^t(\rho_2))$ is a monotonically decreasing function of time, the states $\Phi^t(\rho_1)$ and $\Phi^t(\rho_2)$ become less and less distinguishable as time goes on, and the dynamics described by $\Phi^t$ is Markovian in terms of the BLP measure. On the other hand, if $D_{tr} (\Phi^t(\rho_1),\Phi^t(\rho_2))$ increases at some times, the distinguishability increases and thus the dynamics is non-Markovian.
Here, we note that for the pure dephasing dynamics of the form Eq.~\eqref{dephasing} the BLP non-Markovianity is equivalent to many other indicators of non-Markovianity, such as revivals of quantum channel capacity, entanglement assisted classical channel capacity \cite{BCM}, violation of CP-divisibility, Bloch volume, and $l_1$ coherence norm \cite{teittinen2018}, so our results apply directly to them too. For the pure dephasing dynamics, BLP non-Markovianity is critical for tightness of the quantum speed limit bound. Thus, our results have also implications on the optimality of the speed of the state dynamics, while the connection does not exist for the set of all qubit dynamical maps \cite{deffnerlutz,xuspeed,teittinen2019}.
For dynamics of the form Eq.~\eqref{dephasing}, the optimal pair of initial states in the BLP non-Markovianity measure can be chosen as $\rho_1^{opt} = \ket{+}\bra{+}$ and $\rho_2^{opt} = \ket{-}\bra{-}$, where $\ket{\pm} = \frac{1}{\sqrt{2}} ( \ket{H} \pm \ket{V})$ \cite{wissmankarlsson}. The dynamics is BLP non-Markovian if and only if $\frac{d}{d t} |\kappa(t)| > 0$ at some time and otherwise it is BLP Markovian. By analytically solving the decoherence function induced by the frequency in state $\xi_1$, we see that the polarization dynamics is BLP non-Markovian if and only if
\begin{align}\label{nmcondforA} \begin{aligned} h(A) :=(1-A) A &> \frac{ 2\pi \Delta nt \sigma ^2} {\theta ( \Delta n t, \sigma, \Delta\mu )}\,,\text{ and} \\ \theta ( \Delta n t, \sigma, \Delta\mu ) & > 0\,, \end{aligned} \end{align} for some $t > 0$, where we have denoted \begin{align}\nonumber \theta ( \Delta n t, \sigma, \Delta\mu ) := & 4 \pi \Delta nt \sigma ^2 - 4 \pi \Delta nt \sigma ^2 \cos \left( 2 \pi \Delta nt \Delta\mu\right) \\ & - \Delta\mu \sin \left( 2 \pi \Delta n t \Delta\mu \right)\,,\text{ and} \\
\Delta\mu := & |\mu_2 - \mu_1|\,. \end{align}
\iffalse \begin{align}\label{nmcondforA} h(A) &> \frac{ 2\pi \text{$\Delta $nt} \sigma ^2} {4 \pi \text{$\Delta $nt} \sigma ^2 - 4 \pi \text{$\Delta $nt} \sigma ^2 \text{Cos}\left( 2 \pi \text{$\Delta $nt} (\text{$\mu $1}-\text{$\mu $2})\right) -(\text{$\mu $1}-\text{$\mu $2}) \text{Sin}\left( 2 \pi \text{$\Delta $nt} (\text{$\mu $1}-\text{$\mu $2})\right)}\,,~~~~\text{and} \\ 0 &< 4 \pi \text{$\Delta $nt} \sigma ^2 - 4 \pi \text{$\Delta $nt} \sigma ^2 \text{Cos}\left( 2 \pi \text{$\Delta $nt} (\text{$\mu $1}-\text{$\mu $2})\right) - (\text{$\mu $1}-\text{$\mu $2}) \text{Sin}\left( 2 \pi \text{$\Delta $nt} (\text{$\mu $1}-\text{$\mu $2})\right)\,, \end{align} \fi
\iffalse \begin{figure}
\caption{
\textbf{The role of the amplitude parameter $A$ in the Markovian-non-Markovian transisitions of the polarization dynamics.} The qubit dynamics is non-Markovian if $A \in [A_{crit} , 1 - A_{crit} ]$ and Markovian otherwise. The value of $A_{crit}$ is determined by the other parameters of the double-peaked Gaussian spectrum.
}
\label{acrit}
\end{figure} \fi
\iffalse \begin{figure}
\caption{
\textbf{The role of the amplitude parameter $A$ in the Markovian-to-non-Markovian transitions of the polarization dynamics.} \textbf{a)} The qubit dynamics is non-Markovian if $A \in [A_{crit} , 1 - A_{crit} ]$ and Markovian otherwise. The value of $A_{crit}$ depends on the other parameters of the double-peaked Gaussian spectrum. \textbf{b)} The boundary values between Markovian and non-Markovian regions as a function of the rescaled distance of the Gaussian peaks $|\mu_2 - \mu_1| / \sigma$.
}
\label{acrit}
\end{figure}
\begin{figure*}
\caption{
\textbf{The experimental setup.} The photon source PS produces a pair of photons. The idler photon arrives at single photon detector D and triggers the coincidence counter CC to expect the signal photon. First, the frequency-polarization state of the signal photon is prepared in the state preparation stage. The photon goes through a beamsplitter BS1 allowing to manipulate the photon independently in two distinct spatial paths. Interference filters IF1 and IF2 in their own branches are used to filter the frequency spectrum to narrower Gaussians $G_1$ and $G_2$. IF2 can be tilted to shift the central frequency of the Gaussian spectrum in the lower branch. Polarizers P1 and P2 are used to filter the polarizations coming to beamsplitter BS2, which combines the paths resulting to a double-peaked frequency spectrum. As the recombined photon goes through polarizer P3, the photon's initial polarization state $\rho$ is fixed. The relative amplitude $A$ between the Gaussian peaks is controlled by rotating polarizers P1 and P2. Then, the photon arrives at the measurement stage where it first goes through the quartz plate combination (QPs) corresponding to the unitary coupling $U$, where the probe (polarization) and the system (frequency) interact. After the interaction, the photon goes through a combination of a quarter-wave plate (QWP), half-wave plate (HWP), and polarizer P4 before finally arriving at a single photon detector D, which are used together to perform full polarization state tomography to extract $\Phi(\rho)$.
}
\label{experimental_setup}
\end{figure*}
Once the other parameters of the Gaussian distributions are fixed, the convex coefficient $A$ determines directly whether the polarization dynamics is Markovian or non-Markovian in the following way: If we first restrict to $A\in[0,1/2]$, $h(A)$ is a monotonically increasing function in $A$. This means, that there exists $A_{crit}\in [0,1/2]$ such that the dynamics is Markovian for all $A<A_{crit}$ and non-Markovian for all $A \in [A_{crit},1/2]$. On the other hand, if $A\in[1/2,1]$, $h(A)$ is a monotonically decreasing function in $A$ and the opposite holds. Thus, we conclude that the dynamics is non-Markovian if $A \in [A_{crit} , 1 - A_{crit} ]$ and Markovian otherwise.
Motivated by the analysis of pure dephasing channels in \cite{tukiainen,lyyrasiltanen2020}, we choose the initial probe states of the polarization system as $\rho_1 = \rho_2 = \rho_3 = \ket{+}\bra{+}$. After the polarization has interacted with the frequency for an unknown time, we perform tomographic measurements on the evolved states $\Phi_1(\rho_1)$, $\Phi_2(\rho_2)$, and $\Phi_3(\rho_3)$ and calculate the $\alpha$-fidelities $F_{\alpha} \big(\Phi_1(\rho_1), \Phi_2(\rho_2)\big)$ and $F_{\alpha} \big(\Phi_1(\rho_1), \Phi_3(\rho_3)\big)$. As $F_{\alpha} (\rho, \rho) = 1$ and $D_{tr} (\rho, \rho) = 0$ for any state $\rho$, Equations \eqref{ainequalityfull}, and \eqref{trainequality} with the measurement data give us simplified bounds for the amplitude parameter as \begin{align}\label{ainequality3} \begin{aligned} 1 - F_{\alpha_3} \big( \Phi_1(\rho), \Phi_3(\rho) \big)^{1/\alpha_3} \leq
A \leq F_{\alpha_2} \big( \Phi_1(\rho), \Phi_2(\rho) \big)^{1/\alpha_2}\,,\\
D_{tr} \big( \Phi_1( \rho ), \Phi_3( \rho ) \big)
\leq A \leq 1 - D_{tr} \big( \Phi_1( \rho ), \Phi_2( \rho ) \big)\,. \end{aligned} \end{align} If any of the experimentally dertermined upper bounds is below $A_{crit}$, we immediately know that the polarization dynamics is Markovian, and similarly if any of the lower bounds is above $1 - A_{crit}$. On the other hand, if any of the lower bounds is larger than $A_{crit}$ and any upper bound is smaller than $1-A_{crit}$, the dynamical map of the polarization qubit is non-Markovian. The Markovian and non-Markovian regions of the $A$ parameter are illustrated in Fig.~\ref{acrit} \textbf{a)} .
In Fig.~\ref{acrit} \textbf{b)}, we have numerically calculated $A_{crit}$ from the non-Markovianity condition Eq.~\eqref{nmcondforA} as a function of the ratio between the other free parameters $\Delta\eta:=\Delta\mu / \sigma$.
The numerical analysis suggests that whenever $\Delta\eta > 0$, there exists a non-empty interval $[A_{crit}, 1 - A_{crit}]$ such that the polarization dynamics is non-Markovian for all $A$ within the interval.
\iffalse Let us now conentrate on the non-Markovian region for the extremal values of $\Delta\eta$. We notice that with small values of the rescaled control parameter $\Delta\eta$, the non-Markovianity condition in Eq.~\eqref{nmcondforA} takes the form \begin{equation} h(A)>\frac{1}{\Delta\eta^2(\tau^2-1)}, \label{small_cp} \end{equation} where $\tau:=2\pi\sigma\Delta nt$. The inequality holds only, when enough time has passed, i.e., when \begin{equation} \tau>\sqrt{1+\frac{1}{\Delta\eta^2h(A)}}\approx\frac{1}{\Delta\eta\sqrt{h(A)}}. \label{nm_time} \end{equation}
By the time Eq. \eqref{nm_time} holds, the revivals of $|\kappa(t)|$ are negligible \textbf{(doesn't this mean, that the dynamics would be Markovian then?)}. Still, Eqs. \eqref{small_cp} and \eqref{nm_time} illustrate the sensitivity of the non-Markovian dynamics with respect to the control parameter; Even with small values of $\Delta\eta$, it is possible to find a convex coefficient $A$ that leads to non-Markovian polarization dynamics. \textbf{This paragraph should be clarified.} \fi
It seems that when $\Delta\eta$ is large, even small values of $A$ produce non-Markovian dynamics. The function fitted to Eq.~\eqref{nmcondforA} illustrates this well. The non-Markovian region $[A_{crit},1-A_{crit}]$ is given by
\begin{equation} A_{crit}(\Delta\eta) = 0.0885553 e^{-0.0870419 \Delta \eta ^2} + \frac{0.411445}{0.0845395 \Delta \eta ^2+1}\,. \label{fit} \end{equation}
Thus, $A_{crit}$ decreases monotonically as a function of $\Delta\eta$ and consequently, the larger $|\mu_2-\mu_1|$ (or the smaller $\sigma$), the larger the non-Markovian set of $A$ values $[A_{crit},1-A_{crit}]$, and vice versa.
\begin{figure*}\label{resultset1_2_3}
\end{figure*}
Recently, a method for probing the lower bound for $\Delta\eta$ was experimentally implemented in the cases of unknown coupling between the system (frequency) and probe (polarization) \cite{lyyrasiltanen2020}. For our notation, the probed bound is given by \begin{equation}\label{etabounds}
\Delta\eta \ge \sqrt{ \frac{ 2 \ln \left[ F_\alpha \left( \Phi_2 (\rho_2), \Phi_3 (\rho_3) \right) / F_\alpha \left( \rho_2, \rho_3 \right) \right] }{ \alpha(\alpha - 1) } }\,. \end{equation}
Since $A_{crit}$ decreases monotonically in $\Delta\eta$, probing a lower bound for $\Delta\eta$ and inserting it in Eq.~\eqref{fit} would give us a pessimistic upper bound $\tilde A_{crit}$ for $A_{crit}$. If our measured bounds of $A$ are between $\tilde A_{crit}$ and $1 - \tilde A_{crit}$, then $A$ is guaranteed to be between the actual values $A_{crit}$ and $1 - A_{crit}$. Thus, the unknown coupling probing protocol can be used to extract appropriate bounds for each parameter to confirm that the dynamics is non-Markovian.
\iffalse
\begin{figure}
\caption{(Color online) The tightest possible upper and lower bounds probed with the $\alpha$-fidelity (blue and red dots) optimized w.r.t. time, $\alpha$'s, and $A$ independently, $A_{crit}$ and $1-A_{crit}$ (black diamonds and squares), and the corresponding fitted curves (cyan and pink solid lines) as functions of $\frac{| \mu_2-\mu_1 |}{\sigma}$.}
\label{numerical_plots}
\end{figure}
Now, let us discuss the limitations of our probing measurements when using the unitary coupling of aligned quartz plates in confirming Markovianity of the polarization dynamics. To conclude whether the dynamics is Markovian, the experimentally obtained upper bound for the convex coefficient must be less than the convex coefficient's critical value, i.e., \begin{equation} F_{\alpha_2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^{1/\alpha_2}<A_{crit}. \label{concl_Mark1} \end{equation} Alternatively, the experimentally obtained lower bound must satisfy \begin{equation} 1-F_{\alpha_3}\big(\Phi_1(\rho),\Phi_3(\rho)\big)^{1/\alpha_3}>1-A_{crit}. \label{concl_Mark2} \end{equation} Numerical analysis shows that $F_{\alpha_2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^{1/\alpha_2}$ grows with $\alpha_2$ and $A$. Since smaller $F_{\alpha_2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^{1/\alpha_2}$ and $F_{\alpha_3}\big(\Phi_1(\rho),\Phi_3(\rho)\big)^{1/\alpha_3}$ lead to tighter bounds in Eq.~\eqref{ainequality3}, we use $\alpha_2 = \alpha_3 = 1/2$ in the measurement data analysis when probing the bounds of $A$. To see if Eq.~\eqref{concl_Mark1} is ever satisfied, we may fix $\alpha_2=1/2$, $A=0$, and only consider $\min_t \Big\{F_{1/2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^2\big\vert_{A=0}\Big\}$, where the minimum is taken over all system-probe interaction times $t$. We have plotted the values of $\min_t \Big\{F_{1/2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^2\big\vert_{A=0}\Big\}$ as a function of $\Delta\eta$ in Fig.~\ref{numerical_plots}, where the values of $A_{crit}$ are shown for comparison. We notice that the values of $A_{crit}$ are in excellent agreement with $\min_t\Big\{F_{1/2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^2\big\vert_{A=0}\Big\}/2$ for all choices of $\Delta\eta$. Thus, we can estimate that \begin{equation} F_{\alpha_2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^{1/{\alpha_2}} \geq\min_t \Big\{F_{1/2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^2\big\vert_{A=0}\Big\} \approx2A_{crit}>A_{crit}. \label{estimation1} \end{equation} A similar analysis for the lower bound results in the estimation \begin{equation} 1-F_{\alpha_3}\big(\Phi_1(\rho),\Phi_3(\rho)\big)^{1/{\alpha_3}}\leq\max_t \Big\{1-F_{1/2}\big(\Phi_1(\rho),\Phi_3(\rho)\big)^2\big\vert_{A=1}\Big\}\approx1-2A_{crit}<1-A_{crit}. \label{estimation2} \end{equation} Combining Eqs.~\eqref{estimation1} and~\eqref{estimation2}, we conclude that, for any choice of parameters $A$, $\mu_1$, $\mu_2$, $\sigma$, $\alpha_2$, $\alpha_3$, and $t$, we get \begin{equation} [1-F_{\alpha_3}\big(\Phi_1(\rho),\Phi_3(\rho)\big)^{1/{\alpha_3}},F_{\alpha_2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^{1/{\alpha_2}}]\cap[A_{crit},1-A_{crit}]\neq\emptyset, \label{conclusion} \end{equation} meaning that the bounds obtained by the $\alpha$-fidelity cannot be used to confirm Markovianity of the qubit dynamics when the quartz plates are aligned in the same orientation.
We can derive the same result for the bounds obtained by the trace distance by using the estimation $A_{crit}\approx\frac{1}{2}\min_t \Big\{F_{1/2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^2\big\vert_{A=0}\Big\}$. Below, we show that the experimentally obtained upper bound satisfies $1-D_{tr}\big(\Phi_1(\rho),\Phi_2(\rho)\big)\geq A_{crit}$ and thus cannot confirm Markovianity. Using the shorthand notation $\tau=2\pi\sigma\Delta nt$, we get \begin{align} \Big[e^{-\frac{1}{2}\tau^2}\sin^2\Big(\frac{\Delta\eta\tau}{2}\Big)-e^{\frac{1}{2}\tau^2}\Big]^2&\geq 0 ~~\forall~\tau \ge 0\\ \Leftrightarrow 2\sin^2\Big(\frac{\Delta\eta\tau}{2}\Big)&\leq e^{\tau^2}+e^{-\tau^2}\sin^4\Big(\frac{\Delta\eta\tau}{2}\Big) ~~\forall~\tau \ge 0\\ \Leftrightarrow 2-2\cos(\Delta\eta\tau)&\leq e^{\tau^2}-\cos(\Delta\eta\tau)+1+\frac{1}{4}e^{-\tau^2}\big[\cos(\Delta\eta\tau)-1\big]^2 ~~\forall~\tau \ge 0\\ \Leftrightarrow\sqrt{2-2\cos(\Delta\eta\tau)}&\leq e^{\frac{1}{2}\tau^2}-\frac{1}{2}e^{-\frac{1}{2}\tau^2}\big[\cos(\Delta\eta\tau)-1\big] ~~\forall~\tau \ge 0\\ \Leftrightarrow 1-\frac{1}{2}e^{-\frac{1}{2}\tau^2}\sqrt{2-2\cos(\Delta\eta\tau)}&\geq\frac{1}{2}\Big\{1+\frac{1}{2}e^{-\tau^2}\big[\cos(\Delta\eta\tau)-1\big]\Big\} ~~\forall~\tau \ge 0\\ \Rightarrow 1-\frac{1-A}{2}e^{-\frac{1}{2}\tau^2}\sqrt{2-2\cos(\Delta\eta\tau)}&\geq\frac{1}{2}\min_t \Big\{1+\frac{1}{2}e^{-\tau^2}\big[\cos(\Delta\eta\tau)-1\big]\Big\}\,,~~\forall\,A\in[0,1]\\ \Leftrightarrow1-D_{tr}\big(\Phi_1(\rho),\Phi_2(\rho)\big)&\geq\frac{1}{2}\min_t \Big\{F_{1/2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^2\big\vert_{A=0}\Big\}\\ \Leftrightarrow 1-D_{tr}\big(\Phi_1(\rho),\Phi_2(\rho)\big)&\geq A_{crit} \end{align} \iffalse \begin{align} 1-D_{tr}\big(\Phi_1(\rho),\Phi_2(\rho)\big)&\geq A_{crit}\\
\Leftrightarrow1-D_{tr}\big(\Phi_1(\rho),\Phi_2(\rho)\big)&\geq\frac{1}{2}\min_t \Big\{F_{1/2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^2\big\vert_{A=0}\Big\}\\
\Leftrightarrow 1-\frac{1-A}{2}e^{-\frac{1}{2}\tau^2}\sqrt{2-2\cos(\Delta\eta\tau)}&\geq\frac{1}{2}\min_t \Big\{1+\frac{1}{2}e^{-\tau^2}\big[\cos(\Delta\eta\tau)-1\big]\Big\}\\
\Leftarrow 1-\frac{1}{2}e^{-\frac{1}{2}\tau^2}\sqrt{2-2\cos(\Delta\eta\tau)}&\geq\frac{1}{2}\Big\{1+\frac{1}{2}e^{-\tau^2}\big[\cos(\Delta\eta\tau)-1\big]\Big\}\\
\Leftrightarrow\sqrt{2-2\cos(\Delta\eta\tau)}&\leq e^{\frac{1}{2}\tau^2}-\frac{1}{2}e^{-\frac{1}{2}\tau^2}\big[\cos(\Delta\eta\tau)-1\big]\\
\Leftrightarrow 2-2\cos(\Delta\eta\tau)&\leq e^{\tau^2}-\cos(\Delta\eta\tau)+1+\frac{1}{4}e^{-\tau^2}\big[\cos(\Delta\eta\tau)-1\big]^2\\
\Leftrightarrow 2\sin^2\Big(\frac{\Delta\eta\tau}{2}\Big)&\leq e^{\tau^2}+e^{-\tau^2}\sin^4\Big(\frac{\Delta\eta\tau}{2}\Big)\\
\Leftrightarrow\Big[e^{-\frac{1}{2}\tau^2}\sin^2\Big(\frac{\Delta\eta\tau}{2}\Big)-e^{\frac{1}{2}\tau^2}\Big]^2&\geq 0, \end{align} \fi which proves the claim. Similar calculation, using $A_{crit}\approx\frac{1}{2}\min\Big\{F_{1/2}\big(\Phi_1(\rho),\Phi_3(\rho)\big)^2\big\vert_{A=1}\Big\}$, holds for the lower bounds.
Thus, we conclude that combinations of quartz plates in the same orientation cannot be used as the system-probe coupling to confirm the Markovianity of the probe dynamics. In this analysis, we concentrated on the global Markovianity, meaning that there are no revivals of the trace distance at any time $t\in [0,\infty)$. If instead we are more interested in local Markovianity, meaning monotonicity of trace distance on some finite interval $[t_1,t_2]$, the amount of trace distance revivals decreases - and consequently - the value of $A_{crit}$ increases. Thus, by restricting our interest to shorter intervals, we can confirm the Markovianity of the dynamics. In this approach, the same measurement data and Eq.~\eqref{ainequality3} can be used, but only the values of $A_{crit}$ should be calculated again according to the time interval when interpreting the results. Similar approach can be applied to cases where we are interested in the non-Markovianity at some specific time intervals. \fi
In the Appendix we analyze the limitations of using multiple quartz plates in the same orientation as system-probe coupling to verify that the probe dynamics is Markovian at all times $t>0$, or in other words, the global Markovianity of the probe dynamics. We conclude that such system-probe coupling always fails in that task. Even though that coupling cannot be used to determine the global Markovianity, we may use the same approach of unknown couplings as in \cite{lyyrasiltanen2020} and choose the rotation angles of each plate randomly. In \cite{lyyrasiltanen2020} random rotation angles improved the precision of the probing. In such a situation, the analysis of the measurement data is exactly the same, as one needs to just use the measured probe states in Eqs.~\eqref{ainequality3} and \eqref{etabounds} to extract the bounds for $A$ and $\Delta\eta$, respectively.
Contrary to the analysis of global Markovianity, the polarization's global non-Markovianity and Markovian time intervals can be conclusively determined with quartz plates in the same orientation, as we will see in section \ref{probing_results}. Next, we present our quantum optical experimental setup and use it to obtain the bounds for the parameters $A$ and $\Delta \eta$ and eventually determine if the polarization dynamics is non-Markovian.
\section{The experimental setup}
Our experimental setup is presented in Fig.~\ref{experimental_setup}. First, a pair of photons with wide frequency spectra is produced in a photon source (PS) by spontaneous parametric down-conversion process when a type I beta-barium borate crystal is pumped with a tightly focused continuous wave laser at the wavelength 405 nm. One of the photons, the idler, is guided directly to a single photon detector D which sends a trigger to the coincidence counting electronics to expect the signal photon at the other detector.
The signal photon arrives at a beamsplitter (BS1), which turns the photon into a spatial superposition of branch 1 and branch 2. In branches 1 and 2, the photon passes through the interference filters IF1 and IF2 with FWHM of 3 nm, respectively. The transmission bands of IF1 and IF2 have different central frequencies $\mu_1$ and $\mu_2$, which filters the frequency spectrum in each branch to the Gaussian distribution $G_1$ and $G_2$. IF2 can be also tilted, and thus the distance of the Gaussians $\Delta\mu$ can be adjusted.
In the branches, the photon goes through the polarizers P1 and P2 placed in rotation stations. Then, the branches are recombined with another beamsplitter (BS2), after which the photon passes through a third polarizer that prepares the initial polarization states $\rho_1 = \rho_2 = \rho_3 = \ket{+}\bra{+}$. Together with the polarizers in branches 1 and 2, the third polarizer P3 controls the amplitudes of the Gaussians $G_1$ and $G_2$ by dimming the Gaussian from each branch. The dimming of each Gaussian, and thus the value of the amplitude parameter $A$, is determined by the relative rotation angles between the branch polarizers P1 and P2 and the third polarizer, giving us high control of the state of interest $\xi_1$. Each of the branches can also be independently blocked, resulting to the reference states $\xi_2$ and $\xi_3$.
After the polarizer P3, the photon goes through a combination of quartz plates (QPs). In the quartz plates, the system (frequency) and probe (polarization) are coupled by unitary $U$, causing the probe to experience dephasing dynamics described by Eq.~\eqref{dephasing}. In the experiment, the interaction time is given by the thickness of the used QP combination.
Once the interaction ends, the photon goes through a combination of a quarter-wave plate (QWP), half-wave plate (HWP), and polarizer P4, after which it is guided into a single photon detector. The waveplate combination and the polarizer P4 are used to perform tomography for the polarization qubit to obtain the evolved probe states $\Phi_1 (\rho)$, $\Phi_2 (\rho)$, and $\Phi_3 (\rho)$.\\
\iffalse
\begin{figure}
\caption{
\textbf{The probed values of the convex coefficient $A$.} The blue crosses are the lower and upper bounds obtained from the $\alpha$-fidelity and the red slanted crosses are the bounds obtained from trace distance. The blue solid and red dashed lines are the theoretical predictions for the bounds of the matching color, calculated taking into account dispersion in quartz. The black dashdotted lines are the values of $A_{crit}$ and $1 - A_{crit}$, limiting the non-Markovian region. The $A_{crit}$ values were obtained by probing of $\Delta \mu / \sigma$, using the probing method in \cite{lyyrasiltanen2020}. Here, $\lambda_1 = 810$ nm, $\lambda_2 = 830$ nm, and $A = 0.5122$.
}
\label{resultset1}
\end{figure}
\begin{figure}
\caption{
\textbf{The probed values of the convex coefficient $A$.} The blue crosses are the lower and upper bounds obtained from the $\alpha$-fidelity and the red slanted crosses are the bounds obtained from the trace distance. The blue solid and red dashed lines are the theoretical predictions for the bounds of the matching color, calculated taking into account dispersion in quartz. The black dashdotted lines are the values of $A_{crit}$ and $1 - A_{crit}$, limiting the non-Markovian region. The $A_{crit}$ values were obtained by probing of $\Delta \mu / \sigma$, using the probing method in \cite{lyyrasiltanen2020}. Here, $\lambda_1 = 810$ nm, $\lambda_2 = 830$ nm, and $A = 0.6377$.
}
\label{resultset2}
\end{figure}
\begin{figure}
\caption{
\textbf{The probed values of the convex coefficient $A$.} The blue crosses are the lower and upper bounds obtained from the $\alpha$-fidelity and the red slanted crosses are the bounds obtained from the trace distance. The blue solid and red dashed lines are the theoretical predictions for the bounds of the matching color, calculated taking into account dispersion in quartz. The black dashdotted lines are the values of $A_{crit}$ and $1 - A_{crit}$, limiting the non-Markovian region. The $A_{crit}$ values were obtained by probing of $\Delta \mu / \sigma$, using the probing method in \cite{lyyrasiltanen2020}. Here, $\lambda_1 = 810$ nm, $\lambda_2 = 820$ nm, and $A = 0.6377$.
}
\label{resultset3}
\end{figure}
\begin{figure}
\caption{
\textbf{The probed values of the convex coefficient $A$.} The blue crosses are the lower and upper bounds obtained from the $\alpha$-fidelity and the red slanted crosses are the bounds obtained from the trace distance. The blue solid and red dashed lines are the theoretical predictions for the bounds of the matching color, calculated taking into account dispersion in quartz. The black dashdotted lines are the values of $A_{crit}$ and $1 - A_{crit}$, limiting the non-Markovian region. The $A_{crit}$ values were obtained by probing of $\Delta \mu / \sigma$, using the probing method in \cite{lyyrasiltanen2020}. Here, $\lambda_1 = 810$ nm, $\lambda_2 = 818$ nm, and $A = 0.7$.
}
\label{resultset5}
\end{figure}
\begin{figure}
\caption{
\textbf{The probed values of the convex coefficient $A$.} The blue crosses are the lower and upper bounds obtained from the $\alpha$-fidelity and the red slanted crosses are the bounds obtained from the trace distance. The blue solid and red dashed lines are the theoretical predictions for the bounds of the matching color, calculated taking into account dispersion in quartz. The black dashdotted lines are the values of $A_{crit}$ and $1 - A_{crit}$, limiting the non-Markovian region. The $A_{crit}$ values were obtained by using the known $\Delta \mu / \sigma$. Here, $\lambda_1 = 810$ nm, $\lambda_2 = 818$ nm, and $A = 0.7$.
}
\label{resultset5-known_eta}
\end{figure}
\begin{figure}
\caption{
\textbf{The probed values of the convex coefficient $A$.} Probed bounds obtained with unknown QP rotation angles. The blue crosses are the lower and upper bounds obtained from the $\alpha$-fidelity and the red slanted crosses are the bounds obtained from the trace distance. The black dashdotted lines are the values of $A_{crit}$ and $1 - A_{crit}$, limiting the non-Markovian region. The $A_{crit}$ values were obtained by probing of $\Delta \mu / \sigma$, using the probing method in \cite{lyyrasiltanen2020}. Here, $\lambda_1 = 810$ nm, $\lambda_2 = 818$ nm, and $A = 0.7$.
}
\label{resultset6}
\end{figure}
\begin{figure}
\caption{
\textbf{The probed values of the convex coefficient $A$.} Probed bounds obtained with unknown QP rotation angles. The blue crosses are the lower and upper bounds obtained from the $\alpha$-fidelity and the red slanted crosses are the bounds obtained from the trace distance. The black dashdotted lines are the values of $A_{crit}$ and $1 - A_{crit}$, limiting the non-Markovian region. The $A_{crit}$ values were obtained by using the known $\Delta \mu / \sigma$. Here, $\lambda_1 = 810$ nm, $\lambda_2 = 818$ nm, and $A = 0.7$.
}
\label{resultset6-known_eta}
\end{figure} \fi
\section{Measurement results}\label{probing_results}
\subsection{Global non-Markovianity}\label{global_n-m}
We present our experimental results in Figs.~\ref{resultset1_2_3}-\ref{resultset5_6}. In each case, we have used different choices for the central frequencies $\mu_k = c/\lambda_k$ of the Gaussians or the convex coefficient $A$, but the standard deviation $\sigma$ of the Gaussians is kept fixed. For each choice of the parameters, we let the probe (polarization) interact with the system (frequency) and performed full tomography of the evolved probes for each initial system state. This was repeated for multiple system-probe couplings, corresponding to different thicknesses of QP combinations, as illustrated by the horizontal axes of the figures. We note here that the thicknesses are shown only to compare the measurement data with the theoretical predictions and they were not used to make any conclusions about non-Markovianity.
For each coupling, the probed lower and upper bounds for $A$ were obtained by using the results of evolved probe state tomographies $\Phi_1 (\rho)$, $\Phi_2 (\rho)$, $\Phi_3 (\rho)$ in Eq.~\eqref{ainequality3} corresponding to both the trace distance (red slanted crosses) and the $\alpha$-fidelity (blue crosses). For the $\alpha$-fidelity bounds, we used $\alpha_2 = \alpha_3 = 1/2$, since numerical tests showed it to result to tightest probed bounds for $A$.
In the measurements presented in Figs.~\ref{resultset1_2_3}, \ref{resultset5_6} \textbf{a)}, and \ref{resultset5_6} \textbf{c)} , we used the method of \cite{lyyrasiltanen2020} to extract lower bounds for $ \Delta\mu /\sigma$, which we used to obtain upper bounds for $A_{crit}$ corresponding to Eq.~\eqref{fit}. For each of these cases, we used the same measurement data as when probing bounds of $A$, but the $\alpha$ values were optimized independently for each used coupling to obtain the tightest bound, as done in \cite{lyyrasiltanen2020}. Like in \cite{lyyrasiltanen2020}, the optimal $\alpha$ value varied and it was most commonly near $\alpha = 1$.
\begin{figure}
\caption{
\textbf{The probed bounds for the convex coefficient $A$.} \textbf{a) - d)}: $\lambda_1 = 810$ nm, $\lambda_2 = 818$ nm, $A = 0.7$. The x axis is the thickness of the quartz plate combination. The blue crosses are the lower and upper bounds obtained from the $\alpha$-fidelity and the red slanted crosses are the bounds obtained from the trace distance. The blue solid and red dashed lines are the theoretical predictions for the bounds of the matching color, calculated taking into account dispersion in quartz. The black dashdotted lines are the values of $A_{crit}$ and $1 - A_{crit}$, limiting the non-Markovian region. In \textbf{a)} and \textbf{b)} the fast axes of all the quartz plates were aligned. In \textbf{c)} and \textbf{d)} the rotation angle of each quartz plate was chosen randomly. In \textbf{a)} and \textbf{c)} the $A_{crit}$ values were obtained by probing of $\Delta \mu / \sigma$ with Eq.~\eqref{etabounds}.
In \textbf{b)} and \textbf{d)} the known value of $A_{crit}$ was used.
The error bars are due to the photon-counting statistics, and they are standard deviations of the bound values calculated by the Monte Carlo method.
}
\label{resultset5_6}
\end{figure}
The panels in Fig.~\ref{resultset1_2_3} show that our probing approach managed to extract tight enough bounds for the unknown parameters $A$ and $ \Delta\mu /\sigma$ to verify the non-Markovianity of the dynamics: in each case, at least for one system-probe coupling the probed lower and upper bounds are between the probed $A_{crit}$ and $1 - A_{crit}$ which define the non-Markovian region. Thus in each case we could determine that the dynamics was non-Markovian from the outcome of a single snapshot measurement.
On the other hand, for some couplings the result is inconclusive, as either the probed lower bound is below $A_{crit}$ or the probed upper bound is above $1 - A_{crit}$. We note that in this case the bounds obtained using trace distance led to a lot tighter bounds than the ones derived from the $\alpha$-fidelities.
The probed upper bound of $A_{crit}$ in Fig.~\ref{resultset5_6} \textbf{a)} was too large leading to inconclusive result in verifying non-Markovianity. Figure \ref{resultset5_6} \textbf{b)} shows that when the exact value of $A_{crit}$ was known, the probed lower and upper bounds of $A$ were tight enough to verify the non-Markovianity of the polarization dynamics.
For the measurements presented in Figs.~\ref{resultset5_6} \textbf{c)} and \textbf{d)} , we used a completely unknown system-probe coupling by fixing the quartz plates in randomly chosen rotation angles, like done earlier in \cite{lyyrasiltanen2020}. As above, we used this unknown coupling to probe if the probe's dynamics would be non-Markovian if all the quartz plates in the combination were aligned. In Fig.~\ref{resultset5_6} \textbf{c)} , we used the measurement data to extract bounds for both $A$ and $A_{crit}$. The data shows that as in the case of Fig.~\ref{resultset5_6} \textbf{a)},
this measurement led to inclonclusive verification of non-Markovianity. In Fig.~\ref{resultset5_6} \textbf{d)}
the exact value of $A_{crit}$ was assumed as known and we see that this time the measurement data led to conclusive verification of non-Markovianity. This means that quantum probing measurements with unknown system-probe interactions can be used to limit the convex coefficient to a non-trivial interval which in turn can be exploited to make conclusions on the characteristics of the BLP non-Markovianity.
\iffalse \textbf{The experimental results here.} \begin{table}[]
\begin{tabular}{ccc|cc|cc|c} $x [mm]$ & $\Delta\eta$ & $A$ & $A_{crit}$ & $1 - A_{crit}$ & Measured $A_{lower}$ & Measured $A_{upper}$ & Conclusion \\ \hline 2 & 16.3093 & 0.5122 & 0.0168826 & 0.9831174 & 0.277423 & 0.636309 & non-M. \\ 5 & 16.3093 & 0.5122 & 0.0168826 & 0.9831174 & 0.221538 & 0.768618 & non-M. \\ 7 & 16.3093 & 0.5122 & 0.0168826 & 0.9831174 & 0.135133 & 0.887806 & non-M. \\ 10 & 16.3093 & 0.5122 & 0.0168826 & 0.9831174 & 0.0603243 & 0.943655
& non-M. \end{tabular} \end{table} \fi
To summarize, in each measurement we succesfully verified the non-Markovianity of the polarization dynamics by probing $A$ at a single unknown interaction time. Additionally, in the measurements presented in Fig.~\ref{resultset1_2_3} we managed to exploit the same measurement data to probe small enough upper bounds for $A_{crit}$ that the non-Markovianity could be verified without assuming any of the parameters in the frequency states.
We emphasize that in order to make conclusions of the non-Markovianity, we only need to measure the probe system evolved with the map of interest only at one unknown time and compare it to our reference maps with the same unknown interaction time. The protocol itself does not require any information on the actual interaction time, or more generally, any knowledge of the system-probe coupling $U$ as demonstrated by the results in Fig.~\ref{resultset5_6} \textbf{d)}
. Even though the general form of the dephasing dynamics in this model is well-known \cite{liu2011,liu2018}, measuring the evolved polarization state at an unknown interaction time cannot tell anything about the Markovianity or non-Markovianity of the qubit dynamics, as many choices of $A$ can lead to the same value $|\kappa(t)|$, corresponding to the distinguishability of the optimal pair of states, even if the other parameters are fixed.
\begin{figure}
\caption{\textbf{Probing the Markovian and non-Markovian time intervals.} (Color online) \textbf{a):} The time and $\Delta\eta$ dependence of the critical amplitude $A_{crit}$. The dynamics is Markovian in the white regions for all $A$. The horizontal line shows the $\Delta\eta$ in panel \textbf{b)} where we have fixed $\lambda_1 = 810$ nm, $\lambda_2 = 818$ nm, $A = 0.7$. In \textbf{b)}, the lower and upper bound of $A$ are the tightest bounds in Fig.~\ref{resultset5_6} \textbf{a)}. Since the probed lower and upper bounds are in $[A_{crit},1-A_{crit}]$ during the time intervals marked with red dashed lines, we verify that the dynamics at those times is non-Markovian. During the intervals marked with green dotted lines neither the lower nor upper bound is in $[A_{crit},1-A_{crit}]$, and we verify that the dynamics is Markovian at these time intervals. Vertical gray lines are guide for the eye. The x axis is the rescaled interaction time inside the quartz plate combination. Here, $\Delta\eta$ is assumed as known.}
\label{acritt}
\end{figure}
\subsection{Markovian and non-Markovian time intervals}\label{n-m_intervals}
Finally, we show how our probing results can be exploited to identify the time intervals where the dynamics is guaranteed to be Markovian or non-Markovian. Figure \ref{acritt} \textbf{a)} shows the time dependence of $A_{crit}$ for different values of $\Delta\eta$, which was determined numerically with Eq.~\eqref{nmcondforA}. Inside the white areas, the dynamics is Markovian for all $A$ in $[0,1]$. We see that as $\Delta\eta$ decreases, the first non-Markovian period appears later. The plot also shows that $A_{crit}$ is always smallest during the first non-Markovian period and larger $\Delta\eta$ leads to smaller $A_{crit}$. These observations are in good agreement with the non-Markovian behavior of the frequency-polarization model, since $\Delta\mu \propto \Delta\eta$ gives rise to the revivals and $\sigma \propto \Delta\eta^{-1}$ corresponds to the damping rate of the trace distance \cite{liu2011,liu2018}. The horizontal black line highlights the fixed value of $\Delta\eta$ in Figs.~\ref{resultset5_6} and \ref{acritt} \textbf{b)}.
In Fig.~\ref{acritt} \textbf{b)} we plot the time dependence of $A_{crit}$ for the value of $\Delta\eta$ in the measurements of Fig.~\ref{resultset5_6}. As in Fig.~\ref{acritt} \textbf{a)}, we see time intervals where $A_{crit}$ is not defined, corresponding to times when there does not exist such $A \in [0,1]$ that would satisfy Eq.~\eqref{nmcondforA}. Thus we know that the dynamics on all those intervals is Markovian, marked with green dotted x axis. The blue and red solid curves limit the non-Markovian region $[A_{crit},1-A_{crit}]$ and the black dashdotted line marks the real value of $A$ in the experiment.
We see that for the rescaled interaction time $2\pi\sigma\Delta n t$ in $[0,3]$ the real value of $A$ is between $A_{crit}$ and $1-A_{crit}$ on three time intervals, meaning that the probe dynamics is really non-Markovian at those times. Looking at the black solid lines, corresponding to the tightest probed lower and upper bounds for $A$ in Fig.~\ref{resultset5_6} \textbf{a)}, we see that for the first two non-Markovian time intervals the lower and upper bound are on the interval $[A_{crit},1-A_{crit}]$. This means that our probed bounds of $A$ combined with our analysis on the model's non-Markovianity lets us verify that the probe dynamics was non-Markovian on those intervals. For these confirmed non-Markovian intervals the x axis is marked with red dashing.
For the three potentially non-Markovian intervals for $2\pi \sigma \Delta n t\in[2,5]$, the probed upper bound is above $1-A_{crit}$ while the probed lower bound is between $A_{crit}$ and $1-A_{crit}$, so our probing measurement leads to inconclusive result on the non-Markovianity on those intervals, marked with black dotted x axis. We note also the very small inconclusive intervals around the two confirmed non-Markovian intervals.
Here, we probed the Markovian and non-Markovian time intervals only for the measurement in Fig.~\ref{resultset5_6}. The same analysis can be directly applied also to the rest of our measurements.
\section{Conclusions and Outlook}
In this paper we applied the generalized data processing inequalities of $\alpha$-fidelities and trace distance to construct a quantum measurement strategy for probing lower and upper bounds for the convex coefficients in mixtures of commuting states. The measurement strategy does not require any knowledge of the used system-probe coupling and it can be directly applied with no modifications if the coupling is changed. We first discussed briefly some possible applications. Then, we explored in detail a specific task, namely the verification of the probe's non-Markovianity, a useful property for certain quantum information protocols \cite{panalgorithm,karlssonEPL,liu2020,laine2014, huelgarivasplenio2012, Xiang2014,utagi}, by snapshot probing measurement at an unknown time and with a completely unknown system-probe coupling.
We showed that when a single photon's polarization interacts with the photon's double peaked Gaussian frequency spectrum in quartz, the non-Markovian behavior of the polarization dynamics is fully contained in an intact and well-defined area in the two-dimensional $(A,\Delta\mu/\sigma)$ parameter space. Here, $A$ is the convex coefficient in the mixture of the Gaussian peaks, $\Delta\mu$ is the difference of their central frequencies, and $\sigma$ is their standard deviation. We applied our strategy in probing lower and upper bounds for the convex coefficient $A$ and exploited the same measurement data in the recently proposed strategy to probe a lower bound for $\Delta\mu/\sigma$ \cite{lyyrasiltanen2020}. The probing strategies were implemented in an all-optical single photon experiment where we were able to restrict the unknown parameters $A$ and $\Delta\mu/\sigma$ within an area where the non-Markovianity of the polarization dynamics is guaranteed. By assuming $\Delta\mu/\sigma$ as known, we applied our probing results of $A$ to identify the Markovian and non-Markovian time intervals of the polarization dynamics. Thus, our results illustrate that quantum probing measurements with unknown system-probe couplings can be constructed and implemented to obtain useful qualitative information on the characteristics of the probe's dynamical map.
Even though we concentrated here in non-Markovianity in terms of revivals of trace distance, in our case of dephasing dynamics, these results apply directly to multiple other definitions of non-Markovianity as well, namely violation of CP-divisibility, Bloch volume oscillations, and increases of $l_1$ coherence norm \cite{teittinen2018}. In the dephasing dynamics, revivals of trace distance also imply that the quantum speed limit bound is not reached \cite{deffnerlutz,xuspeed,teittinen2019}. Thus, our results can be directly used to conclude that the probe dynamics is not on its fastest trajectory when the dynamics is verified as non-Markovian.
Our verification strategy is based on quantum probing measurements. It has been recently shown that using probes initially entangled with an ancillary system can achieve higher precision in quantum probing \cite{girolami}. However, adding an ancillary system increases the total probe-ancilla Hilbert space dimension, which makes the required tomography more demanding. Recently, experimentally estimating the fidelity between two-photon polarization states was shown to be more efficient than full tomography \cite{teiko,lyyra}. The results and experimental implementation can be directly generalized to $\alpha$-fidelities. Future studies will show if using entangled ancillary polarization system can be exploited to obtain better sensitivity in verification of non-Markovianity without the need of increasing the amount of measurements significantly.
\section{Acknowledgments}
H.~L. acknowledges the fruitful discussions with Erkka Haapasalo and Juha-Pekka Pellonpää. O.S. acknowledges the financial support from Magnus Ehrnrooth Foundation.
\appendix
\renewcommand{A\arabic{equation}}{A\arabic{equation}}
\setcounter{equation}{0}
\section*{Appendix: Analysis of probing global Markovianity}
\begin{figure}
\caption{(Color online) The tightest possible upper and lower bounds probed with the $\alpha$-fidelity (blue and red dots) optimized w.r.t. time, $\alpha$'s, and $A$ independently, $A_{crit}$ and $1-A_{crit}$ (black diamonds and squares), and the corresponding fitted curves (cyan and pink solid lines) as functions of $| \mu_2-\mu_1 | / \sigma$.}
\label{numerical_plots}
\end{figure}
Here we discuss the limitations of our probing measurements when using aligned quartz plates as the unitary coupling in verifying Markovianity of the polarization dynamics. To conclude whether the dynamics is Markovian, the experimentally obtained upper bound for the convex coefficient must be less than the convex coefficient's critical value, i.e., \begin{equation} F_{\alpha_2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^{1/\alpha_2}<A_{crit}. \label{concl_Mark1} \end{equation} Alternatively, the experimentally obtained lower bound must satisfy \begin{equation} 1-F_{\alpha_3}\big(\Phi_1(\rho),\Phi_3(\rho)\big)^{1/\alpha_3}>1-A_{crit}. \label{concl_Mark2} \end{equation}
Numerical analysis shows that $F_{\alpha_2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^{1/\alpha_2}$ grows with $\alpha_2 \in [1/2,1)$ and $A \in [0,1]$. Since smaller $F_{\alpha_2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^{1/\alpha_2}$ and $F_{\alpha_3}\big(\Phi_1(\rho),\Phi_3(\rho)\big)^{1/\alpha_3}$ lead to tighter bounds in Eq.~\eqref{ainequality3}, we use $\alpha_2 = \alpha_3 = 1/2$ in the measurement data analysis when probing the bounds of $A$. To see if Eq.~\eqref{concl_Mark1} is ever satisfied, we may fix $\alpha_2=1/2$, $A=0$, and only consider $\min_t \Big\{F_{1/2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^2\big\vert_{A=0}\Big\}$, where the minimum is taken over all system-probe interaction times $t$. We have plotted the values of $\min_t \Big\{F_{1/2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^2\big\vert_{A=0}\Big\}$ as a function of $\Delta\eta$ in Fig.~\ref{numerical_plots}, where the values of $A_{crit}$ are shown for comparison. We notice that the values of $A_{crit}$ are in excellent agreement with $\min_t\Big\{F_{1/2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^2\big\vert_{A=0}\Big\}/2$ for all choices of $\Delta\eta$. Thus, we can estimate that \begin{widetext} \begin{equation} F_{\alpha_2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^{1/{\alpha_2}} \geq\min_t \Big\{F_{1/2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^2\big\vert_{A=0}\Big\} \approx2A_{crit}>A_{crit}. \label{estimation1} \end{equation} A similar analysis for the lower bound results in the estimation \begin{equation} 1-F_{\alpha_3}\big(\Phi_1(\rho),\Phi_3(\rho)\big)^{1/{\alpha_3}}\leq\max_t \Big\{1-F_{1/2}\big(\Phi_1(\rho),\Phi_3(\rho)\big)^2\big\vert_{A=1}\Big\}\approx1-2A_{crit}<1-A_{crit}. \label{estimation2} \end{equation} Combining Eqs.~\eqref{estimation1} and~\eqref{estimation2}, we conclude that, for any choice of parameters $A$, $\mu_1$, $\mu_2$, $\sigma$, $\alpha_2$, $\alpha_3$, and $t$, we get \begin{equation} [1-F_{\alpha_3}\big(\Phi_1(\rho),\Phi_3(\rho)\big)^{1/{\alpha_3}},F_{\alpha_2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^{1/{\alpha_2}}]\cap[A_{crit},1-A_{crit}]\neq\emptyset, \label{conclusion} \end{equation} \end{widetext} meaning that the bounds obtained by the $\alpha$-fidelity cannot be used to confirm the global Markovianity of the qubit dynamics when the quartz plates are aligned in the same orientation.
We can derive the same result for the bounds obtained by the trace distance by using the estimation $A_{crit}\approx\frac{1}{2}\min_t \Big\{F_{1/2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^2\big\vert_{A=0}\Big\}$. Below, we show that the experimentally obtained upper bound satisfies $1-D_{tr}\big(\Phi_1(\rho),\Phi_2(\rho)\big)\geq A_{crit}$ and thus cannot confirm Markovianity. Using the shorthand notation $\tau=2\pi\sigma\Delta nt$, we get \begin{widetext} \begin{align} \Big[e^{-\frac{1}{2}\tau^2}\sin^2\Big(\frac{\Delta\eta\tau}{2}\Big)-e^{\frac{1}{2}\tau^2}\Big]^2&\geq 0 ~~\forall~\tau \ge 0\\ \Leftrightarrow 2\sin^2\Big(\frac{\Delta\eta\tau}{2}\Big)&\leq e^{\tau^2}+e^{-\tau^2}\sin^4\Big(\frac{\Delta\eta\tau}{2}\Big) ~~\forall~\tau \ge 0\\ \Leftrightarrow 2-2\cos(\Delta\eta\tau)&\leq e^{\tau^2}-\cos(\Delta\eta\tau)+1+\frac{1}{4}e^{-\tau^2}\big[\cos(\Delta\eta\tau)-1\big]^2 ~~\forall~\tau \ge 0\\ \Leftrightarrow\sqrt{2-2\cos(\Delta\eta\tau)}&\leq e^{\frac{1}{2}\tau^2}-\frac{1}{2}e^{-\frac{1}{2}\tau^2}\big[\cos(\Delta\eta\tau)-1\big] ~~\forall~\tau \ge 0\\ \Leftrightarrow 1-\frac{1}{2}e^{-\frac{1}{2}\tau^2}\sqrt{2-2\cos(\Delta\eta\tau)}&\geq\frac{1}{2}\Big\{1+\frac{1}{2}e^{-\tau^2}\big[\cos(\Delta\eta\tau)-1\big]\Big\} ~~\forall~\tau \ge 0\\ \Rightarrow 1-\frac{1-A}{2}e^{-\frac{1}{2}\tau^2}\sqrt{2-2\cos(\Delta\eta\tau)}&\geq\frac{1}{2}\min_t \Big\{1+\frac{1}{2}e^{-\tau^2}\big[\cos(\Delta\eta\tau)-1\big]\Big\}~~\forall\,A\in[0,1]\\ \Leftrightarrow1-D_{tr}\big(\Phi_1(\rho),\Phi_2(\rho)\big)&\geq\frac{1}{2}\min_t \Big\{F_{1/2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^2\big\vert_{A=0}\Big\}\\ \Leftrightarrow 1-D_{tr}\big(\Phi_1(\rho),\Phi_2(\rho)\big)&\geq A_{crit} \end{align} \end{widetext} \iffalse \begin{align} 1-D_{tr}\big(\Phi_1(\rho),\Phi_2(\rho)\big)&\geq A_{crit}\\
\Leftrightarrow1-D_{tr}\big(\Phi_1(\rho),\Phi_2(\rho)\big)&\geq\frac{1}{2}\min_t \Big\{F_{1/2}\big(\Phi_1(\rho),\Phi_2(\rho)\big)^2\big\vert_{A=0}\Big\}\\
\Leftrightarrow 1-\frac{1-A}{2}e^{-\frac{1}{2}\tau^2}\sqrt{2-2\cos(\Delta\eta\tau)}&\geq\frac{1}{2}\min_t \Big\{1+\frac{1}{2}e^{-\tau^2}\big[\cos(\Delta\eta\tau)-1\big]\Big\}\\
\Leftarrow 1-\frac{1}{2}e^{-\frac{1}{2}\tau^2}\sqrt{2-2\cos(\Delta\eta\tau)}&\geq\frac{1}{2}\Big\{1+\frac{1}{2}e^{-\tau^2}\big[\cos(\Delta\eta\tau)-1\big]\Big\}\\
\Leftrightarrow\sqrt{2-2\cos(\Delta\eta\tau)}&\leq e^{\frac{1}{2}\tau^2}-\frac{1}{2}e^{-\frac{1}{2}\tau^2}\big[\cos(\Delta\eta\tau)-1\big]\\
\Leftrightarrow 2-2\cos(\Delta\eta\tau)&\leq e^{\tau^2}-\cos(\Delta\eta\tau)+1+\frac{1}{4}e^{-\tau^2}\big[\cos(\Delta\eta\tau)-1\big]^2\\
\Leftrightarrow 2\sin^2\Big(\frac{\Delta\eta\tau}{2}\Big)&\leq e^{\tau^2}+e^{-\tau^2}\sin^4\Big(\frac{\Delta\eta\tau}{2}\Big)\\
\Leftrightarrow\Big[e^{-\frac{1}{2}\tau^2}\sin^2\Big(\frac{\Delta\eta\tau}{2}\Big)-e^{\frac{1}{2}\tau^2}\Big]^2&\geq 0, \end{align} \fi which proves the claim. Similar calculation, using $A_{crit}\approx\frac{1}{2}\min\Big\{F_{1/2}\big(\Phi_1(\rho),\Phi_3(\rho)\big)^2\big\vert_{A=1}\Big\}$, holds for the lower bounds.
Thus, we conclude that combinations of quartz plates in the same orientation cannot be used as the system-probe coupling to confirm the Markovianity of the probe dynamics. In this analysis, we concentrated on the global Markovianity, meaning that there are no revivals of the trace distance at any time $t\in [0,\infty)$.
If instead we are more interested in local Markovianity, meaning monotonicity of trace distance on some finite interval $[t_1,t_2]$, the amount of trace distance revivals decreases - and consequently - the value of $A_{crit}$ increases. Thus, by restricting our interest to shorter intervals, we can confirm the Markovianity of the dynamics. In this approach, the same measurement data and Eq.~\eqref{ainequality3} can be used, but only the values of $A_{crit}$ should be calculated again according to the time interval when interpreting the results. Similar approach can be applied to cases where we are interested in the non-Markovianity at some specific time intervals. This was successfully implemented in Section \ref{n-m_intervals} of the main article.
\end{document} | arXiv |
Energy efficiency maximization of full-duplex two-way relay-assisted device-to-device communications underlaying cellular networks
Yiliang Chang1,
Hongbin Chen ORCID: orcid.org/0000-0003-4008-37041,2 &
Feng Zhao1
EURASIP Journal on Wireless Communications and Networking volume 2016, Article number: 222 (2016) Cite this article
With the substantial progress in self-interference (SI) cancelation, the full-duplex (FD) technique, which allows the communication user to transmit and receive signals over the same frequency band simultaneously, enables a significant enhancement of spectral efficiency (SE) in comparison with the traditional half-duplex (HD) technique. Recently, relay-assisted device-to-device (D2D) communications underlaying cellular networks have aroused a great deal of research interests due to its high SE. For the new meaningful paradigm of the combination of the FD and the amplify-and-forward (AF) relay-assisted D2D communications, analyzing the SE and energy efficiency (EE) is crucial, which have not been investigated in the existing works. In this paper, we focus on the EE of the FD two-way (FDTW) relay-assisted D2D communications with uplink channel reuse by considering the residual SI at the D2D users and compare it with the HD counterpart. Our goal is to find the optimal transmission powers and amplification gain to maximize the system EE while guaranteeing SE requirements and maximum transmission power constraints. A new two-tier alternative iteration optimization algorithm is proposed to solve the optimization problem. Simulation results show that (1) the results obtained by the proposed algorithm is very close to those obtained by the exclusive searching method, (2) smaller residual power of SI leads to better performance of EE and SE, (3) the SE obtained by FDTW relay-assisted D2D networks is higher than the SE obtained by the HD counterpart, and (4) the EE comparison of FDTW relay-assisted D2D networks and its HD counterpart depends on the residual power of SI. The EE obtained by FDTW relay-assisted D2D is higher than the EE obtained by HD counterpart only when the residual power of SI is sufficiently small.
It is expected that mobile data traffic will increase a thousand-fold over the next decade which will be driven by the expected 50 billion mobile devices connected to the cloud, anywhere and anytime. With a rapid increase in the number of connected devices, some challenges for next 5G networks appear which will be responded by increasing capacity and by improving energy efficiency (EE), coverage, spectrum utilization, and so on [1]. As the key techniques to address the requirements for 5G networks, direct device-to-device (D2D) communication and in-band full-duplex (FD) have attracted a great deal of research interests [2–4].
D2D communications allow proximate cellular users to communicate with each other directly under the control of base station (BS) with lower transmit power requirements. The high channel quality of short-range D2D links facilitate high data rates for local services, prolong users' battery lives, and offload heavy traffic of BS [5]. In addition, D2D links can underlay cellular links by reusing the same time and frequency resources; thus, the spectral efficiency (SE) of cellular networks can be improved obviously. However, the introduction of D2D communications into cellular networks poses new challenge in the resource allocation design due to co-channel interference caused by spectrum reuse, which has marked impacts on the performance of communication reliability and network throughput. Thus, effective power allocation is required to alleviate the interference problem. Besides, another efficient way to reduce the impact of interference is providing a relay user between D2D pair [6]. We refer to this as relay-assisted D2D communications which can be an efficient approach to provide a better Quality of Service (QoS) and lower transmission powers for communication between distant D2D users.
Relay communication, in which the relay forwards the signal received by a source to a destination, has been envisaged as a spectral- and energy-efficient technology for cellular networks [7]. Based on the directions of signal transmission, relaying systems can be classified into three transmission types (i.e., one-way, two-way, and multiway). Compared with one-way relaying, two-way relaying can achieve higher SE, which allows a relay user to simultaneously communicate with two end users. Existing studies also show that the two-way relay-assisted D2D communications have significant improvement on system performance (e.g., outage probability, system sum rate) [8, 9]. Thus, with the assistance of two-way relaying, the performance advantage of D2D communications can be further improved. Multiway relaying, in which multiple users exchange information with the assistance of a relay, can achieve better performance and flexibility than two-way relaying. However, its protocol and encoder design are rather complicated, making it a potential choice for relay-assisted D2D communications in the future.
On the other hand, the in-band FD transmission, which allows transmitting and receiving at the same frequency and the same time, enables an enormous enhancement of SE than half-duplex (HD) [10–12]. The main limitation impacting FD transmission is the strong self-interference (SI) signal imposed by the transmit antenna on the receive antenna within the same transceiver. Both academia and industry reached consensus that SI cancelation would play the most pivotal role in implementing FD communication systems. There are usually two main categories: passive suppression (PS) and active cancelation [13]. Therein, the active cancelation also includes analog cancelation (AC) and digital cancelation (DC). Since none of the individual cancelation techniques is capable of satisfying the system requirements in terms of the attainable SI cancelation capability, a high-capability cancelation scheme by combining the active and passive techniques is necessarily developed. Among all the SI cancelation techniques, the PS has an important position. The primary advantage to performing PS in the propagation domain is that the downstream receiver hardware can process signals more accurately with a large dynamic range [10]. In addition, DC techniques can be performed in the digital domain, which are the lowest-complexity active SI cancelation techniques in FD systems [10, 14]. As the design of AC usually requires a complex and large-size hardware circuits with respect to mobile devices [11, 14, 15], a combination of the latest PS and DC (referred to as PSDC) techniques will be an appropriate way to implement the mobile FD system [13, 14].
Hence, it can be practically beneficial applying the FD techniques into two-way relay-assisted D2D communications with certain SI cancelation where the relay user and D2D users can transmit and receive signals simultaneously, and thus, the SE gets further improved. Since FD transmission possesses strong SI, it generally consumes more power than HD. Therefore, there is an urgent demand for maintaining high system throughput while limiting energy consumption. EE, defined as the ratio of throughput to power consumption, is an important measure of green communication solutions [7]. It is urgent and interesting to investigate the EE of FD two-way (FDTW) relay-assisted D2D communications underlaying cellular networks.
In the existing literature, many works have been done on the EE of D2D communications underlaying cellular networks. For example, the EE of D2D communications without channel reuse was investigated in [16]. The authors showed the EE improvement with the deployment of D2D communications in heterogeneous networks (HetNets) compared with the full small-cell deployment, thus provided a greener alternative to cellular network deployment. Correspondingly, [17] and [18] presented the analysis of the EE of D2D systems with cellular channel reuse. Therein, [17] revealed the tradeoff between EE and delay in D2D communications, where stochastic traffic arrivals and time-varying channel conditions were both considered. Given the SE requirement and maximum transmission power constraints, the authors in [18] investigated the EE and SE, in which each user is self-interested and wants to maximize its own EE. Inspired by the performance gain of a combination of relay and D2D communications, the authors in [19] extended radio resource management algorithms into one-way relay-assisted D2D communications to balance SE and EE while considering mode selection and resource allocation constraints. In [6], the maximum achievable transmission capacity was studied in one-way relay-assisted D2D communications while guaranteeing the outage probability of both cellular and D2D links. For a higher performance improvement, two-way relaying with network coding technique which can improve transmission efficiency will be a promising choice in D2D communications. The authors in [20] demonstrated the performance gain of D2D communications assisted by two-way relaying over Rayleigh fading channels. The achievable rate of different two-way transmission schemes of relay-assisted D2D communications were presented in [9], while considering the interference due to spectrum sharing. Compared with the traditional cellular communications through BS, the EE and SE of two-way relay-assisted D2D communications were studies in [21]. The optimal user equipment (UE) transmission powers to maximize the EE were also derived. However, there is little work addressing the EE in FDTW relay-assisted D2D communications underlaying cellular networks, which is the focus of this paper.
In this paper, we provide a comprehensive EE analysis of FDTW relay-assisted D2D communications underlaying cellular networks, in comparison to its HD counterpart. The contributions of this paper are summarized as follows.
To the best of our knowledge, this is the first work to investigate the EE of FDTW relay-assisted D2D communications underlaying cellular networks. Moreover, the comparison between the FDTW relay-assisted D2D network and the HD two-way (HDTW) relay-assisted D2D network in terms of SE and EE is made. Finally, we demonstrate that the proposed two-tier alternative iteration optimization algorithm can converge very quickly.
From the perspective of practical implementation, SI cannot be fully canceled due to technical challenges. Therefore, we consider remaining SI after cancelation at the FD users in a more reasonable and relatively tractable way to study EE and SE in the FDTW relay-assisted D2D communications.
The joint optimization problem is non-convex. To solve this issue, the primal optimization problem is first decomposed into two subproblems, and then, we solve the two subproblems alternatively. For the first subproblem, we prove it is convex and then solve it by the bisection method. For the second subproblem, we decomposed it into two sub-subproblems again, and then, we solve the two sub-subproblems alternatively. For the first sub-subproblem, we transform it into concave function by using the nonlinear fractional programming, which can be effectively solved by the Dinkelbach method [22]. For the second sub-subproblem, we first prove it is convex and then solve it by the bisection method.
A full range of power consumption sources is considered in the power consumption model. In particular, the power consumed by SI cancelation is considered. The impact of imperfect SI cancelation is also considered in the analysis.
The rest of this paper is organized as follows. Section 2 describes the system model and optimization problem for FDTW relay-assisted D2D communications underlaying cellular networks. The elaborate solving process of the optimization problem is presented in Section 3. In Section 4, simulation results are presented. Finally, some concluding remarks are given in Section 5.
In this paper, we consider the FDTW relay-assisted D2D communications underlaying the cellular networks where the uplink radio resource is shared by the D2D link as depicted in Fig. 1. The cellular link consists of a cellular user equipment (CUE) and a receiver base station (BS), while the D2D link consists of a D2D pair (including source user equipment (SUE) and destination user equipment (DUE)) and a relay user equipment (RUE). In the FDTW relay-assisted D2D link, SUE intends to exchange information with DUE under the assistance of RUE, where SUE and DUE both transmit their signals to RUE, and at the same time, RUE broadcasts the previous received signals to SUE and DUE. Each user of the D2D link operates in the FD mode (with one antenna dedicated to transmission and another to reception, in order to increase the isolation of the SI [23]), which can double the spectral resource utilization potentially, and the CUE operates in traditional cellular networks with a single antenna. There is no direct link between SUE and DUE (e.g., due to the shadowing effect or the poor direct link condition).
FDTW relay-assisted D2D communications
D2D transmissions will cause interference to the BS receiver, and D2D receivers also suffer interference from the cellular uplink transmission. Moreover, the users in FD D2D link transmit and receive at the same time and frequency, causing SI to the receivers from its transmitters. It can be suppressed by the methods in [13, 14], which is beyond the scope of this paper. However, due to the practical constraint, the residual SI after interference suppression still exists and limits the performance of FDTW relay-assisted D2D network, which becomes the main concern in this paper. Hence, in order to analyze the effect of SI on system performance, we assume the SI in the FDTW relay-assisted D2D communications can be reduced but cannot be completely eliminated by SI cancelation techniques. The existence of SI will make the optimization problem formulation complicated as shown later. Let ρ denotes the average residual power of SI after cancelation at each user in D2D link. Based on the experimental results in [23], the residual SI power model general includes two cases: the complicated case that SI power increase linearly with the transmission power [24], and the other one in which the SI power is a constant and not a function of the transmission power [25]. Similar to our previous work in [26], we consider the second case that SI power is modeled as an invariable value. There are two main reasons for us to employ this model. One is that if ρ is regarded as a function of the transmission power of the D2D users, both the primal problem (16) and the transformed subproblem (P2) (defined in Sections 2 and 3.2, respectively) may become non-convex. The second reason is that the power of SI can be controlled in a range after cancelation [27]. Thus, it is reasonable to regard ρ to be fixed. The issue that ρ may be associated with the transmission power of D2D users will be considered in our future work. The SI channel coefficients at SUE, RUE, and DUE are denoted by h ss , h rr , and h dd respectively, which are generally modeled by the zero-mean circularly symmetric complex Gaussian (CSCG) random variables.
Assume that the channel coefficient between SUE and RUE and the channel coefficient between RUE and DUE are both reciprocal and denoted by h sr and h rd , respectively. Herein, the subscripts s, r, d represent SUE, RUE, and DUE, respectively. At the time instant t, CUE transmits its symbol to BS, and SUE and DUE transmit their respective symbols to RUE, respectively. Simultaneously, RUE broadcasts another symbol r(t) to SUE and DUE. We consider quasi-static fading in which the channel coefficients are constant within one defined frame but may change independently from one frame to another. Therefore, at the time instant t, the received signal at RUE can be expressed as
$$ {y}_r(t)=\sqrt{P_s}{h}_{sr}{x}_s(t)+\sqrt{P_d}{h}_{rd}{x}_d(t)+\sqrt{\rho_r}{h}_{rr}r(t)+\sqrt{P_u}{h}_{ur}{x}_u(t)+{N}_r(t) $$
where x s (t), x d (t), and x u (t) are the transmitted signals of SUE, DUE, and CUE, respectively, with Ε{|x s (t)|2} = Ε{|x d (t)|2} = Ε{|x u (t)|2} = 1. Here, Ε{x} defines the expectation of the random variable x, and P s , P d , and P u are the transmission power of SUE, DUE, and CUE, respectively. ρ r is the residual SI power of RUE which depends on the amount of SI cancelation, and h ur denotes the channel coefficient between the CUE and RUE. N r (t) is additive white Gaussian noise (AWGN) with zero mean and variance σ 2. The expression (1) reveals that for the HDTW relay-assisted D2D communications, the SI is zero, and thus y r (t) has no relationship with its own transmitted symbol r(t). However, for the FDTW relay-assisted D2D communications, y r (t) is decided by r(t), due to the existence of residual SI.
RUE broadcasts the received signal from both SUE and DUE at the time instant t − 1 with the amplify-and-forward (AF) protocol, namely, r(t) = y r (t − 1). To meet the transmission power constraint of RUE, the average power of r(t) is normalized to 1. Hence, with network coding technique [28], each user knows its transmitted signal, the received signals after subtracting interference at SUE and DUE are, respectively, given by
$$ \begin{array}{l}{y}_s(t)=\sqrt{\beta }{h}_{sr}r(t)+\sqrt{P_u}{h}_{us}{x}_u(t)+\sqrt{\rho_s}{h}_{ss}{x}_s(t)+{N}_s(t)\\ {}\kern1.75em =\sqrt{\beta {P}_d}{h}_{sr}{h}_{rd}{x}_d\left(t-1\right)+\sqrt{\beta {\rho}_r}{h}_{sr}{h}_{rr}r\left(t-1\right)+\sqrt{\beta {P}_u}{h}_{sr}{h}_{ur}{x}_u\left(t-1\right)\\ {}\kern2.5em +\sqrt{\beta }{h}_{sr}{N}_r\left(t-1\right)+\sqrt{P_u}{h}_{us}{x}_u(t)+\sqrt{\rho_s}{h}_{ss}{x}_s(t)+{N}_s(t)\end{array} $$
$$ \begin{array}{l}{y}_d(t)=\sqrt{\beta }{h}_{rd}r(t)+\sqrt{P_u}{h}_{ud}{x}_u(t)+\sqrt{\rho_d}{h}_{dd}{x}_d(t)+{N}_d(t)\\ {}\kern1.75em =\sqrt{\beta {P}_s}{h}_{sr}{h}_{rd}{x}_s\left(t-1\right)+\sqrt{\beta {\rho}_r}{h}_{rd}{h}_{rr}r\left(t-1\right)+\sqrt{\beta {P}_u}{h}_{rd}{h}_{ur}{x}_u\left(t-1\right)\\ {}\kern2.5em +\sqrt{\beta }{h}_{rd}{N}_r\left(t-1\right)+\sqrt{P_u}{h}_{ud}{x}_u(t)+\sqrt{\rho_d}{h}_{dd}{x}_d(t)+{N}_d(t)\end{array} $$
where h us and h ud denote the channel coefficient between CUE and SUE and the channel coefficient between CUE and DUE, respectively. β is the amplification gain of RUE, which should satisfy the constraint \( 0\le \beta \le \overline{\beta} \) with \( \overline{\beta}={P}_{r \max }/\left({P}_s{\left|{h}_{sr}\right|}^2+{P}_d{\left|{h}_{rd}\right|}^2+{\rho}_r{\left|{h}_{rr}\right|}^2+{P}_u{\left|{h}_{ur}\right|}^2+{\sigma}^2\right) \), and P r max is the maximum allowed transmission power of RUE. ρ s and ρ d are the residual SI power of SUE and DUE which depend on the amount of SI cancelation, respectively. N s (t) and N d (t) are AWGN with zero mean and variance σ 2.
At the same time, cellular uplink transmission will suffer interference both from the D2D pair and the RUE. Therefore, at the time instant t, the received signal at BS can be expressed as
$$ \begin{array}{l}{y}_u(t)=\sqrt{P_u}{h}_{ub}{x}_u(t)+\sqrt{P_s}{h}_{sb}{x}_s(t)+\sqrt{P_d}{h}_{db}{x}_d(t)+\sqrt{\beta }{h}_{rb}r(t)+{N}_b(t)\\ {}=\sqrt{P_u}{h}_{ub}{x}_u(t)+\sqrt{P_s}{h}_{sb}{x}_s(t)+\sqrt{P_d}{h}_{db}{x}_d(t)+\sqrt{\beta {P}_s}{h}_{sr}{h}_{rb}{x}_s\left(t-1\right)+\sqrt{\beta {P}_d}{h}_{rd}{h}_{rb}{x}_d\left(t-1\right)\\ {}+\sqrt{\beta {\rho}_r}{h}_{rb}{h}_{rr}r\left(t-1\right)+\sqrt{\beta {P}_u}{h}_{rb}{h}_{ur}{x}_u\left(t-1\right)+\sqrt{\beta }{h}_{rb}{N}_r\left(t-1\right)+{N}_b(t)\end{array} $$
where h ub , h sb , h rb , and h db denote the channel coefficient between BS and CUE, SUE, RUE, and DUE, respectively. N b (t) is AWGN with zero mean and variance σ 2.
For a more concise expression, the channel gain between the transmitter i and j is denoted as G ij = |h ij |2, where h ij is CSCG random variable with variance \( {\sigma}_{ij}={d}_{ij}^{-v} \). Herein, d ij is the normalized distance between nodes i and j, and v is the path-loss exponent.
According to formulas (2), (3), and (4), the instantaneous received signal-to-interference-plus-noise ratios (SINRs) at SUE, DUE, and CUE can be respectively calculated as
$$ {\gamma}_s=\frac{\beta {P}_d{G}_{sr}{G}_{rd}}{\beta {P}_u{G}_{sr}{G}_{ur}+\beta {\rho}_r{G}_{sr}{G}_{rr}+\beta {G}_{sr}{\sigma}^2+{P}_u{G}_{us}+{\rho}_s{G}_{ss}+{\sigma}^2} $$
$$ {\gamma}_d=\frac{\beta {P}_s{G}_{sr}{G}_{rd}}{\beta {P}_u{G}_{rd}{G}_{ur}+\beta {\rho}_r{G}_{rd}{G}_{rr}+\beta {G}_{rd}{\sigma}^2+{P}_u{G}_{ud}+{\rho}_d{G}_{dd}+{\sigma}^2} $$
$$ {\gamma}_u=\frac{P_u{G}_{ub}}{P_s{G}_{sb}+{P}_d{G}_{db}+\beta {P}_s{G}_{sr}{G}_{rb}+\beta {P}_d{G}_{rd}{G}_{rb}+\beta {\rho}_r{G}_{rb}{G}_{rr}+\beta {P}_u{G}_{ur}{G}_{rb}+\beta {G}_{rb}{\sigma}^2+{\sigma}^2} $$
The SE of cellular link and FDTW relay-assisted D2D link are written as
$$ {\eta}_{u,SE}\left({P}_s,{P}_d,\beta, {P}_u\right)={R}_u $$
$$ {\eta}_{d,SE}\left({P}_s,{P}_d,\beta, {P}_u\right)={R}_s+{R}_d $$
where R u = log2(1 + γ u ), R s = log2(1 + γ s ), and R d = log2(1 + γ d ).
Hence, the overall system SE which is the sum of cellular link and FDTW relay-assisted D2D link can be given by
$$ \begin{array}{l}{\eta}_{SE}\left({P}_s,{P}_d,\beta, {P}_u\right)={\eta}_{u,SE}\left({P}_s,{P}_d,\beta, {P}_u\right)+{\eta}_{d,SE}\left({P}_s,{P}_d,\beta, {P}_u\right)\\ {}\kern6.25em ={R}_u+{R}_s+{R}_d\end{array} $$
The power consumption in a general wireless node for communications usually contains two main parts: power amplifier and circuit power [18]. Power amplifier is related to the power amplifier drain efficiency and the transmission power level and is usually modeled as the ratio of the transmission power to the power amplifier drain efficiency [10]. Circuit power consumption is usually considered to be independent of the data rate and is regarded as a constant [29]. We consider both power amplifier and circuit power consumption in the transmitters. In addition, to be more practical, the circuit power consumption in the receivers is also considered. In a practical system, the circuit power consumption of BS in uplink is huge that can be supported by various efficient energy sources including renewable energy. Therefore, the EE of BS would be less critical as compared to that in the users since mobile devices are typically hand-held devices with limited battery life and can quickly run out of battery. So, it is reasonable to neglect the circuit power consumption of BS in uplink in the system EE [17, 18]. In addition, in FD communications, applying PS generally does not consume additional power; however, the power consumed by active DC is non-negligible. The power consumed by DC, P DC, is regarded as a constant due to the power consumed by the involved chip components which are not related to the throughput state.
Therefore, the power consumption of cellular link and FDTW relay-assisted D2D link can be respectively expressed as
$$ {P}_{u,\mathrm{total}}\left({P}_u\right)=\xi {P}_u+{P}_c $$
$$ {P}_{d,\mathrm{total}}\left({P}_s,{P}_d,\beta, {P}_u\right)=\xi {P}_s+\xi {P}_d+\beta \left({P}_s{G}_{sr}+{P}_d{G}_{rd}+{\rho}_r{G}_{rr}+{P}_u{G}_{ur}+{\sigma}^2\right)+3{P}_{DC}+3{P}_c $$
where 1/ξ is the power amplifier drain efficiency of the CUE, SUE, and DUE, respectively. P c is the static circuit power.
The EE of cellular link and FDTW relay-assisted D2D link is written as
$$ {\eta}_{u,EE}\left({P}_s,{P}_d,\beta, {P}_u\right)=\frac{W{\eta}_{u,SE}\left({P}_s,{P}_d,\beta, {P}_u\right)}{P_{u,\mathrm{total}}\left({P}_u\right)} $$
$$ {\eta}_{d,EE}\left({P}_s,{P}_d,\beta, {P}_u\right)=\frac{W{\eta}_{d,SE}\left({P}_s,{P}_d,\beta, {P}_u\right)}{P_{d,\mathrm{total}}\left({P}_s,{P}_d,\beta, {P}_u\right)} $$
respectively, where W denotes the channel bandwidth. Since powers of different users cannot be shared and so are their throughput and EE, a novel definition of system EE based on summation of EE of all transmitters rather than the ratio of sum network throughput to sum network power consumption was proposed in [30]. Inspired of this novel definition, we define the system EE of the FDTW relay-assisted D2D communications underlaying cellular networks as the sum of the EE of cellular link and FDTW relay-assisted D2D link. Therefore, the EE of the whole system can be expressed as
$$ {\eta}_{EE}={\eta}_{d,EE}\left({P}_s,{P}_d,\beta, {P}_u\right)+{\eta}_{u,EE}\left({P}_s,{P}_d,\beta, {P}_u\right) $$
Our objective is to find the optimal transmission powers P s , P d , and P u and the optimal relay amplification gain β to maximize η EE while keeping η d,SE and η u,SE above the thresholds and satisfying the maximum power constraints. The optimization problem can be formulated as
$$ \begin{array}{l}\underset{\left\{{P}_s,{P}_d,\beta, {P}_u\right\}}{ \max}\kern0.5em {\eta}_{\mathrm{EE}}\\ {}\mathrm{s}.\mathrm{t}.\left\{\begin{array}{l}{\eta}_{u,\mathrm{S}\mathrm{E}}\ge {\overline{\eta}}_{u,\mathrm{S}\mathrm{E}}\ \\ {}{\eta}_{d,SE}\ge {\overline{\eta}}_{d,\mathrm{S}\mathrm{E}}\\ {}0\le {P}_u\le {P}_{u, \max }\ \\ {}0\le {P}_s\le {P}_{s, \max}\kern1.25em \\ {}0\le {P}_d\le {P}_{d, \max}\\ {}0\le \beta \le \overline{\beta}\end{array}\right.\end{array} $$
where \( {\overline{\eta}}_{u,\mathrm{S}\mathrm{E}} \) and \( {\overline{\eta}}_{d,SE} \) denote the minimum SE requirements of the cellular link and the D2D link and \( {\overline{\eta}}_{u,\mathrm{S}\mathrm{E}}={R}_{u, min} \) and \( {\overline{\eta}}_{d,\mathrm{S}\mathrm{E}}={R}_{s, min}+{R}_{d, min} \), here R u,min , R s,min , and R d,min are the minimum data rate requirements of CUE, SUE, and DUE, respectively. P u,max , P s,max , and P d,max are the maximum allowed transmission power of the CUE, SUE, and DUE, respectively.
Energy efficiency maximization problem
In this section, we formulate the EE optimization problem for FDTW relay-assisted D2D communications underlaying the cellular networks. Specifically, we will seek the optimal transmission powers P s , P d , P u , and the optimal relay amplification gain β to maximize EE while keeping SE above a threshold and guaranteeing transmission power constraints. Unfortunately, joint optimization over P s , P d , P u , and β is very hard to be found due to that η EE is not concave in P s , P d , P u , and β jointly. So, it cannot be solved by the general convex optimization methods. To overcome this difficulty, a new two-tier alternative iteration optimization algorithm is proposed in this section. The basic idea is to first optimize the EE of the cellular link by the bisection method and the FDTW relay-assisted D2D link with iterative optimization algorithm, respectively, and then alternately iterate the cellular link and the FDTW relay-assisted D2D link until convergence to produce the optimal system EE.
EE maximization of cellular link with fixed P s , P d , and β
In the EE optimization problem of the cellular link, given fixed P s , P d , and β, our objective is to seek the optimal transmission power P u of CUE to maximize η u,EE while keeping η u,SE above a threshold and satisfying the maximum power constraint. The sub-optimization problem can be formulated as
$$ \begin{array}{l}\underset{P_u}{ \max }\ {\eta}_{u,\mathrm{E}\mathrm{E}}\\ {}s.t.\kern0.5em C1:{\eta}_{u,\mathrm{S}\mathrm{E}}\ge {\overline{\eta}}_{u,\mathrm{S}\mathrm{E}}\ \\ {}\kern1.25em C2:0\le {P}_u\le {P}_{u, \max}\end{array} $$
(P1)
It is easy to prove that η u,SE is a monotonically increasing function of P u ∈ [0, + ∞), and according to the restricted conditions of η u,SE (i.e., η u,SE ≥ 0, \( {\eta}_{u,SE}\ge {\overline{\eta}}_{u,SE} \)), we have \( {P}_u\in \left\{{P}_u\left|{P}_u\in \right.\left({\overset{\smile }{P}}_u,+\infty \right)\right\} \), where \( {\overset{\smile }{P}}_u={G}_2\left({2}^{{\overline{\eta}}_{u,SE}}-1\right)/{G}_{ub}-\left({2}^{{\overline{\eta}}_{u,SE}}-1\right){G}_1 \), G 1 = βG ur G rb , and G2 = P s G sb + P d G db + β(P s G sr G rb + P d G rd G rb + ρ r G rb G rr + G rb σ 2) + σ 2.
Since P u ∈ [0, P u,max] and \( {P}_u\in \left\{{P}_u\left|{P}_u\in \right.\left({\overset{\smile }{P}}_u,+\infty \right)\right\} \), the feasible region of the subproblem (P1) can be rewritten as \( {P}_u\in \left[{\overset{\smile }{P}}_u,{P}_{u, \max}\right] \).
Theorem 1
Given P s , P d , β ∈ [0, + ∞), η u,EE is strictly quasiconcave in P u for P u ∈ [0, + ∞).
Refer to Appendix 1.
From Theorem 1, there are only three cases for the curve η u,EE vs. P u for \( {P}_u\in \left[{\overset{\smile }{P}}_u,{P}_{u, max}\right] \), as illustrated in Fig. 2.
Three cases of the η u,EE versus P u curve
Case 1: η u,EE strictly increases with P u for \( {P}_u\in \left[{\overset{\smile }{P}}_u,{P}_{u, max}\right] \), if \( \frac{d{\eta}_{u,\mathrm{E}\mathrm{E}}}{d{P}_u}\left|{}_{P_u={P}_{u, max}}\right.\ge 0 \), where \( \frac{d{\eta}_{u,\mathrm{E}\mathrm{E}}}{d{P}_u} \) is given by (29) in Appendix 1. In this case, the optimal solution to the subproblem (P1) is achieved at \( {\widehat{P}}_u={P}_{u, max} \).
Case 2: η u,EE strictly decreases with P u for \( {P}_u\in \left[{\overset{\smile }{P}}_u,{P}_{u, max}\right] \), if \( \frac{d{\eta}_{u,\mathrm{E}\mathrm{E}}}{d{P}_u}\left|{}_{P_u={\overset{\smile }{P}}_u}\right.\le 0 \). The optimal solution to the subproblem (P1) is achieved at \( {\widehat{P}}_u={\overset{\smile }{P}}_u \).
Case 3: η u,EE first strictly increases and then strictly decreases with P u for \( {P}_u\in \left[{\overset{\smile }{P}}_u,{P}_{u, max}\right] \), if \( \frac{d{\eta}_{u,\mathrm{E}\mathrm{E}}}{d{P}_u}\left|{}_{P_u={P}_{u, \max }}\right.<0 \) and \( \frac{d{\eta}_{u,\mathrm{E}\mathrm{E}}}{d{P}_u}\left|{}_{P_u={\overset{\smile }{P}}_u}\right.>0 \). The optimal solution to the subproblem (P1) is achieved at \( {\widehat{P}}_u={P_u}^{*} \), where P u * is the point at which η u,EE reaches its maximum for P u ∈ [0, + ∞) and is obtained by solving the equation \( \frac{d{\eta}_{u,\mathrm{E}\mathrm{E}}}{d{P}_u}=0 \), as shown in Appendix 1. The exact expression of P u * is not easily obtained and can be found by the numerical methods, such as the bisection method.
EE maximization of FDTW relay-assisted D2D link with a fixed P u
In the EE optimization problem of FDTW relay-assisted D2D link, given a fixed P u , our objective is to seek the optimal transmission powers P s , P d and the optimal relay amplification gain β to maximize η d,EE while keeping η d,SE above a threshold and satisfying the maximum power constraints. The sub-optimization problem can be formulated as
$$ \begin{array}{l}\underset{P_s,{P}_d,\beta }{ \max }\ {\eta}_{d,EE}\\ {}s.t.\kern0.5em C3:{\eta}_{d,SE}\ge {\overline{\eta}}_{d,SE}\ \\ {}\kern1.25em C4:0\le {P}_s\le {P}_{s, max}\kern1.25em \\ {}\kern1.25em C5:0\le {P}_d\le {P}_{d, max}\\ {}\kern1.25em C6:0\le \beta \le \overline{\beta}\end{array} $$
In the sub-optimization problem (P2), both η d,EE and η d,SE are non-convex with respect to P s , P d , and β, so that directly solving problem (P2) introduces enormous computational complexity. To address this issue, the iterative optimization algorithm is adopted in this section. The basic idea is to first optimize the objective function over a portion of variables when the others are fixed and then joint optimize these variables by utilizing the separate optimization results. Specifically, we first optimize the problem (P2) over P s and P d with a fixed β and then optimize the problem (P2) over β with fixed P s and P d . This process will be repeated until convergence.
Optimization over P s and P d with a fixed β
Given a fixed β, our objective is to seek the optimal transmission powers P s and P d to maximize η d,EE while satisfying the required constraints. Thus, given β, the subproblem (P2) can be rewritten as
$$ \begin{array}{l}\underset{P_s,{P}_d}{ \max }\ \frac{\eta_{d,SE}}{P_{d,\mathrm{total}}}\\ {}s.t.\kern0.5em C3:{\eta}_{d,SE}\ge {\overline{\eta}}_{d,SE}\kern0.5em \\ {}\kern1.25em C4:0\le {P}_s\le {P}_{s, max}\kern1.25em \\ {}\kern1.25em C5:0\le {P}_d\le {P}_{d, max}\end{array} $$
(P2-1)
For any given β, η d,SE is a concave and strictly increasing function of P s and P d for P s , P d ∈ [0, + ∞).
Although the constraints in the subproblem (P2-1) are convex according to Theorem 2, the objective function is still nonconvex. To solve this issue, we transform the objective function into a concave one by using the fractional programming technique developed in [22]. We define the maximum EE of the FDTW relay-assisted D2D link as \( {q}_d^{*} \), which is given by
$$ {q}_d^{\ast }={\eta}_{d,\mathrm{S}\mathrm{E}}\left({P}_s^{\ast },{P}_d^{\ast}\right)/{P}_{d,\mathrm{total}}\left({P}_s^{\ast },{P}_d^{\ast}\right) $$
where P s * and P d * are the optimal values at the maximum EE. The following theorem can be proved:
The maximum EE \( {q}_d^{*} \) is achieved if and only if \( {\eta}_{d,\mathrm{S}\mathrm{E}}\left({P}_s,{P}_d\right)-{q}_d^{\ast }{P}_{d,\mathrm{total}}\left({P}_s,{P}_d\right)={\eta}_{d,\mathrm{S}\mathrm{E}}\left({P}_s^{\ast },{P}_d^{\ast}\right)-{q}_d^{\ast }{P}_{d,\mathrm{total}}\left({P}_s^{\ast },{P}_d^{\ast}\right)=0 \) and for a given q d , the transformed function in subtractive form is a concave function.
Theorem 3 shows that the transformed problem with an objective function in subtractive form is equivalent to the non-convex problem (P2-1) in fractional form, i.e., they lead to the same optimum solution P s * and P d *. Thus, there exists a non-fractional expression that is equal to a fractional function. Searching \( {q}_d^{*} \) can be done by the Dinkelbach method that can converge to the optimal value at a superlinear rate [22]. Therefore, for any feasible q d , the corresponding solution of subproblem (P2-1) can be obtained by solving the following equivalent transformed optimization subproblem (18):
$$ \begin{array}{l}\underset{P_s,{P}_d}{ \max}\kern0.5em {\eta}_{d,\mathrm{S}\mathrm{E}}\left({P}_s,{P}_d\right)-{q}_d{P}_{d,\mathrm{total}}\left({P}_s,{P}_d\right)\\ {}\mathrm{s}.\mathrm{t}.\kern0.5em {C}_3,{C}_4,{C}_5\end{array} $$
The partial Lagrangian function associated with the subproblem (18) can be expressed as
$$ {L}_{d,\mathrm{E}\mathrm{E}}\left({P}_s,{P}_d,{\lambda}_1\right)={\eta}_{d,\mathrm{S}\mathrm{E}}\left({P}_s,{P}_d\right)-{q}_d{P}_{d,\mathrm{total}}\left({P}_s,{P}_d\right)+{\lambda}_1\left({\eta}_{d,\mathrm{S}\mathrm{E}}-{\overline{\eta}}_{d,\mathrm{S}\mathrm{E}}\right) $$
where λ 1 is the Lagrangian multiplier of the inequality constraint C1 and is required to be nonnegative. Since the subproblem (18) is in a standard concave form with differentiable objective and constraint functions, the Karush-Kuhn-Tucker (KKT) conditions are used to find the optimum solutions. Therefore, the first-order derivative of L d,EE with respect to P s and P d is required to be zero, i.e.,
$$ \left(1+{\lambda}_1\right)\frac{\eta }{\eta {P}_s+{\varphi}_s}={q}_d \ln 2\left(\xi +\beta {G}_{sr}\right) $$
$$ \left(1+{\lambda}_1\right)\frac{\eta }{\eta {P}_d+{\varphi}_d}={q}_d \ln 2\left(\xi +\beta {G}_{rd}\right) $$
In addition, since λ 1 is required to be a nonnegative one, the first-order derivative of L d,EE with respect to λ 1 should satisfy \( {\lambda}_1\frac{\partial {L}_{d,\mathrm{E}\mathrm{E}}}{\partial {\lambda}_1}=0 \), i.e.,
$$ {\lambda}_1\left(\frac{\eta }{\varphi_d}{P}_d+\frac{\eta }{\varphi_s}{P}_s+\frac{\eta^2}{\varphi_d{\varphi}_s}{P}_d{P}_s-\varpi \right)=0 $$
where η = βG sr G rd , φ s = βP u G rd G ur + βρ r G rd G rr + βG rd σ 2 + P u G ud + ρ d G dd + σ 2, and φ d = βP u G sr G ur + βρ r G sr G rr + βG sr σ 2 + P u G us + ρ s G ss + σ 2, \( \varpi ={2}^{{\overline{\eta}}_{d,\mathrm{S}\mathrm{E}}}-1 \).
According to formula (20) and (21), we can obtain
$$ {P}_s^{\ast }={\left[\frac{1+{\lambda}_1}{q_d \ln 2\left(\xi +\beta {G}_{sr}\right)}-\frac{\varphi_s}{\eta}\right]}^{+} $$
$$ {P}_d^{\ast }={\left[\frac{1+{\lambda}_1}{q_d \ln 2\left(\xi +\beta {G}_{rd}\right)}-\frac{\varphi_d}{\eta}\right]}^{+} $$
where [x]+ denotes max(0, x), which guarantees P s ≥ 0 and P d ≥ 0.
Notice that the optimal transmission power terms of SUE and DUE given by (23) and (24) are similar to traditional water-filling solutions, where the heights of the pool are defined as φ s /η and φ d /η, and the water level is partially determined by λ 1 and q d . Since both P s and P d involve the Lagrangian multiplier λ 1 and the KKT conditions of (22) require λ 1 to be nonnegative, we derive λ 1 in the following two cases.
Case 1: λ 1 > 0
In this case, according to formula (22), we have (η/φ d )P d + (η/φ s )P s + (η 2/φ d φ s )P d P s − ϖ = 0. After substituting (23) and (24), we have (λ 1 + 1)2 S 1 − S 2 = 0, where \( {S}_1={\eta}^2/{\left( \ln 2\right)}^2{q}_d^2{\varphi}_d{\varphi}_s\left(\xi +\beta {G}_{sr}\right)\left(\xi +\beta {G}_{rd}\right) \) and S 2 = ϖ + 1. Since λ 1 > 0, we retain the positive root, i.e., \( {\lambda}_1^{\ast }=\sqrt{S_2/{S}_1}-1 \). Substituting this positive root back to (23) and (24) yields the optimal solution of the subproblem (18) as
$$ {P}_s^{\ast }={\left[\frac{\sqrt{S_2}}{q_d \ln 2\sqrt{S_1}\left(\xi +\beta {G}_{sr}\right)}-\frac{\varphi_s}{\eta}\right]}^{+} $$
$$ {P}_d^{\ast }={\left[\frac{\sqrt{S_2}}{q_d \ln 2\sqrt{S_1}\left(\xi +\beta {G}_{rd}\right)}-\frac{\varphi_d}{\eta}\right]}^{+} $$
Case 2: λ 1 = 0
In this case, the condition (22) always holds true. Hence, we have
$$ {P}_s^{\ast }={\left[\frac{1}{q_d \ln 2\left(\xi +\beta {G}_{sr}\right)}-\frac{\varphi_s}{\eta}\right]}^{+} $$
$$ {P}_d^{\ast }={\left[\frac{1}{q_d \ln 2\left(\xi +\beta {G}_{rd}\right)}-\frac{\varphi_d}{\eta}\right]}^{+} $$
Note that the solutions of case 2 may break the SE constraint. If this happens, the solution (25), (26) should be adopted. Otherwise, both solutions (25), (26) and (27), (28) are feasible, and the one leading to greater EE is the final solution. Considering the transmission power constraints, the final solution can be expressed as \( {\widehat{P}}_s= min\left({P}_{d, max},{P}_s^{\ast}\right) \) and \( {\widehat{P}}_d= min\left({P}_{d, max},{P}_d^{\ast}\right) \). Since the transformed problem (P2-1) is convex with a given q d , the optimal EE can be obtained with a given β by the Dinkelbach method. According to the above analysis, the detailed procedure of the Dinkelbach method is listed in Table 1.
Table 1 Dinkelbach method
Optimization over β with fixed P s and P d
In this subproblem, our objective is to maximize η d,EE while satisfying the minimum required η d,SE and the constraint of β, which can be formulated as
$$ \begin{array}{l}\underset{\beta }{ \max }\ {\eta}_{d,EE}\\ {}s.t.\ \mathrm{C}3,\mathrm{C}6\end{array} $$
It is easy to prove that η d,SE is monotonically increasing function of β ∈ [0, + ∞), and according to the restricted conditions of η d,SE (i.e., η d,SE ≥ 0, \( {\eta}_{d,SE}\ge {\overline{\eta}}_{d,SE} \)), we have \( \beta \in \left\{\beta \left|\beta \in \left[-\infty, {\overset{\smile }{\beta}}^{\prime}\right]\right.\cup \left[\overset{\smile }{\beta },+\infty \right]\right\} \), where \( {\overset{\smile }{\beta}}^{\prime }=\left(-b-\sqrt{b^2-4ac}\right)/2a \), \( \overset{\smile }{\beta }=\left(-b+\sqrt{b^2-4ac}\right)/2a \), a = c 2 c 3 + c 1 c 4 + c 1 c 3 − ϖc 2 c 4, b = c 3 σ 1 + c 1 σ 2 − ϖc 2 σ 2 − ϖc 4 σ 1, and c = − ϖσ 1 σ 2. Since \( \beta \in \left[0,\overline{\beta}\right] \) and \( \beta \in \left\{\beta \left|\beta \in \left[-\infty, {\overset{\smile }{\beta}}^{\prime}\right]\right.\cup \left[\overset{\smile }{\beta },+\infty \right]\right\} \), the feasible region of the subproblem (P2-2) can be rewritten as \( \beta \in \left[\overset{\smile }{\beta },\overline{\beta}\right] \).
Given P s , P d ∈ [0, + ∞), η d,EE is strictly quasiconcave in β for β ∈ [0, + ∞).
From Theorem 4, similar to the optimization problem of cellular link, there are only three cases for the curve η d,EE vs. β for \( \beta \in \left[\overset{\smile }{\beta },\overline{\beta}\right] \).
Case 1: η d,EE strictly increases with β for \( \beta \in \left[\overset{\smile }{\beta },\overline{\beta}\right] \), if \( \frac{d{\eta}_{d,\mathrm{E}\mathrm{E}}}{d\beta}\left|{}_{\beta =\overline{\beta}}\right.\ge 0 \), where \( \frac{d{\eta}_{d,\mathrm{E}\mathrm{E}}}{d\beta } \) is given by (42) in Appendix 4. In this case, the optimal solution to the subproblem (P2-2) is achieved at \( \widehat{\beta}=\overline{\beta} \).
Case 2: η d,EE strictly decreases with β for \( \beta \in \left[\overset{\smile }{\beta },\overline{\beta}\right] \), if \( \frac{d{\eta}_{d,\mathrm{E}\mathrm{E}}}{d\beta}\left|{}_{\beta =\overset{\smile }{\beta }}\right.\le 0 \). The optimal solution to the subproblem (P2-2) is achieved at \( \widehat{\beta}=\overset{\smile }{\beta } \).
Case 3: η d,EE first strictly increases and then strictly decreases with β for \( \beta \in \left[\overset{\smile }{\beta },\overline{\beta}\right] \) if \( \frac{d{\eta}_{d,\mathrm{E}\mathrm{E}}}{d\beta}\left|{}_{\beta =\overline{\beta}}\right.<0 \) and \( \frac{d{\eta}_{d,\mathrm{E}\mathrm{E}}}{d\beta}\left|{}_{\beta =\overset{\smile }{\beta }}\right.>0 \). The optimal solution to the subproblem (P2-2) is achieved at \( \widehat{\beta}={\beta}^{*} \), where β* is the point at which η d,EE reaches its maximum for β ∈ [0, + ∞) and is obtained by solving the equation \( \frac{d{\eta}_{d,\mathrm{E}\mathrm{E}}}{d\beta }=0 \), as shown in Appendix 4. The exact expression of β* is not easily obtained and can be found by the numerical methods, such as the bisection method.
Iterative optimization algorithm for the FDTW relay-assisted D2D link
According to the previous analysis, by combining the solution processes in Sections 3.2.1 and 3.2.2, the maximum EE of the FDTW relay-assisted D2D link can be achieved. The basic idea is that the subproblem (P2-1) and the subproblem (P2-2) are solved repeated alternately by letting the output of one optimization be the input of the other to reach the solution of the primal problem (P2). The detailed procedure of the iterative optimization algorithm is listed in Table 2.
Table 2 Iterative optimization algorithm
Two-tier alternative iteration optimization algorithm
In the formulated EE optimization problem, due to the existence of interference between cellular and D2D users, the best response of D2D link depends of cellular link; conversely, the best response of cellular link depends of D2D link. Therefore, by combining the solution processes in Section 3.1 and 3.2, the maximum EE of the whole system can be achieved. The basic idea is that the subproblem (P1) and the subproblem (P2) can be repeated alternately by letting the output of one optimization be the input of the other to reach the solution of the primal problem (16). For clarity, the detailed procedure of the two-tier alternative iteration optimization algorithm is listed in Table 3, the first-tier iteration optimization indicates the D2D link optimization problem, and the second-tier iteration optimization indicates the primal optimization problem.
Table 3 Two-tier alternative iteration optimization algorithm
Firstly, under the given initial transmission power of CUE, to solve the EE optimization problem of D2D link, the algorithm list in Table 2 was used. Then, according to the obtained optimal power values and amplification gain value of D2D link, to solve the cellular link EE optimization problem, the bisection method was used. Repeat this process until convergence is reached, i.e., \( \left|{\eta}_{\mathrm{EE}}^{\ast (n)}-{\eta}_{\mathrm{EE}}^{\ast \left(n-1\right)}\right| \) is equal or smaller than a predefined threshold Δ.
Simulation results
In this section, we evaluate the performance of the proposed algorithm by using the MATLAB tool. The maximum transmission powers of SUE, DUE, RUE, and CUE are set to P s,max = 1 W, P d,max = 1 W, P r,max = 1 W and P u,max = 1 W, respectively. The minimum data rate requirements of SUE, DUE, and CUE are set to R s,min = 1 bit/s/Hz, R d,min = 1 bit/s/Hz, and R u,min = 1 bit/s/Hz, respectively. The circuit power consumption is set to P c = 0.1 W, and the power amplifier efficiency of users is set to 1/ξ = 1/0.35. The power consumed by active DC in FD mode is set to 0.1 W. The parameters of path-loss exponent and noise power are set to 4 and −50 dBm, respectively. The bandwidth W is normalized to 1, i.e., W = 1 Hz. These values of simulation parameters and channel gains are inspired by [18] and [26].
Figure 3 shows the locations of D2D users and cellular user. The BS is located at the center with coordinate (0, 0), the coordinates of SUE, DUE, and CUE are (−50, − 300), (50, − 300), and (0, 100), respectively. We mainly consider the effect of RUE on system performance.
The locations of D2D users and cellular user
Figure 4 shows the convergence of the proposed algorithm in terms of average optimal EE. The RUE is located in the middle of SUE and DUE, i.e., d s,r = 50 m. It is shown that the proposed algorithm can achieve convergence quickly. Besides, we can find that the results obtained by the proposed optimization algorithm are very close to those obtained by the exclusive searching method, which indicates that the two-tier alternative iteration optimization algorithm can find a suboptimal solution.
Convergence of the proposed algorithm in terms of average optimal EE (ρ = − 3 dB)
Figures 5 and 6 show the comparison of the proposed algorithm and the exhaustive searching method when the power of SI is ρ = − 3 dB. From the two figures, one can observe that average optimal EE first increases and then decreases as d s,r increases and reaches its maximum value at d s,r = 50 m (i.e., the RUE is located in the center between SUE and the DUE). For average optimal SE, it can satisfy the constraint, which also first increases and then decreases as d s,r increases. In addition, simulation results in Fig. 7 also show that the amplification gain β reaches its peak value at d s,r = 50 m. Hence, according to the above analysis, when the D2D pair needs the relay to assist the transmission, the optimal transmission scheme is that the relay is located in the middle position of the D2D pair.
Average optimal EE versus distance d s,r
Average optimal SE versus distance d s,r
Average β versus distance d s,r
The exhaustive searching method needs to evaluate all the possible values of P s , P d , β, P u , so it needs \( O\left(\frac{P_{s, max}{P}_{d, max}{P}_{u, max}}{\varepsilon^3}\times \frac{\overline{\beta}-\overset{\smile }{\beta }}{\varepsilon}\right) \) steps to obtain the optimal value, where ε is the searching accuracy. In the proposed algorithm, both the bisection method and the Dinkelbach method are adopted alternatively, which takes \( O\left({M}_2\left({M}_1\left({M}_0+{ \log}_2\left(\left(\overline{\beta}-\overset{\smile }{\beta}\right)/\varepsilon \right)\right)+{ \log}_2\left({P}_{u, max}/\varepsilon \right)\right)\right) \) steps, where M 0, M 1, and M 2 are the average iterative steps of the Dinkelbach method, first-tier and second-tier, respectively, and are both set to 10 in our simulation. Therefore, the proposed algorithm significantly reduces the computational complexity compared with the exhaustive search method.
Figures 8 and 9 show the average optimal EE and the average optimal SE versus different residual power of SI under the proposed two-tier alternative iteration optimization algorithm. The EE and the SE decrease with the increase of residual power of SI. Smaller residual power of SI leads to better performance of the system.
Average optimal EE versus different residual power of SI
Average optimal SE versus different residual power of SI
Figures 10 and 11 show the comparison of SE and EE in terms of different residual power of SI. It can be observed in Fig. 10 that the SE obtained by FDTW relay-assisted D2D networks is higher than the SE obtained by HD counterpart, and this is mainly because the FD network can achieve higher SE compared with HD network. However, it also can be seen in Fig. 11 that the EE obtained by FDTW relay-assisted D2D networks is higher than the EE obtained by HD counterpart only when the residual power of SI is sufficiently small. Otherwise, the EE obtained by FDTW relay-assisted D2D networks is lower than the EE obtained by HD counterpart. These phenomena can be intuitively explained as follows. Although the FDTW relay-assisted D2D network can achieve higher SE compared with the HD counterpart, the FD transmission mode has to overcome SI, which needs a larger power to transmit the signal. That is to say, it results in more power consumption than the HD network. Therefore, we can choose the appropriate transmission mode according to the SI cancelation techniques.
Comparison of average optimal SE for FDTW and HDTW relay-assisted D2D networks
Comparison of average optimal EE for FDTW and HDTW relay-assisted D2D networks
In this paper, the EE in FDTW relay-assisted D2D communications has been studied. By taking the SI at D2D users into consideration, the optimization problem in which the goal is to maximize EE while satisfying the SE requirement and the transmission power constraints was formulated. Since the optimization problem (16) is non-concave with respect to the variables P u , P s , P d , and β, we decomposed it into two subproblems, i.e., cellular link optimization problem and D2D link optimization problem. As for the cellular link, the bisection method is used to seek the optimal solution. As for the D2D link, an iterative optimization algorithm was used to find the optimal solution. Then, a new two-tier alternative iteration optimization algorithm is proposed to solve the primal optimization problem. Simulation results show that the results of proposed algorithm are consistent with exhaustive searching results. The simulation results also indicate that smaller SI leads to higher SE and EE. Furthermore, the FDTW relay-assisted D2D network can achieve higher SE than the HDTW relay-assisted D2D network. However, the EE obtained by FDTW relay-assisted D2D underlaying cellular networks is higher than the EE obtained by HD counterpart only when the residual power of SI is sufficiently small.
A Gupta, RK Jha, A survey of 5G network: architecture and emerging technologies. IEEE Access 3, 1206–1232 (2015)
P Demestichas, A Georgakopoulos, D Karvounas, K Tsagkaris, V Stavroulaki, J Lu, C Xiong, J Yao, 5G on the horizon: key challenges for the radio-access network. IEEE Veh Tech Maga 8(3), 47–53 (2013)
A Zakrzewska, S Ruepp, MS Berger, Towards converged 5G mobile networks-challenges and current trends. Proc. ITU Kaleidoscope Academic Conf, 2014, pp. 39–45
S Talwar, D Choudhury, K Dimou, E Aryafar, B Bangerter, K Stewart, Enabling technologies and architectures for 5G wireless. Proc. IEEE Microwave Symp, 2014, pp. 1–4
N Reider, G Fodor, A distributed power control and mode selection algorithm for D2D communications. EURASIP J Wireless Commun Netw 2012(1), 1–25 (2012)
Wen, X. Zhu, Y. Lin, Z. Lin, X. Zhang, D. Yang, "Achievable transmission capacity of relay-assisted device-to-device (D2D) communication underlay cellular networks," in Proc. IEEE VTC Fall, pp. 1–5, Sept. 2013
Z Hasan, H Boostanimehr, VK Bhargava, Green cellular networks: a survey, some research issues and challenges. IEEE Commun Surv Tutorials 13(4), 524–540 (2011)
Y Ni, S Jin, KK Wong, H Zhu, S Shao, Outage performances for device-to-device communication assisted by two-way amplify-and-forward relay protocol. Proc. IEEE Wireless Communications and Networking Conf, 2014, pp. 502–507
Y Zhao, Y Li, N Ge, Physical layer network coding aided two-way device-to-device communication underlaying cellular networks. Proc. IEEE Global Commun. Conf, 2015, pp. 1–6
G Liu, FR Yu, H Ji, VCM Leung, X Li, In-band full-duplex relaying: a survey, research issues and challenges. IEEE Commun Surv Tutorials 17(2), 500–524 (2015)
Z Zhang, K Long, AV Vasilakos, L Hanzo, Full-duplex wireless communications: challenges, solutions, and future research directions. Proc IEEE 104(7), 1369–1409 (2016)
L Song, Y Li, Z Han, Resource allocation in full-duplex communications for future wireless networks. IEEE Wireless Commun 22(4), 88–96 (2015)
E Ahmed, AM Eltawil, Z Li, BA Cetiner, Full-duplex systems using multireconfigurable antennas. IEEE Trans Wireless Commun 14(11), 5971–5983 (2015)
E Ahmed, AM Eltawil, All-digital self-interference cancellation technique for full-duplex systems. IEEE Trans Wireless Commun 14(7), 3519–3532 (2015)
S Li, RD Murch, An investigation into baseband techniques for single-channel full-duplex wireless communication systems. IEEE Trans Wireless Commun 13(9), 4794–4806 (2014)
YA Sambo, MZ Shakir, KA Qaraqe, E Serpedin, Energy efficiency improvements in HetNets by exploiting device-to-device communications. Proc. IEEE Signal Process. Conf., 2014, pp. 151–155
M Sheng, Y Li, X Wang, J Li, Y Shi, Energy efficiency and delay tradeoff in device-to-device communications underlaying cellular networks. IEEE J Sel Areas Commun 34(1), 92–106 (2016)
Z Zhou, M Dong, K Ota, J Wu, T Sato, Energy efficiency and spectral efficiency tradeoff in device-to-device (D2D) communications. IEEE Wireless Commun Lett 3(5), 485–488 (2014)
JMB da Silva, G Fodor, TF Maciel, Performance analysis of network-assisted two-hop D2D communications. Proc. IEEE Globecom Workshops, 2014, pp. 1050–1056
Y Cai, Y Ni, H Zhu, Performance for device-to-device communication with three-time-slot two-way amplify-and-forward relay protocol. China Commun 12(11), 1–11 (2015)
L Wei, R Hu, Y Qian, G Wu, Energy-efficiency and spectrum-efficiency of multi-hop device-to-device communications underlaying cellular networks. IEEE Trans Veh Tech 65(1), 367–380 (2015)
S Schaible, T Ibaraki, Fractional programming. Eur J Oper Res 12(4), 325–338 (1983)
M Duarte, C Dick, A Sabharwal, Experiment-driven characterization of full-duplex wireless systems. IEEE Trans Wireless Commun 11(12), 4296–4307 (2012)
L Song, Y Liao, K Bian, L Song, Z Han, Cross-layer protocol design for CSMA/CD in full-duplex WiFi networks. IEEE Commun Lett 20(4), 792–795 (2016)
D Bharadia, E McMilin, S Katti, Full duplex radios. ACM SIGCOMM Computer Commun Review 43(4), 375–386 (2013)
H. Chen, G. Li, J. Cai, "Spectral-energy efficiency tradeoff in full-duplex two-way relay networks," IEEE Syst. J., published online.
M Jain et al., Practical, real-time, full duplex wireless. Proc. ACM MobiCom, 2011, pp. 301–312
RHY Louie, Y Li, B Vucetic, Practical physical layer network coding for two-way relay channels: performance analysis and comparison. IEEE Trans Wireless Commun 9(2), 764–777 (2010)
Y Zhao, Y Li, H Zhang, N Ge, J Lu, Fundamental tradeoffs on energy-aware D2D communication underlaying cellular networks: a dynamic graph approach. IEEE J Sel Areas Commun 34(4), 864–882 (2016)
G Miao, N Himayat, GY Li, S Talwar, Distributed interference-aware energy-efficient power optimization. IEEE Trans Wireless Commun 10(4), 1323–1333 (2011)
This work was supported by the National Natural Science Foundation of China (61471135, 61671165), the Guangxi Natural Science Foundation (2013GXNSFGA019004, 2015GXNSFBB139007), the Fund of Key Laboratory Cognitive Radio and Information Processing, Guilin University of Electronic Technology, China, and Guangxi Key Laboratory of Wireless Wideband Communication and Signal Processing (CRKL150104, CRKL160105), the Innovation Project of Guangxi Graduate Education (YCSZ2015144), and the Innovation Project of GUET Graduate Education (2016YJCX91).
Key Laboratory of Cognitive Radio and Information Processing, Guilin University of Electronic Technology, Guilin, 541004, China
Yiliang Chang
, Hongbin Chen
& Feng Zhao
Guangxi Experiment Center of Information Science, Guilin, 541004, China
Hongbin Chen
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Correspondence to Hongbin Chen.
Proof of Theorem 1
We take the first-order derivative of η u,EE with respect to P u , which is represented by
$$ \frac{d{\eta}_{u,EE}}{d{P}_u}=\frac{f\left({P}_u\right)}{{\left(\xi {P}_u+{P}_c\right)}^2} $$
$$ f\left({P}_u\right)=\frac{\left(\xi {P}_u+{P}_c\right)}{ \ln 2}\frac{G_{ub}{G}_2}{\left({P}_u{G}_1+{G}_2+{P}_u{G}_{ub}\right)\left({P}_u{G}_1+{G}_2\right)}-\xi { \log}_2\left(1+\frac{P_u{G}_{ub}}{P_u{G}_1+{G}_2}\right) $$
Then, the first-order derivative of f(P u ) with respect to P u can be calculated as
$$ \frac{df\left({P}_u\right)}{d{P}_u}=\frac{-\left(\xi {P}_u+{P}_c\right){G}_{ub}{G}_2\left\{\left({G}_1+{G}_{ub}\right)\left({P}_u{G}_1+{G}_2\right)+\left({P}_u{G}_1+{G}_2+{P}_u{G}_{ub}\right){G}_1\right\}}{ \ln 2{\left({P}_u{G}_1+{G}_2+{P}_u{G}_{ub}\right)}^2{\left({P}_u{G}_1+{G}_2\right)}^2}<0 $$
Therefore, we have f(+∞) < f(P u ) < f(0), ∀ P u ∈ [0, + ∞). Since \( \underset{P_u\to +\infty }{ \lim }f\left({P}_u\right)=-\xi { \log}_2\left({G}_{ub}/{G}_1\right)<0 \) and \( \underset{P_u\to 0}{ \lim }f\left({P}_u\right)=\left({P}_c/ \ln 2\right)\left({G}_{ub}/{G}_2\right)>0 \),there exists a single value of P u , which is denoted as P u *, so that f(P u *) = 0. It is obvious that the denominator of (29) is positive, and we have dη u,EE /dP u > 0 when P u < P u * and dη u,EE /dP u < 0 when P u > P u *. It means that η u,EE firstly increases and then decreases when P u increases. As η u,SE increases monotonically as P u increases, η u,SE is a concave function of P u . P u,total is an affine function of P u . Therefore, η u,EE is quasiconcave in P u for P u ∈ [0, + ∞) [18]. Hence, the proof of Theorem 1 is complete.
We take the first-order partial derivatives of γ s and γ d with respect to P s and P d , respectively, which are represented by
$$ \frac{\partial {\gamma}_s}{\partial {P}_d}=\frac{\beta {G}_{sr}{G}_{rd}}{\beta {P}_u{G}_{sr}{G}_{ur}+\beta {\rho}_r{G}_{sr}{G}_{rr}+\beta {G}_{sr}{\sigma}^2+{P}_u{G}_{us}+{\rho}_r{G}_{ss}+{\sigma}^2}\kern3.5em \frac{\partial {\gamma}_s}{\partial {P}_s}=0 $$
$$ \frac{\partial {\gamma}_d}{\partial {P}_s}=\frac{\beta {G}_{sr}{G}_{rd}}{\beta {P}_u{G}_{rd}{G}_{ur}+\beta {\rho}_r{G}_{rd}{G}_{rr}+\beta {G}_{rd}{\sigma}^2+{P}_u{G}_{ud}+{\rho}_r{G}_{dd}+{\sigma}^2}\kern3em \frac{\partial {\gamma}_d}{\partial {P}_d}=0 $$
Then, according to the formula (9), the first and the second partial derivatives of η d,SE with respect to P s and P d can be calculated as
$$ \frac{\partial {\eta}_{d,SE}}{\partial {P}_s}=\frac{1}{ \ln 2}\left(\frac{\beta {G}_{sr}{G}_{rd}}{\beta {P}_u{G}_{rd}{G}_{ur}+\beta {\rho}_r{G}_{rd}{G}_{rr}+\beta {G}_{rd}{\sigma}^2+{P}_u{G}_{ud}+{\rho}_d{G}_{dd}+{\sigma}^2+\beta {P}_s{G}_{sr}{G}_{rd}}\right)>0 $$
$$ \frac{\partial {\eta}_{d,SE}}{\partial {P}_d}=\frac{1}{ \ln 2}\left(\frac{\beta {G}_{sr}{G}_{rd}}{\beta {P}_u{G}_{sr}{G}_{ur}+\beta {\rho}_r{G}_{sr}{G}_{rr}+\beta {G}_{sr}{\sigma}^2+{P}_u{G}_{us}+{\rho}_s{G}_{ss}+{\sigma}^2+\beta {P}_d{G}_{sr}{G}_{rd}}\right)>0 $$
$$ \frac{\partial^2{\eta}_{d,SE}}{\partial {P}_s^2}=\frac{-1}{ \ln 2}{\left(\frac{\beta {G}_{sr}{G}_{rd}}{\beta {P}_u{G}_{rd}{G}_{ur}+\beta {\rho}_r{G}_{rd}{G}_{rr}+\beta {G}_{rd}{\sigma}^2+{P}_u{G}_{ud}+{\rho}_d{G}_{dd}+{\sigma}^2+\beta {P}_s{G}_{sr}{G}_{rd}}\right)}^2<0 $$
$$ \frac{\partial^2{\eta}_{d,SE}}{\partial {P}_d^2}=\frac{-1}{ \ln 2}{\left(\frac{\beta {G}_{sr}{G}_{rd}}{\beta {P}_u{G}_{sr}{G}_{ur}+\beta {\rho}_r{G}_{sr}{G}_{rr}+\beta {G}_{sr}{\sigma}^2+{P}_u{G}_{us}+{\rho}_s{G}_{ss}+{\sigma}^2+\beta {P}_d{G}_{sr}{G}_{rd}}\right)}^2<0 $$
$$ \frac{\partial^2{\eta}_{d,SE}}{\partial {P}_d\partial {P}_s}=\frac{\partial^2{\eta}_{d,SE}}{\partial {P}_s\partial {P}_d}=0 $$
$$ \frac{\partial^2{\eta}_{d,SE}}{\partial {P}_s^2}\frac{\partial^2{\eta}_{d,SE}}{\partial {P}_d^2}-\frac{\partial^2{\eta}_{d,SE}}{\partial {P}_s\partial {P}_d}\frac{\partial^2{\eta}_{d,SE}}{\partial {P}_d\partial {P}_s}>0 $$
From (30) to (33), we can obtain that the Hessian matrix of η d,SE is negatively definite. Thus, η d,SE is a concave function for P s , P d ∈ [0, + ∞). Hence, the proof of Theorem 2 is complete.
Define F(q d ) = η d,SE(P s , P d ) − q d P d,total(P s , P d ). We prove the equivalence and convexity, respectively.
The proof of equivalence is divided into two parts: necessity and sufficiency. Firstly, we prove the necessity. Suppose that the optimal EE of the subproblem (P2-1) is \( {q}_d^{\ast }={\eta}_{d,\mathrm{S}\mathrm{E}}\left({P}_s^{\ast },{P}_d^{\ast}\right)/{P}_{d,\mathrm{total}}\left({P}_s^{\ast },{P}_d^{\ast}\right) \). Then, for any feasible region of P s and P d , we have
$$ {q}_d^{\ast }=\frac{\eta_{d,\mathrm{S}\mathrm{E}}\left({P}_s^{\ast },{P}_d^{\ast}\right)}{P_{d,\mathrm{total}}\left({P}_s^{\ast },{P}_d^{\ast}\right)}\ge \frac{\eta_{d,\mathrm{S}\mathrm{E}}\left({P}_s,{P}_d\right)}{P_{d,\mathrm{total}}\left({P}_s,{P}_d\right)} $$
By rearranging (38), we obtain
$$ {\eta}_{d,\mathrm{S}\mathrm{E}}\left({P}_s,{P}_d\right)-{q}_d^{\ast }{P}_{d,\mathrm{total}}\left({P}_s,{P}_d\right)\le {\eta}_{d,\mathrm{S}\mathrm{E}}\left({P}_s^{\ast },{P}_d^{\ast}\right)-{q}_d^{\ast }{P}_{d,\mathrm{total}}\left({P}_s^{\ast },{P}_d^{\ast}\right)=0 $$
Hence, the maximum value of \( {\eta}_{d,\mathrm{S}\mathrm{E}}\left({P}_s,{P}_d\right)-{q}_d^{\ast }{P}_{d,\mathrm{total}}\left({P}_s,{P}_d\right) \) is equal to zero and can only be achieved at \( {P}_s={P}_s^{\ast } \), \( {P}_d={P}_d^{\ast } \). This completes the necessity proof.
Next, we turn to the sufficiency proof. Assume that \( {\eta}_{d,\mathrm{S}\mathrm{E}}\left({P}_s,{P}_d\right)-{q}_d^{\ast }{P}_{d,\mathrm{total}}\left({P}_s,{P}_d\right)={\eta}_{d,\mathrm{S}\mathrm{E}}\left({P}_s^{\ast },{P}_d^{\ast}\right)-{q}_d^{\ast }{P}_{d,\mathrm{total}}\left({P}_s^{\ast },{P}_d^{\ast}\right)=0 \). Then, for given any feasible region of P s and P d , we have
By rearranging (40), we have
$$ \frac{\eta_{d,\mathrm{S}\mathrm{E}}\left({P}_s,{P}_d\right)}{P_{d,\mathrm{total}}\left({P}_s,{P}_d\right)}\le \frac{\eta_{d,\mathrm{S}\mathrm{E}}\left({P}_s^{\ast },{P}_d^{\ast}\right)}{P_{d,\mathrm{total}}\left({P}_s^{\ast },{P}_d^{\ast}\right)}={q}_d^{\ast } $$
Thus, the maximum value of EE can be obtained at \( {P}_s={P}_s^{\ast } \), \( {P}_d={P}_d^{\ast } \), and this completes the sufficiency proof. Therefore, the optimal solution to the subproblem (P2-1) and the problem \( { \max}_{P_s,{P}_d}F\left({q}_d^{\ast}\right)=0 \) are equivalent.
Now, we prove the convexity. For any given q d , the transformed subtractive form can be written as η d,SE(P s , P d ) − q d P d,total(P s , P d ). The first part of the subtractive form is η d,SE (P s , P d ), which is a concave function for P s , P d ∈ [0, + ∞) according to Theorem 2. The second part is an affine function. Since the sum of a concave function and an affine function is also concave, this completes the convexity proof.
Firstly, we introduce the following equations and inequality:
c 1 = P d G sr G rd , c 2 = P u G sr G ur + ρ r G sr G rr + G sr σ 2, σ 1 = P u G us + ρ s G ss + σ 2
c 3 = P s G sr G rd , c 4 = P u G rd G ur + ρ r G rd G rr + G rd σ 2, σ 2 = P u G ud + ρ d G dd + σ 2
a 1 = P s G sr + P d G rd + ρ r G rr + P u G ur + σ 2, a 2 = ξP s + ξP d + 3P c
k 1 = c 2 c 4 + c 2 c 3 + c 1 c 4 + c 1 c 3, k 2 = c 2 σ 2 + c 1 σ 2 + c 3 σ 1 + c 4 σ 1, k 3 = c 2 c 4, k 4 = c 2 σ 2 + σ 1 c 4
\( {k}_3^2-{k}_1^2<0 \), k 3 k 4 − k 1 k 2 < 0, \( 2{\sigma}_1{\sigma}_2\left({k}_1-{k}_3\right)+{k}_4^2-{k}_2^2<0 \)
We take the first-order derivatives of η d,EE with respect to β, which is given by
$$ \frac{d{\eta}_{d,EE}}{d\beta }=\frac{f\left(\beta \right)}{{\left({a}_1\beta +{a}_2\right)}^2} $$
$$ f\left(\beta \right)=\frac{\left({a}_1\beta +{a}_2\right)}{ \ln 2}\left(\frac{2{k}_1\beta +{k}_2}{k_1{\beta}^2+{k}_2\beta +{\sigma}_1{\sigma}_2}-\frac{2{k}_3\beta +{k}_4}{k_3{\beta}^2+{k}_4\beta +{\sigma}_1{\sigma}_2}\right)-{a}_1{ \log}_2\frac{k_1{\beta}^2+{k}_2\beta +{\sigma}_1{\sigma}_2}{k_3{\beta}^2+{k}_4\beta +{\sigma}_1{\sigma}_2} $$
Then, the first-order derivatives of f(β) with respect to β can be calculated as
$$ \begin{array}{l}\frac{df\left(\beta \right)}{d\beta}\\ {}=\frac{\left({a}_1\beta +{a}_2\right)}{ \ln 2}\left(\frac{2{k}_1\left({k}_1{\beta}^2+{k}_2\beta +{\sigma}_1{\sigma}_2\right)-{\left(2{k}_1\beta +{k}_2\right)}^2}{{\left({k}_1{\beta}^2+{k}_2\beta +{\sigma}_1{\sigma}_2\right)}^2}\right.\\ {}\kern4.75em \left. - \frac{2{k}_3\left({k}_3{\beta}^2+{k}_4\beta +{\sigma}_1{\sigma}_2\right)-{\left(2{k}_3\beta +{k}_4\right)}^2}{{\left({k}_3{\beta}^2+{k}_4\beta +{\sigma}_1{\sigma}_2\right)}^2}\right)\\ {}\le \frac{\left({a}_1\beta +{a}_2\right)}{ \ln 2}\left(\frac{2\left({k}_3^2-{k}_1^2\right){\beta}^2+2\left({k}_3{k}_4-{k}_1{k}_2\right)\beta +2{\sigma}_1{\sigma}_2\left({k}_1-{k}_3\right)+{k}_4^2-{k}_2^2}{{\left({k}_3{\beta}^2+{k}_4\beta +{\sigma}_1{\sigma}_2\right)}^2}\right)<0,\ \beta \in \left[0,+\infty \right)\end{array} $$
Therefore, we have f(+∞) < f(β) < f(0), ∀ β ∈ [0, + ∞). Since \( \underset{\beta \to +\infty }{ \lim }f\left(\beta \right)=-{a}_1{ \log}_2\left({k}_1/{k}_3\right)<0 \) and \( \underset{\beta \to 0}{ \lim }f\left(\beta \right)=\left({a}_2/ \ln 2\right)\left(\left({k}_2-{k}_4\right)/{\sigma}_1{\sigma}_2\right)>0 \), there exists a single value of β, which is denoted as β*, so that f(β*) = 0. It is obvious that the denominator of (42) is positive, we have dη d,EE /dβ > 0 when β < β* and dη d,EE /dβ < 0 when β > β*. It means that η d,EE firstly increases and then decreases when β increases. As η d,SE increases monotonically as β increases, η d,SE is a concave function of β. P d,total is an affine function of β. Therefore, η d,EE is quasiconcave in β for β ∈ [0, + ∞) [18]. Hence, the proof of Theorem 4 is complete.
Chang, Y., Chen, H. & Zhao, F. Energy efficiency maximization of full-duplex two-way relay-assisted device-to-device communications underlaying cellular networks. J Wireless Com Network 2016, 222 (2016) doi:10.1186/s13638-016-0721-2
Relay-assisted device-to-device (D2D) communications
Full-duplex (FD)
Self-interference (SI)
Energy efficiency (EE)
Spectral efficiency (SE)
Radar and Sonar Networks | CommonCrawl |
Dmitri Olegovich Orlov
Dmitri Olegovich Orlov, (Дмитрий Олегович Орлов, born September 19, 1966 in Vladimir, Russia) is a Russian mathematician, specializing in algebraic geometry. He is known for the Bondal-Orlov reconstruction theorem (2001).[1]
Education and career
In 1988 Orlov graduated from the Faculty of Mechanics and Mathematics of Moscow State University. There he received his Candidate of Sciences degree (PhD) 1991 with thesis Производные категории когерентных пучков, моноидальные преобразования и многообразия Фано (Derived categories of coherent sheaves, monoidal transformations and Fano varieties) under Vasilii Alekseevich Iskovskikh (and Alexey Igorevich Bondal).[2] At the Steklov Institute of Mathematics, Orlov was from April 1996 to April 2011 a researcher in the Algebra Department and is since April 2011 the head of the Algebraic Geometry Department.[3] In 2002 Orlov received his Doctor of Sciences degree (habilitation) with thesis Производные категории когерентных пучков и эквивалентности между ними (Derived categories of coherent sheaves and equivalences between them).[4] In 2002 he was, with A. Bondal, an Invited Speaker with talk Derived categories of coherent sheaves at the International Congress of Mathematicians in Beijing.[5]
Orlov's research deals with homological algebra, (derived categories, triangulated categories), algebraic geometry (derived algebraic geometry, homological mirror symmetry, quasicoherent sheaves, and noncommutative geometry.[6]
Orlov is one of the pioneers of the modern emerging categorical framework which unites the commutative and noncommutative algebraic geometry, via the study of enhanced triangulated categories of quasicoherent sheaves.[7]
He was elected on December 20, 2011 a corresponding member and on 15 November 2019 a full member of the Russian Academy of Sciences.
Selected publications
• with A. Bondal: Semi-orthogonal decomposition for algebraic varieties, Arxiv, 1995
• with A. Bondal: Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Math., vol. 125, 2001, pp. 327–344, Arxiv
• with A. I. Bondal: Derived categories of coherent sheaves, Proc. Internat. Congress of Mathematicians, Peking, 2002, Arxiv
• Quasi-coherent sheaves in commutative and non-commutative geometry, Izv. RAN. Ser. Mat., vol. 67, 2003, pp. 119–138
• Derived categories of coherent sheaves and equivalences between them, Russian Mathematical Surveys, vol. 58, 2003, p. 511
• with A. N. Kapustin: Lectures on mirror symmetry, derived categories, and D-branes, Russian Mathematical Surveys, vol. 59, 2004, pp. 907–940, Arxiv
• Derived categories of coherent sheaves and motives, Russian Mathematical Surveys, vol. 60, 2005, pp. 1242–1244, Arxiv
• with V. A. Lunts, A. I. Efimov: Deformation theory of objects in homotopy and derived categories, Part 1, Advances in Mathematics, vol. 222, 2009, pp. 359–401, Arxiv, Part 2, Advances in Mathematics, vol. 224, 2010, pp. 45–102, Arxiv, Part 3, Advances in Mathematics, vol. 226, 2011, pp. 3857–3911, Arxiv
• with Valery A. Lunts: Uniqueness of enhancement for triangulated categories, J. Amer. Math. Soc., vol. 23, 2010, pp. 853–908, doi:10.1090/S0894-0347-10-00664-8 Arxiv
• Formal completions and idempotent completions of triangulated categories of singularities, Advances in Mathematics, vol. 226, 2011, pp. 206–217, Arxiv
• Landau-Ginzburg Models, D-branes, and Mirror Symmetry, Mat. Contemp., vol. 41, 2012, pp. 75–112, Arxiv
• with Mohammed Abouzaid, Denis Auroux, Alexander I. Efimov, and Ludmil Katzarkov: Homological mirror symmetry for punctured spheres, J. Amer. Math. Soc. vol. 26, 2013, pp. 1051-1083 doi:10.1090/S0894-0347-2013-00770-5
• Derived noncommutative schemes, geometric realizations, and finite dimensional algebras, Russian Math. Surveys, vol. 73, 2018, pp. 865–918, Arxiv
References
1. "Bondal-Orlov reconstruction theorem". nLab (ncatlab.org).
2. Dmitri Olegovich Orlov at the Mathematics Genealogy Project
3. "Dmitri Orlov". ResearchGate.
4. "Орлов Дмитрий Олегович (ИС АРАН)". isaran.ru.
5. Bondal, A.; Orlov, D. (2002). "Derived Categories of Coherent Sheaves" (PDF). Proceedings of the International Congress of Mathematicians. Vol. 2. pp. 47–56.
6. "Orlov, Dmitri Olegovich, Member of the Russian Academy of Sciences". mathnet.ru.
7. "Dmitri Orlov". nLab (ncatlab.org).
External links
• "Orlov Dmitri Olegovich (recent publications)". Steklov Institute.
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| Wikipedia |
Uspekhi Mat. Nauk:
Uspekhi Mat. Nauk, 2001, Volume 56, Issue 1(337), Pages 107–146 (Mi umn358)
This article is cited in 164 scientific papers (total in 165 papers)
Borsuk's problem and the chromatic numbers of some metric spaces
A. M. Raigorodskii
M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract: A detailed survey is given of various results pertaining to two well-known problems of combinatorial geometry: Borsuk's problem on partitions of an arbitrary bounded $d$-dimensional set of non-zero diameter into parts of smaller diameter, and the problem of finding chromatic numbers of some metric spaces. Furthermore, a general method is described for obtaining good lower bounds for the minimum number of parts of smaller diameter into which an arbitrary non-singleton set of dimension $d$ can be divided as well as for the chromatic numbers of various metric spaces, in particular, $\mathbb R^d$ and $\mathbb Q^d$. Finally, some new lower bounds are proved for chromatic numbers in low dimensions, and new natural generalizations of the notion of chromatic number are proposed.
DOI: https://doi.org/10.4213/rm358
Russian Mathematical Surveys, 2001, 56:1, 103–139
UDC: 514.17+519.174
MSC: Primary 51M15, 54E35, 51M20, 05C15; Secondary 52A20, 52C10
Citation: A. M. Raigorodskii, "Borsuk's problem and the chromatic numbers of some metric spaces", Uspekhi Mat. Nauk, 56:1(337) (2001), 107–146; Russian Math. Surveys, 56:1 (2001), 103–139
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A. M. Raigorodskii, "The Borsuk and Hadwiger problems and systems of vectors with restrictions on scalar products", Russian Math. Surveys, 57:3 (2002), 606–607
A. M. Raigorodskii, "The Borsuk problem for integral polytopes", Sb. Math., 193:10 (2002), 1535–1556
Raigorodskii A.M., "Borsuk's problem for $(0,1)$-polytopes and cross-polytopes", Dokl. Math., 65:3 (2002), 413–416
Raigorodskii A.M., "The Erdős-Hadwiger problem and the chromatic numbers of finite geometric graphs", Dokl. Math., 68:2 (2003), 216–220
Hinrichs A., Richter Ch., "New sets with large Borsuk numbers", Discrete Math., 270:1-3 (2003), 137–147
Raigorodskii A.M., "The problem of Borsuk, Hadwinger, and Grunbaum for some classes of polytopes and graphs", Dokl. Math., 67:1 (2003), 85–89
A. M. Raigorodskii, Yu. A. Kalnishkan, "On Borsuk"s Problem in $\mathbb R^3$", Math. Notes, 74:1 (2003), 144–146
A. M. Raigorodskii, "On lower bounds for Borsuk and Hadwiger numbers", Russian Math. Surveys, 59:3 (2004), 585–586
A. M. Raigorodskii, "The chromatic number of a space with the metric $l_q$", Russian Math. Surveys, 59:5 (2004), 973–975
Raigorodskii A.M., "The Borsuk partition problem: the seventieth anniversary", Math. Intelligencer, 26:3 (2004), 4–12
Furedi Z., Kang Jeong-Hyun, "Distance graph on $\mathbb Z^n$ with $l_1$ norm", Theoret. Comput. Sci., 319:1-3 (2004), 357–366
Martini H., Swanepoel K.J., "The geometry of Minkowski spaces—a survey. II", Expo. Math., 22:2 (2004), 93–144
Frankl P., Rödl V., "Strong Ramsey properties of simplices", Israel J. Math., 139 (2004), 215–236
A. M. Raigorodskii, "Colorings of spaces, and random graphs", J. Math. Sci., 146:2 (2007), 5723–5730
A. M. Raigorodskii, "The problems of Borsuk and Grünbaum on lattice polytopes", Izv. Math., 69:3 (2005), 513–537
A. M. Raigorodskii, "The connection between the Borsuk and Erdös–Hadwiger problems", Russian Math. Surveys, 60:4 (2005), 796–798
A. M. Raigorodskii, "The Erdős–Hadwiger problem and the chromatic numbers of finite geometric graphs", Sb. Math., 196:1 (2005), 115–146
Raigorodskii A.M., "The Nelson-Erdős-Hadwiger problem and embeddings of random graphs into geometric ones", Dokl. Math., 72:1 (2005), 516–518
Elsholtz C., Klotz W., "Maximal dimension of unit simplices", Discrete Comput. Geom., 34:1 (2005), 167–177
A. M. Raigorodskii, "On the Borsuk and Erdös–Hadwiger numbers", Math. Notes, 79:6 (2006), 854–863
A. M. Raigorodskii, "On the structure of distance graphs with large chromatic numbers", Math. Notes, 80:3 (2006), 451–453
A. M. Raigorodskii, "The Nelson–Erdős–Hadwiger problem and a space realization of a random graph", Russian Math. Surveys, 61:4 (2006), 783–785
L. L. Ivanov, "An estimate for the chromatic number of the space $\mathbb R^4$", Russian Math. Surveys, 61:5 (2006), 984–986
Raigorodskii, AM, "On the chromatic number of a space with two forbidden distances", Doklady Mathematics, 73:3 (2006), 417
Raigorodskii A.M., "On a series of Ramsey-type problems in combinatorial geometry", Dokl. Math., 75:2 (2007), 221
A. M. Raigorodskii, "On distance graphs with large chromatic number but without large simplices", Russian Math. Surveys, 62:6 (2007), 1224–1225
N. G. Moshchevitin, A. M. Raigorodskii, "Colorings of the Space $\mathbb R^n$ with Several Forbidden Distances", Math. Notes, 81:5 (2007), 656–664
A. M. Raigorodskii, "On Ramsey Numbers for Special Complete Distance Graphs", Math. Notes, 82:3 (2007), 426–429
O. I. Rubanov, "Chromatic Numbers of 3-Dimensional Distance Graphs Containing No Tetrahedra", Math. Notes, 82:5 (2007), 718–721
A. M. Raigorodskii, "Around Borsuk's Hypothesis", Journal of Mathematical Sciences, 154:4 (2008), 604–623
A. M. Raigorodskii, "Chromatic Numbers of Metric Spaces", Journal of Mathematical Sciences, 154:4 (2008), 624–627
Kemnitz, A, "Coloring the line", Ars Combinatoria, 85 (2007), 183
Chen, JJ, "Distance graphs on R-n supercript stop with 1-norm", Journal of Combinatorial Optimization, 14:2–3 (2007), 267
Shitova I.M., "On the chromatic number of a space with several forbidden distances", Dokl. Math., 75:2 (2007), 228–230
V. I. Bogachev, A. M. Raigorodskii, A. B. Skopenkov, N. A. Tolmachev, "Studencheskie olimpiady i mezhkafedralnyi seminar na mekhmate Moskovskogo gosudarstvennogo universiteta", Matem. prosv., ser. 3, 12, Izd-vo MTsNMO, M., 2008, 205–222
A. M. Raigorodskii, M. M. Kityaev, "On a Series of Problems Related to the Borsuk and Nelson–Erdős–Hadwiger Problems", Math. Notes, 84:2 (2008), 239–255
A. M. Raigorodskii, I. M. Shitova, "On the Chromatic Number of Euclidean Space and the Borsuk Problem", Math. Notes, 83:4 (2008), 579–582
A. B. Kupavskii, A. M. Raigorodskii, "On the chromatic number of $\mathbb R^9$", J. Math. Sci., 163:6 (2009), 720–731
A. M. Raigorodskii, I. M. Shitova, "Chromatic numbers of real and rational spaces with real or rational forbidden distances", Sb. Math., 199:4 (2008), 579–612
Nagaeva, SV, "Embeddability of finite distance graphs with a large chromatic number in random graphs", Doklady Mathematics, 77:1 (2008), 13
Zong Ch., "The kissing number, blocking number and covering number of a convex body", Surveys on Discrete and Computational Geometry: Twenty Years Later, Contemporary Mathematics Series, 453, 2008, 529–548
A. M. Raigorodskii, I. I. Timirova, "O probleme Nelsona–Erdesha–Khadvigera dlya odnoi serii metricheskikh prostranstv", Chebyshevskii sb., 9:1 (2008), 158–168
A. È. Guterman, V. K. Lyubimov, A. M. Raigorodskii, S. A. Usachev, "On the Independence Number of Distance Graphs with Vertices in $\{-1,0,1\}^n$", Math. Notes, 86:5 (2009), 744–746
K. A. Mikhailov, A. M. Raigorodskii, "On the Ramsey numbers for complete distance graphs with vertices in $\{0,1\}^n$", Sb. Math., 200:12 (2009), 1789–1806
E. S. Gorskaya, I. M. Mitricheva (Shitova), V. Yu. Protasov, A. M. Raigorodskii, "Estimating the chromatic numbers of Euclidean space by convex minimization methods", Sb. Math., 200:6 (2009), 783–801
Lyubimov V.K., Raigorodskii A.M., "Lower bounds for the independence numbers of some distance graphs with vertices in $\{-1,0,1\}^n$", Dokl. Math., 80:1 (2009), 547–549
S. V. Nagaeva, "On the realization of random graphs as distance graphs in spaces of fixed dimension", Dokl. Math., 79:1 (2009), 63–65
Yu Long, Zong Chuanming, "On the blocking number and the covering number of a convex body", Adv. Geom., 9:1 (2009), 13–29
A. B. Kupavskii, "Lifting of a bound for the chromatic number of $\mathbb R^n$ to higher dimensions", Dokl. Math., 80:3 (2009), 833–836
Payne M.S., "Unit distance graphs with ambiguous chromatic number", Electronic Journal of Combinatorics, 16:1 (2009), N31
A.B. Kupavskii, A.M. Raigorodskii, "On the chromatic numbers of small-dimensional Euclidean spaces", Electronic Notes in Discrete Mathematics, 34 (2009), 435
A. R. Yarmukhametov, "O svyaznosti sluchainykh distantsionnykh grafov spetsialnogo vida", Chebyshevskii sb., 10:1 (2009), 95–108
A. B. Kupavskii, A. M. Raigorodskii, "Partition of Three-Dimensional Sets into Five Parts of Smaller Diameter", Math. Notes, 87:2 (2010), 218–229
A. M. Raigorodskii, O. I. Rubanov, "Distance Graphs with Large Chromatic Number and without Large Cliques", Math. Notes, 87:3 (2010), 392–402
V. P. Filimonov, "Covering planar sets", Sb. Math., 201:8 (2010), 1217–1248
Zhukovskii M.E., "The weak zero-one laws for the random distance graphs", Doklady Mathematics, 81:1 (2010), 51–54
Jeong-Hyun Kang, Hiren Maharaj, "Distance Graphs fromp-adic Norms", Integers, 10:4 (2010), 379
M. E. Zhukovskii, "The weak zero-one law for the random distance graphs", Theory Probab. Appl., 55:2 (2011), 356–360
N. G. Moshchevitin, "Density modulo 1 of lacunary and sublacunary sequences: application of Peres–Schlag's construction", J. Math. Sci., 180:5 (2012), 610–625
Raigorodskii A.M., "Counterexamples to Borsuk's Conjecture on Spheres of Small Radius", Doklady Mathematics, 82:2 (2010), 719–721
Kupavskii A.B., "The Chromatic Number of R-n with a Set of Forbidden Distances", Doklady Mathematics, 82:3 (2010), 963–966
Vasantha W.B., Rajkumar R., "A Class of Cayley Graph Interconnection Networks Based on Rosenbloom-Tsfasman Metric", Iccnt 2009: Proceedings of the 2009 International Conference on Computer and Network Technology, 2010, 66–72
V. F. Moskva, A. M. Raigorodskii, "New Lower Bounds for the Independence Numbers of Distance Graphs with Vertices in $\{-1,0,1\}^n$", Math. Notes, 89:2 (2011), 307–308
A. B. Kupavskii, "On the colouring of spheres embedded in $\mathbb R^n$", Sb. Math., 202:6 (2011), 859–886
M. E. Zhukovskii, "On a sequence of random distance graphs subject to the zero-one law", Problems Inform. Transmission, 47:3 (2011), 251–268
Andrey Kupavskiy, "On the chromatic number of with an arbitrary norm", Discrete Mathematics, 311:6 (2011), 437
Ponomarenko E.I., Raigorodskii A.M., "Some analogues of the Borsuk problem in $q^{n}$", Doklady Mathematics, 83:1 (2011), 59–62
Raigorodskii A.M., "Izbrannye zadachi kombinatornoi geometrii i teorii grafov", Trudy Moskovskogo fiziko-tekhnicheskogo instituta, 3:4 (2011), 127–139
I. M. Mitricheva (Shitova), "On the Chromatic Number for a Set of Metric Spaces", Math. Notes, 91:3 (2012), 399–408
M. E. Zhukovskii, "A weak zero-one law for sequences of random distance graphs", Sb. Math., 203:7 (2012), 1012–1044
A. B. Kupavskii, A. M. Raigorodskii, "Distance graphs with large chromatic numbers and small clique numbers", Dokl. Math, 85:3 (2012), 394
A. R. Yarmuhametov, "Gigantic Component in Random Distance Graphs of Special Form", Math. Notes, 92:3 (2012), 426–441
M. E. Zhukovskii, "On the Probability of the Occurrence of a Copy of a Fixed Graph in a Random Distance Graph", Math. Notes, 92:6 (2012), 756–766
A. R. Yarmuhametov, "Gigantic and Small Components in Random Distance Graphs of Special Form", Math. Notes, 93:1 (2013), 201–205
Raigorodskii A.M., "On the Chromatic Numbers of Spheres in R-N", Combinatorica, 32:1 (2012), 111–123
Kupavskii A.B., Raigorodskii A.M., Titova M.V., "On densest sets omitting distance 1 in spaces of small dimensions", Trudy moskovskogo fiziko-tekhnicheskogo instituta, 4 (2012), 111–121
Yarmukhametov A.R., "O nekotorykh svoistvakh sluchainykh distantsionnykh grafov spetsialnogo vida", Trudy moskovskogo fiziko-tekhnicheskogo instituta, 4 (2012), 12–18
Zvonarev A.E., Raigorodskii A.M., "O distantsionnykh grafakh s bolshim khromaticheskim i malym klikovym chislami", Trudy Moskovskogo fiziko-tekhnicheskogo instituta, 4:1-13 (2012), 122–126
Ponomarenko E.I., Raigorodskii A.M., "O khromaticheskom chisle prostranstva q^{n}", Trudy moskovskogo fiziko-tekhnicheskogo instituta, 4 (2012), 127–130
Kokotkin A.A., Raigorodskii A.M., "O realizatsii sluchainykh grafov grafami diametrov", Trudy moskovskogo fiziko-tekhnicheskogo instituta, 4 (2012), 19–28
Raigorodskii A.M., "Predislovie redaktora nomera", Trudy Moskovskogo fiziko-tekhnicheskogo instituta, 4:1-13 (2012), 4–11
Goldshtein V.B., "O probleme gryunbauma dlya (0,1)- i (-1,0,1)-mnogogrannikov v prostranstvakh maloi razmernosti", Trudy moskovskogo fiziko-tekhnicheskogo instituta, 2012, 41–50
Bulankina V.V., "O razbienii ploskikh mnozhestv na pyat chastei bez rasstoyaniya: (kv. koren iz (2 minus kv. koren iz 3))", Trudy Moskovskogo fiziko-tekhnicheskogo instituta, 4:1-13 (2012), 56–72
Kupavskii A.B., Ponomarenko E.I., Raigorodskii A.M., "O nekotorykh analogakh problemy borsuka v prostranstve q^{n}", Trudy moskovskogo fiziko-tekhnicheskogo instituta, 4 (2012), 81–90
Goldshtein V.B., "O probleme borsuka dlya (0, 1)- i (-1, 0, 1)-mnogogrannikov v prostranstvakh maloi razmernosti", Trudy Moskovskogo fiziko-tekhnicheskogo instituta, 4:1-13 (2012), 91–110
A. M. Raigorodskii, D. V. Samirov, "Chromatic Numbers of Spaces with Forbidden Monochromatic Triangles", Math. Notes, 93:1 (2013), 163–171
E. E. Demekhin, A. M. Raigorodskii, O. I. Rubanov, "Distance graphs having large chromatic numbers and containing no cliques or cycles of a given size", Sb. Math., 204:4 (2013), 508–538
A. B. Kupavskii, M. V. Titova, "Distance ramsey numbers", Dokl. Math, 87:2 (2013), 171
A.B.. Kupavskii, A.M.. Raigorodskii, M.V.. Titova, "New bounds for the distance Ramsey number", Discrete Mathematics, 313:22 (2013), 2566
E.I.. Ponomarenko, A.M.. Raigorodskii, "A new intersection theorem and its applications to bounding the chromatic numbers of spaces", Electronic Notes in Discrete Mathematics, 43 (2013), 241
A. B. Kupavskii, A. M. Raigorodskii, "Obstructions to the realization of distance graphs with large chromatic numbers on spheres of small radii", Sb. Math., 204:10 (2013), 1435–1479
E. I. Ponomarenko, A. M. Raigorodskii, "A new lower bound for the chromatic number of the rational space", Russian Math. Surveys, 68:5 (2013), 960–962
E. I. Ponomarenko, A. M. Raigorodskii, "New estimates in the problem of the number of edges in a hypergraph with forbidden intersections", Problems Inform. Transmission, 49:4 (2013), 384–390
A. A. Kokotkin, A. M. Raigorodskii, "On large subgraphs with small chromatic numbers contained in distance graphs", Journal of Mathematical Sciences, 214:5 (2016), 665–674
V. O. Manturov, "On the chromatic numbers of integer and rational lattices", Journal of Mathematical Sciences, 214:5 (2016), 687–698
M. V. Titova, "One problem on geometric Ramsey numbers", J. Math. Sci., 201:4 (2014), 527–533
A. B. Kupavskii, "Explicit and probabilistic constructions of distance graphs with small clique numbers and large chromatic numbers", Izv. Math., 78:1 (2014), 59–89
V. V. Bulankina, A. B. Kupavskii, A. A. Polyanskii, "On Schur's conjecture in ℝ4", Dokl. Math, 89:1 (2014), 88
S. N. Popova, "Zero-one law for random distance graphs with vertices in $\{-1,0,1\}^n$", Problems Inform. Transmission, 50:1 (2014), 57–78
Andrey Kupavskii, "Diameter Graphs in
$${\mathbb R}^4$$
R 4", Discrete Comput Geom, 2014
E. I. Ponomarenko, A. M. Raigorodskii, "New Upper Bounds for the Independence Numbers of Graphs with Vertices in $\{-1,0,1\}^n$ and Their Applications to Problems of the Chromatic Numbers of Distance Graphs", Math. Notes, 96:1 (2014), 140–148
D. V. Samirov, A. M. Raigorodskii, "New bounds for the chromatic number of a space with forbidden isosceles triangles", Dokl. Math, 89:3 (2014), 313
A. A. Kokotkin, "On Large Subgraphs of a Distance Graph Which Have Small Chromatic Number", Math. Notes, 96:2 (2014), 298–300
V. P. Filimonov, "Covering sets in $\mathbb{R}^m$", Sb. Math., 205:8 (2014), 1160–1200
A. E. Zvonarev, A. M. Raigorodskii, D. V. Samirov, A. A. Kharlamova, "On the chromatic number of a space with forbidden equilateral triangle", Sb. Math., 205:9 (2014), 1310–1333
A. E. Zvonarev, A. M. Raigorodskii, D. V. Samirov, A. A. Kharlamova, "Improvement of the Frankl-Rödl theorem on the number of edges in hypergraphs with forbidden cardinalities of edge intersections", Dokl. Math, 90:1 (2014), 432
L. I. Bogolyubskii, A. S. Gusev, M. M. Pyaderkin, A. M. Raigorodskii, "Independence numbers and chromatic numbers of random subgraphs in some sequences of graphs", Dokl. Math, 90:1 (2014), 462
A. V. Berdnikov, A. M. Raigorodskii, "On the Chromatic Number of Euclidean Space with Two Forbidden Distances", Math. Notes, 96:5 (2014), 827–830
Venkataraman Yegnanarayanan, "Chromatic number of graphs with special distance sets, I", Algebra Discrete Math., 17:1 (2014), 135–160
Kokotkin A.A., "Realization of Subgraphs of Random Graphs By Graphs of Diameters in Euclidean Spaces", Dokl. Math., 89:3 (2014), 362–364
Ponomarenko E.I. Raigorodskii A.M., "An Improvement of the Frankl-Wilson Theorem on the Number of Edges in a Hypergraph With Forbidden Intersections of Edges", Dokl. Math., 89:1 (2014), 59–60
V. V. Bulankina, A. B. Kupavskii, A. A. Polyanskii, "On Schur's Conjecture in $\mathbb R^4$", Math. Notes, 97:1 (2015), 21–29
E. I. Ponomarenko, A. M. Raigorodskii, "New Lower Bound for the Chromatic Number of a Rational Space with One and Two Forbidden Distances", Math. Notes, 97:2 (2015), 249–254
M. E. Zhukovskii, A. M. Raigorodskii, "Random graphs: models and asymptotic characteristics", Russian Math. Surveys, 70:1 (2015), 33–81
A. S. Gusev, "New Upper Bound for the Chromatic Number\of a Random Subgraph of a Distance Graph", Math. Notes, 97:3 (2015), 326–332
A. E. Zvonarev, A. M. Raigorodskii, "Improvements of the Frankl–Rödl theorem on the number of edges of a hypergraph with forbidden intersections, and their consequences in the problem of finding the chromatic number of a space with forbidden equilateral triangle", Proc. Steklov Inst. Math., 288 (2015), 94–104
A. A. Kokotkin, A. M. Raigorodskii, "On the Realization of Subgraphs of a Random Graph by Diameter Graphs in Euclidean Spaces", Math. Notes, 97:5 (2015), 709–724
V. V. Utkin, "Hamiltonian Paths in Distance Graphs", Math. Notes, 97:6 (2015), 919–929
A. V. Krot, A. M. Raigorodskii, "O realizatsii sluchainykh grafov grafami rasstoyanii i diametrov v evklidovykh prostranstvakh", Chebyshevskii sb., 16:2 (2015), 133–143
A. V. Bobu, O. A. Kostina, A. E. Kupriyanov, "Independence numbers and chromatic numbers of some distance graphs", Problems Inform. Transmission, 51:2 (2015), 165–176
M. M. Pyaderkin, "On the stability of the Erdös-Ko-Rado theorem", Dokl. Math, 91:3 (2015), 290
A. M. Raigorodskii, "Lovász' Theorem on the Chromatic Number of Spheres Revisited", Math. Notes, 98:3 (2015), 522–524
L. I. Bogolubsky, A. S. Gusev, M. M. Pyaderkin, A. M. Raigorodskii, "Independence numbers and chromatic numbers of the random subgraphs of some distance graphs", Sb. Math., 206:10 (2015), 1340–1374
A. V. Burkin, "Small subgraphs in random distance graphs", Theory Probab. Appl., 60:3 (2016), 367–382
A. V. Burkin, "The threshold probability for the property of planarity of a random subgraph of a regular graph", Russian Math. Surveys, 70:6 (2015), 1170–1172
Bogolubsky L.I., Raigorodskii A.M., "on the Measurable Chromatic Number of a Space of Dimension N a Parts Per Thousand Currency Sign 24", 92, no. 3, 2015, 761–763
Bobu A.V., Kupriyanov A.E., Raigorodskii A.M., "on the Maximal Number of Edges in a Uniform Hypergraph With One Forbidden Intersection", 92, no. 1, 2015, 401–403
Krot A.V., "on Threshold Probabilities For the Realization of a Random Graph By a Geometric Graph", 92, no. 1, 2015, 480–481
Kostina O.A., Raigorodskii A.M., "on Lower Bounds For the Chromatic Number of Sphere", 92, no. 1, 2015, 500–502
Ph. A. Pushnyakov, "A new estimate for the number of edges in induced subgraphs of a special distance graph", Problems Inform. Transmission, 51:4 (2015), 371–377
M. M. Pyaderkin, "Independence Numbers of Random Subgraphs of a Distance Graph", Math. Notes, 99:2 (2016), 312–319
S. N. Popova, "Zero-one law for random subgraphs of some distance graphs with vertices in $\mathbb Z^n$", Sb. Math., 207:3 (2016), 458–478
Ph. Pushnyakov, "On the Number of Edges in Induced Subgraphs of a Special Distance Graph", Math. Notes, 99:4 (2016), 545–551
M. M. Pyaderkin, "Independence Numbers of Random Subgraphs of Distance Graphs", Math. Notes, 99:4 (2016), 556–563
A. V. Bobu, A. E. Kupriyanov, A. M. Raigorodskii, "Asymptotic study of the maximum number of edges in a uniform hypergraph with one forbidden intersection", Sb. Math., 207:5 (2016), 652–677
A. V. Berdnikov, "Estimate for the Chromatic Number of Euclidean Space with Several Forbidden Distances", Math. Notes, 99:5 (2016), 774–778
S. N. Popova, "Zero-one laws for random graphs with vertices in a Boolean cube", Siberian Adv. Math., 27:1 (2017), 26–75
A. V. Bobu, A. E. Kupriyanov, "On chromatic numbers of close-to-Kneser distance graphs", Problems Inform. Transmission, 52:4 (2016), 373–390
Tikhomirov M., "On computational complexity of length embeddability of graphs", Discrete Math., 339:11 (2016), 2605–2612
Tikhomirov M.I., "On the distance and multidistance graph embeddability problem", Dokl. Math., 93:3 (2016), 280–281
Raigorodskii A.M., "Combinatorial Geometry and Coding Theory*", Fundam. Inform., 145:3 (2016), 359–369
A. B. Kupavskii, A. A. Poljanskij, "On Simplices in Diameter Graphs in $\mathbb R^4$", Math. Notes, 101:2 (2017), 265–276
A. Sagdeev, "Lower Bounds for the Chromatic Numbers of Distance Graphs with Large Girth", Math. Notes, 101:3 (2017), 515–528
A. Sagdeev, "The Chromatic Number of Space with Forbidden Regular Simplex", Math. Notes, 102:4 (2017), 541–546
Yu. A. Demidovich, "Lower Bound for the Chromatic Number of a Rational Space with Metric $l_u$ and with One Forbidden Distance", Math. Notes, 102:4 (2017), 492–507
A. Ya. Kanel-Belov, V. A. Voronov, D. D. Cherkashin, "On the chromatic number of infinitesimal plane layer", St. Petersburg Math. J., 29:5 (2018), 761–775
Bobu A.V., Kupriyanov A.E., Raigorodskii A.M., "On the Number of Edges in a Uniform Hypergraph With a Range of Permitted Intersections", Dokl. Math., 96:1 (2017), 354–357
Cherkashin D.D. Raigorodskii A.M., "On the Chromatic Numbers of Low-Dimensional Spaces", Dokl. Math., 95:1 (2017), 5–6
Raigorodskii A.M. Sagdeev A.A., "On the Chromatic Number of a Space With a Forbidden Regular Simplex", Dokl. Math., 95:1 (2017), 15–16
A. V. Bobu, A. E. Kupriyanov, A. M. Raigorodskii, "On the number of edges of a uniform hypergraph with a range of allowed intersections", Problems Inform. Transmission, 53:4 (2017), 319–342
A. Sokolov, "On the Chromatic Numbers of Rational Spaces", Math. Notes, 103:1-2 (2018), 111–117
A. V. Burkin, M. E. Zhukovskii, "Small subgraphs and their extensions in a random distance graph", Sb. Math., 209:2 (2018), 163–186
R. I. Prosanov, "Upper Bounds for the Chromatic Numbers of Euclidean Spaces with Forbidden Ramsey Sets", Math. Notes, 103:2 (2018), 243–250
Frankl P., Kupayskii A., "Erdos-Ko-Rado Theorem For (0, +/- 1)-Vectors", J. Comb. Theory Ser. A, 155 (2018), 157–179
Cherkashin D. Kulikov A. Raigorodskii A., "On the Chromatic Numbers of Small-Dimensional Euclidean Spaces", Discrete Appl. Math., 243 (2018), 125–131
Frankl P., "An Exact Result For (0, +/- 1)-Vectors", Optim. Lett., 12:5 (2018), 1011–1017
Frankl P., Kupavskii A., "Families of Vectors Without Antipodal Pairs", Stud. Sci. Math. Hung., 55:2 (2018), 231–237
E. S. Gorskaya, I. M. Mitricheva, "The chromatic number of the space $(\mathbb R^n, l_1)$", Sb. Math., 209:10 (2018), 1445–1462
A. Sagdeev, "On the Frankl–Rödl theorem", Izv. Math., 82:6 (2018), 1196–1224
A. A. Sagdeev, "Exponentially Ramsey sets", Problems Inform. Transmission, 54:4 (2018), 372–396
O. A. Kostina, "On Lower Bounds for the Chromatic Number of Spheres", Math. Notes, 105:1 (2019), 16–27
L. I. Bogolubsky, A. M. Raigorodskii, "A Remark on Lower Bounds for the Chromatic Numbers of Spaces of Small Dimension with Metrics $\ell_1$ and $\ell_2$", Math. Notes, 105:2 (2019), 180–203
A. V. Bobu, A. E. Kupriyanov, "Uluchshenie nizhnikh otsenok khromaticheskogo chisla prostranstva s zapreschennymi odnotsvetnymi treugolnikami", Matem. zametki, 105:3 (2019), 349–363
F. A. Pushnyakov, "O kolichestvakh reber v porozhdennykh podgrafakh nekotorykh distantsionnykh grafov", Matem. zametki, 105:4 (2019), 592–602
R. I. Prosanov, "Kontrprimery k gipoteze Borsuka, imeyuschie bolshoi obkhvat", Matem. zametki, 105:6 (2019), 890–898
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References: 117 | CommonCrawl |
I am a bit confused about the definition of weak del pezzo surface. Can someone give an example that what kind of weak del pezzo surface is not a del pezzo surface?
A surface $S$ is del Pezzo if $-K_S$ is ample. It is weak del Pezzo if $-K_S$ is nef and big.
To get examples of (true) weak del Pezzos, remember that a del Pezzo of degree $d$ is the blowup of $\mathbf P^2$ in $9-d$ general points.
If $S$ is the blowup of $\mathbf P^2$ in points $p_1,\ldots,p_r$, then $-K_S=3H-E_1-\cdots-E_r$ (in the obvious notation).
So the trick is to choose the points so that $-K_S$ is nef and big, but has degree $0$ on some curve. For example, choose 6 points in $\mathbf P^2$ such that 3 of them lie on a line. Then on the blowup, $-K_S \cdot L=0$ where $L$ is the proper transform of the line. However, one can verify that $-K_S$ is still basepoint-free, hence nef, and has 4-dimensional space of sections, giving a birational map onto the image of $S$ in $\mathbf P^3$, hence is big.
The simplest example is the second Hirzebruch surface.
Not the answer you're looking for? Browse other questions tagged algebraic-geometry surfaces or ask your own question.
Del Pezzo surface of degree 4 is intersection of two quadrics?
Why are Del Pezzo surfaces rational?
Is it true that if the anticanonical divisor has positive self-intersection then it is ample?
How exactly do vectors, normals and faces relate in surfaces?
Is it always possible to construct a rigid vector bundles with given chern character? | CommonCrawl |
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A new class of integrable Lotka–Volterra systems
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Deep learning as optimal control problems: Models and numerical methods
December 2019, 6(2): 199-222. doi: 10.3934/jcd.2019010
Algebraic structure of aromatic B-series
Geir Bogfjellmo
Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway
Received April 2019 Revised September 2019 Published November 2019
Full Text(HTML)
Aromatic B-series are a generalization of B-series. Some of the algebraic structures on B-series can be defined analogically for aromatic B-series. This paper derives combinatorial formulas for the composition and substitution laws for aromatic B-series.
Keywords: Aromatic series, aromatic trees, composition law, substitution law, coalgebra of aromatic trees, coproduct.
Mathematics Subject Classification: 37C10, 41A58, 16T05.
Citation: Geir Bogfjellmo. Algebraic structure of aromatic B-series. Journal of Computational Dynamics, 2019, 6 (2) : 199-222. doi: 10.3934/jcd.2019010
G. Bogfjellmo and A. Schmeding, The Lie group structure of the Butcher group, Found. Comput. Math., 17 (2017), 127-159. doi: 10.1007/s10208-015-9285-5. Google Scholar
C. Brouder, Runge–Kutta methods and renormalization, European Physical J. C-Particles and Fields, 12 (2000), 521-534. doi: 10.1007/s100529900235. Google Scholar
J. C. Butcher, Coefficients for the study of Runge-Kutta integration processes, J. Austral. Math. Soc., 3 (1963), 185-201. doi: 10.1017/S1446788700027932. Google Scholar
J. C. Butcher, An algebraic theory of integration methods, Math. Comp., 26 (1972), 79-106. doi: 10.1090/S0025-5718-1972-0305608-0. Google Scholar
D. Calaque, K. Ebrahimi-Fard and D. Manchon, Two interacting Hopf algebras of trees: A Hopf-algebraic approach to composition and substitution of B-series, Adv. in Appl. Math., 47 (2011), 282-308. doi: 10.1016/j.aam.2009.08.003. Google Scholar
P. Chartier, E. Hairer and G. Vilmart, Algebraic structures of B-series, Found. Comput. Math., 10 (2010), 407-427. doi: 10.1007/s10208-010-9065-1. Google Scholar
P. Chartier and A. Murua, Preserving first integrals and volume forms of additively split systems, IMA J. Numer. Anal., 27 (2007), 381-405. doi: 10.1093/imanum/drl039. Google Scholar
A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys., 199 (1998), 203-242. doi: 10.1007/s002200050499. Google Scholar
K. Feng and Z. J. Shang, Volume-preserving algorithms for source-free dynamical systems, Numer. Math., 71 (1995), 451-463. doi: 10.1007/s002110050153. Google Scholar
J. B. Fraleigh, A First Course in Abstract Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967. Google Scholar
E. Hairer and G. Wanner, On the Butcher group and general multi-value methods, Computing (Arch. Elektron. Rechnen), 13 (1974), 1-15. doi: 10.1007/BF02268387. Google Scholar
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-30666-8. Google Scholar
A. Iserles, G. R. W. Quispel and P. S. P. Tse, B-series methods cannot be volume-preserving, BIT, 47 (2007), 351-378. doi: 10.1007/s10543-006-0114-8. Google Scholar
D. Manchon, Algebraic background for numerical methods, control theory and renormalization, preprint, arXiv: math/1501.07205. Google Scholar
R. I. McLachlan, K. Modin, H. Munthe-Kaas and O. Verdier, Butcher series: A story of rooted trees and numerical methods for evolution equations, Asia Pac. Math. Newsl., 7 (2017), 1-11. Google Scholar
R. I. McLachlan, K. Modin, H. Munthe-Kaas and O. Verdier, B-series methods are exactly the affine equivariant methods, Numer. Math., 133 (2016), 599-622. doi: 10.1007/s00211-015-0753-2. Google Scholar
H. Munthe-Kaas and O. Verdier, Aromatic Butcher series, Found. Comput. Math., 16 (2016), 183-215. doi: 10.1007/s10208-015-9245-0. Google Scholar
A. Murua, Formal series and numerical integrators, part Ⅰ: Systems of ODEs and symplectic integrators, Appl. Numer. Math., 29 (1999), 221-251. doi: 10.1016/S0168-9274(98)00064-6. Google Scholar
G. M. Poore, Reproducible documents with pythontex, in Proceedings of the 12th Python in Science Conference, 2013, 78–84. Google Scholar
G. R. W. Quispel, Volume-preserving integrators, Phys. Lett. A, 206 (1995), 26-30. doi: 10.1016/0375-9601(95)00586-R. Google Scholar
J. M. Sanz-Serna and A. Murua, Formal series and numerical integrators: Some history and some new techniques, in Proceedings of the 8th International Congress on Industrial and Applied Mathematics, Higher Ed. Press, Beijing, 2015, 311-331. Google Scholar
M. E. Sweedler, Hopf Algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. Google Scholar
Table 1. The coproduct $\Delta_{\mathcal{AT}}$
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Table 2. Substitution law (1 root)
Table 3. Substitution law (0 roots)
Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020464
Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108
Shanding Xu, Longjiang Qu, Xiwang Cao. Three classes of partitioned difference families and their optimal constant composition codes. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020120
Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107
Kateřina Škardová, Tomáš Oberhuber, Jaroslav Tintěra, Radomír Chabiniok. Signed-distance function based non-rigid registration of image series with varying image intensity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1145-1160. doi: 10.3934/dcdss.2020386
Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, Stock price fluctuation prediction method based on time series analysis. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 915-915. doi: 10.3934/dcdss.2019061
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Convergence issues with infinite product of formal series
Question first:
Show that if $s_1 < s_2 < \dots$ is an increasing sequence of positive integers and $P(x)$ is a nonzero polynomial then we cannot have $$ P(x) \equiv \prod_{j=1}^\infty (1 - x^{s_j}) $$ as formal series.
The right-hand side really means $\lim_{N \to \infty} \prod_{j=1}^N (1 - x^{s_j})$ where the notion of convergence is as described in this math.SE answer by Bill.
The intuition is that if we take $x \to 1^-$ then the left-hand side tends to zero at a slower rate than the right-hand side, since $P$ has the root $1$ with only finite multiplicity, while every factor on the RHS tends to zero. However, since formal and functional power series aren't actually the same thing (see link above), I'm not sure how to make this precise, even after formally inverting the $1-x$ factors on the left-hand side (by multiplying both sides by $(1+x+x^2+\dots+)^k$).
Does anyone know the correct incantations?
(For context, this came up in a solution to a recent IMO proposal. Since it was for high school students I swept the convergence issues under the rug, but myself I'd like to know exactly what the right thing to do is.)
co.combinatorics generating-functions enumerative-combinatorics
122 silver badges33 bronze badges
Evan ChenEvan Chen
$\begingroup$ If they are equal then the constant term of $P(x)$ is $1$, so it can be factored as $\prod_{i=1}^m (1 - \alpha_iX)$ over its reciprocal roots in $\mathbf C$. Every formal power series in $1 + X{\mathbf C}[[X]]$ has a unique decomposition as a product $\prod_{k\geq 1} (1 + a_kX^k)$ with $a_k \in \mathbf C$ (you can solve for the $a_k$'s recursively). In particular, $P(X)$ arises like this with a product having finitely many terms, so it is not also such a product with infinitely many terms. $\endgroup$ – KConrad Jul 7 '16 at 20:33
$\begingroup$ Why must $P(X)$ be a finite product? For example, $P(X) := 1+X+X^2$ doesn't appear to be such a finite product, unless I'm missing something. (It appears $P(X) = (1+X)(1+X^2)(1-X^3)\dots$.) $\endgroup$ – Evan Chen Jul 7 '16 at 20:48
$\begingroup$ Ah, I see my error: the unique decomposition I described has terms $1 + a_kX^k$ with distinct $k$, but when you factor a polynomial over its reciprocal roots you will have parts all of degree 1, e.g., $1-X^2 = (1-X)(1+X)$ is not a decomposition into the form $\prod_{k \geq 1} (1 + a_kX^k)$. $\endgroup$ – KConrad Jul 7 '16 at 23:48
Equivalently, we'll show that we cannot have
$$\frac{1}{P(x)} = \frac{1}{\prod_{j=1}^{\infty} (1 - x^{s_j})}$$
as formal power series. The idea is that the LHS has a pole of finite order at $x = 1$ while the RHS has an essential singularity at $x = 1$. Precisely, the coefficients on the LHS have asymptotic growth a polynomial times an exponential. On the other hand, the coefficients of the RHS can be shown to have growth both strictly larger than any polynomial (by truncating the product) and strictly smaller than any exponential (by comparing to the growth rate in the case where $s_j = j$, which is known). So the two rates of growth can't match.
Qiaochu YuanQiaochu Yuan
Just define the RHS as $\sum c_ix^i$, where $$ c_i = \sum_{\substack{i_1 < \dots < i_k \\ s_{i_1} + \dots s_{i_k} = i}} (-1)^k. $$ Then $c_i$ is a finite sum. It's clear that $|c_i| \le p(i) \ll (1+\epsilon)^i$ for all $\epsilon > 0$, where $p(i)$ is the partition function. Therefore, the RHS as defined converges absolutely for all $|x| < 1.$ This allows us to substitute values $x = 1 - \delta_i$ for $\delta_i \rightarrow 0$, and compute the RHS as written. As $x \rightarrow 1$, the RHS $\rightarrow 0$, so $P(x) \rightarrow 0$, so $P(1) = 0.$ Let $P(x) = (1-x)Q(x).$ Considering only values $|x| < 1$ still, we can multiply both sides by $(1+x+\dots)$ formally, and continue taking limits as $x \rightarrow 1.$ Repeating this argument, we can finish.
yangpliuyangpliu
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Truncated sums of symmetric polynomials; reference request for an algebraic derivation
Generating functions for objects with irrational sizes
mod 5 partition identity proof
sum of squares of Schur polynomials indexed over partition valued functions on a set
Is this bound uniform in $N$?
An identity for polynomials over partitions | CommonCrawl |
Double exponential function
A double exponential function is a constant raised to the power of an exponential function. The general formula is $f(x)=a^{b^{x}}=a^{(b^{x})}$ (where a>1 and b>1), which grows much more quickly than an exponential function. For example, if a = b = 10:
• f(x) = 1010x
• f(0) = 10
• f(1) = 1010
• f(2) = 10100 = googol
• f(3) = 101000
• f(100) = 1010100 = googolplex.
Factorials grow faster than exponential functions, but much more slowly than doubly exponential functions. However, tetration and the Ackermann function grow faster. See Big O notation for a comparison of the rate of growth of various functions.
The inverse of the double exponential function is the double logarithm log(log(x)).
Doubly exponential sequences
A sequence of positive integers (or real numbers) is said to have doubly exponential rate of growth if the function giving the nth term of the sequence is bounded above and below by doubly exponential functions of n. Examples include
• The Fermat numbers
$F(m)=2^{2^{m}}+1$
• The harmonic primes: The primes p, in which the sequence 1/2 + 1/3 + 1/5 + 1/7 + ⋯ + 1/p exceeds 0, 1, 2, 3, …
The first few numbers, starting with 0, are 2, 5, 277, 5195977, ... (sequence A016088 in the OEIS)
• The Double Mersenne numbers
$MM(p)=2^{2^{p}-1}-1$
• The elements of Sylvester's sequence (sequence A000058 in the OEIS)
$s_{n}=\left\lfloor E^{2^{n+1}}+{\frac {1}{2}}\right\rfloor $
where E ≈ 1.264084735305302 is Vardi's constant (sequence A076393 in the OEIS).
• The number of k-ary Boolean functions:
$2^{2^{k}}$
• The prime numbers 2, 11, 1361, ... (sequence A051254 in the OEIS)
$a(n)=\left\lfloor A^{3^{n}}\right\rfloor $
where A ≈ 1.306377883863 is Mills' constant.
Aho and Sloane observed that in several important integer sequences, each term is a constant plus the square of the previous term. They show that such sequences can be formed by rounding to the nearest integer the values of a doubly exponential function with middle exponent 2.[1] Ionaşcu and Stănică describe some more general sufficient conditions for a sequence to be the floor of a doubly exponential sequence plus a constant.[2]
Applications
Algorithmic complexity
In computational complexity theory, 2-EXPTIME is the class of decision problems solvable in doubly exponential time. It is equivalent to AEXPSPACE, the set of decision problems solvable by an alternating Turing machine in exponential space, and is a superset of EXPSPACE.[3] An example of a problem in 2-EXPTIME that is not in EXPTIME is the problem of proving or disproving statements in Presburger arithmetic.[4]
In some other problems in the design and analysis of algorithms, doubly exponential sequences are used within the design of an algorithm rather than in its analysis. An example is Chan's algorithm for computing convex hulls, which performs a sequence of computations using test values hi = 22i (estimates for the eventual output size), taking time O(n log hi) for each test value in the sequence. Because of the double exponential growth of these test values, the time for each computation in the sequence grows singly exponentially as a function of i, and the total time is dominated by the time for the final step of the sequence. Thus, the overall time for the algorithm is O(n log h) where h is the actual output size.[5]
Number theory
Some number theoretical bounds are double exponential. Odd perfect numbers with n distinct prime factors are known to be at most $2^{4^{n}}$, a result of Nielsen (2003).[6]
The maximal volume of a polytope in a d-dimensional integer lattice with k ≥ 1 interior lattice points is at most
$k\cdot (8d)^{d}\cdot 15^{d\cdot 2^{2d+1}},$
a result of Pikhurko (2001).[7]
The largest known prime number in the electronic era has grown roughly as a double exponential function of the year since Miller and Wheeler found a 79-digit prime on EDSAC1 in 1951.[8]
Theoretical biology
In population dynamics the growth of human population is sometimes supposed to be double exponential. Varfolomeyev and Gurevich[9] experimentally fit
$N(y)=375.6\cdot 1.00185^{1.00737^{y-1000}}\,$
where N(y) is the population in millions in year y.
Physics
In the Toda oscillator model of self-pulsation, the logarithm of amplitude varies exponentially with time (for large amplitudes), thus the amplitude varies as doubly exponential function of time.[10]
Dendritic macromolecules have been observed to grow in a doubly-exponential fashion.[11]
References
1. Aho, A. V.; Sloane, N. J. A. (1973), "Some doubly exponential sequences", Fibonacci Quarterly, 11: 429–437.
2. Ionaşcu, Eugen-Julien; Stănică, Pantelimon (2004), "Effective asymptotics for some nonlinear recurrences and almost doubly-exponential sequences" (PDF), Acta Mathematica Universitatis Comenianae, LXXIII (1): 75–87.
3. Christos Papadimitriou, Computational Complexity (1994), ISBN 978-0-201-53082-7. Section 20.1, corollary 3, page 495.
4. Fischer, M. J., and Michael O. Rabin, 1974, ""Super-Exponential Complexity of Presburger Arithmetic. Archived 2006-09-15 at the Wayback Machine" Proceedings of the SIAM-AMS Symposium in Applied Mathematics Vol. 7: 27–41
5. Chan, T. M. (1996), "Optimal output-sensitive convex hull algorithms in two and three dimensions", Discrete and Computational Geometry, 16 (4): 361–368, doi:10.1007/BF02712873, MR 1414961
6. Nielsen, Pace P. (2003), "An upper bound for odd perfect numbers", INTEGERS: The Electronic Journal of Combinatorial Number Theory, 3: A14.
7. Pikhurko, Oleg (2001), "Lattice points in lattice polytopes", Mathematika, 48 (1–2): 15–24, arXiv:math/0008028, Bibcode:2000math......8028P, doi:10.1112/s0025579300014339
8. Miller, J. C. P.; Wheeler, D. J. (1951), "Large prime numbers", Nature, 168 (4280): 838, Bibcode:1951Natur.168..838M, doi:10.1038/168838b0.
9. Varfolomeyev, S. D.; Gurevich, K. G. (2001), "The hyperexponential growth of the human population on a macrohistorical scale", Journal of Theoretical Biology, 212 (3): 367–372, Bibcode:2001JThBi.212..367V, doi:10.1006/jtbi.2001.2384, PMID 11829357.
10. Kouznetsov, D.; Bisson, J.-F.; Li, J.; Ueda, K. (2007), "Self-pulsing laser as oscillator Toda: Approximation through elementary functions", Journal of Physics A, 40 (9): 1–18, Bibcode:2007JPhA...40.2107K, doi:10.1088/1751-8113/40/9/016, S2CID 53330023.
11. Kawaguchi, Tohru; Walker, Kathleen L.; Wilkins, Charles L.; Moore, Jeffrey S. (1995). "Double Exponential Dendrimer Growth". Journal of the American Chemical Society. 117 (8): 2159–2165. doi:10.1021/ja00113a005.
| Wikipedia |
PHYSIOLOGY AND BIOTECHNOLOGY
Application of Functional Genomics to Pathway Optimization for Increased Isoprenoid Production
Lance Kizer, Douglas J. Pitera, Brian F. Pfleger, Jay D. Keasling
Lance Kizer
Department of Chemical Engineering, University of California, Berkeley, California 94720-1462
Douglas J. Pitera
Brian F. Pfleger
Jay D. Keasling
Department of Chemical Engineering, University of California, Berkeley, California 94720-1462Department of Bioengineering, University of California, Berkeley, California 94720-1762Synthetic Biology Department, Physical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720
For correspondence: [email protected]
Producing complex chemicals using synthetic metabolic pathways in microbial hosts can have many advantages over chemical synthesis but is often complicated by deleterious interactions between pathway intermediates and the host cell metabolism. With the maturation of functional genomic analysis, it is now technically feasible to identify modes of toxicity associated with the accumulation of foreign molecules in the engineered bacterium. Previously, Escherichia coli was engineered to produce large quantities of isoprenoids by creating a mevalonate-based isopentenyl pyrophosphate biosynthetic pathway (V. J. J. Martin et al., Nat. Biotechnol. 21:796-802, 2003). The engineered E. coli strain produced high levels of isoprenoids, but further optimization led to an imbalance in carbon flux and the accumulation of the pathway intermediate 3-hydroxy-3-methylglutaryl-coenzyme A (HMG-CoA), which proved to be cytotoxic to E. coli. Using both DNA microarray analysis and targeted metabolite profiling, we have studied E. coli strains inhibited by the intracellular accumulation of HMG-CoA. Our results indicate that HMG-CoA inhibits fatty acid biosynthesis in the microbial host, leading to generalized membrane stress. The cytotoxic effects of HMG-CoA accumulation can be counteracted by the addition of palmitic acid (16:0) and, to a lesser extent, oleic acid (cis-Δ9-18:1) in the growth medium. This work demonstrates the utility of using transcriptomic and metabolomic methods to optimize synthetic biological systems.
Production of chemicals via synthetic enzymatic pathways in heterologous hosts has proven useful for many important classes of molecules, including isoprenoids (61), polyketides (49, 50), nonribosomal peptides (61), bioplastics (1), and chemical building blocks (45). Due to the inherent modularity of biological information, synthetic biology holds great potential for expanding this list of microbially produced compounds even further. Yet embedding a novel biochemical pathway in the metabolic network of a host cell can disrupt the subtle regulatory mechanisms that the cell has evolved over the millennia. Indeed, the final yield of a compound is often limited by deleterious effects on the engineered cell's metabolism that are difficult to predict due to our limited understanding of the complex interactions that occur within the cell. The unregulated consumption of cellular resources (18, 37), metabolic burden of heterologous protein production (19, 23, 24), and accumulation of pathway intermediates/products that are inhibitory (5, 62, 63) or toxic (6) to the host are all significant issues that may limit overall yield. A systematic, bottom-up survey to identify potential interactions that limit titer can be time-consuming and requires extensive iteration. Here we report on a top-down, systems biology approach to characterize the toxicity associated with the accumulation of a synthetic pathway intermediate, using DNA microarray analysis coupled with targeted metabolite detection.
Previously, Escherichia coli was engineered to produce amorpha-4,11-diene, the sesquiterpene precursor to the antimalarial compound artemisinin, using a multigene, heterologous pathway (44). Artemisinin, a sesquiterpene lactone endoperoxide extracted from Artemisia annua L (family Asteraceae; commonly known as sweet wormwood), is highly effective against multidrug-resistant Plasmodium spp. and is the key component of the artemisinin-based combination therapies now endorsed as the frontline therapy to treat the disease (2). However, the drug is in short supply and unaffordable to most malaria sufferers. Although total synthesis of artemisinin is difficult and costly, the semisynthesis of artemisinin or any derivative from microbially sourced artemisinic acid, its immediate precursor, represents a cost-effective, environmentally friendly, high-quality, and reliable source of artemisinin for use in artemisinin-based combination therapies. During the design optimization of the amorpha-4,11-diene pathway in E. coli, it was discovered that high expression of the first three genes in this synthetic pathway (atoB, ERG13, and tHMG1) resulted in a severe growth defect. The enzymes encoded by these three genes convert acetyl-coenzyme A (acetyl-CoA) to mevalonate via two intermediate compounds, acetoacetyl-CoA and 3-hydroxy-3-methylglutaryl-CoA (HMG-CoA) (Fig. 1A). While acetoacetyl-CoA is a compound endogenous to E. coli, HMG-CoA is not, and the growth inhibition was shown to correlate with the intracellular accumulation of this exogenous small molecule (51). However, the reason for its toxicity, and thus a solution to the problem, was not known.
(A) The heterologous pathway produces mevalonate from acetyl-CoA in three biochemical steps. The data presented in this article show that the growth inhibition associated with the pathway intermediate HMG-CoA is due to inhibition of malonyl-CoA:ACP transacylase (FabD) in E. coli's type II FAB pathway. The inhibition of the FAB pathway invokes a generalized stress response in the mevalonate-producing strain. As can be seen, the heterologous intermediate is similar in structure to malonyl-CoA, the native substrate for FabD. Abbreviations: A-CoA, acetyl-CoA; AA-CoA, acetoacetyl-CoA; HMG-CoA, hydroxymethylglutaryl-CoA; G6P, glucose-6-phosphate; FDP, fructose-1,6-bisphosphate; G3P, glyceraldehyde-3-phosphate; DHAP, dihydroxyacetone phosphate; PEP, phosphoenolpyruvate. (B) Genes of the heterologous mevalonate-producing pathway in E. coli as well as the inactive-pathway control, which due to a point mutation in ERG13, cannot produce HMG-CoA or mevalonate. Abbreviations: atoB, acetoacetyl-CoA thiolase gene; Erg13, HMG-CoA synthase gene; tHMGR, truncated HMG-CoA reductase gene; Erg13(C159A), inactive HMG-CoA synthase gene.
The toxicity associated with the accumulation of HMG-CoA provided an ideal model system with which to study the utility of top-down systems biology analysis in metabolic engineering. With the maturation of DNA microarray analysis, it is now possible to study the complex interplay between a synthetic biochemical pathway and the endogenous metabolic network of the heterologous host. Since a DNA microarray study gathers data on the entire transcriptome, it is not limited by any initial hypothesis and can identify potential deleterious interactions between an engineered pathway and the host metabolism that would have been difficult to predict a priori. Thus, transcriptional profiling, especially combined with other targeted methods such as metabolite analysis, is a powerful tool for diagnosing problems in the design of an engineered microorganism. Using this approach to study the toxicity caused by accumulation of HMG-CoA in bacteria, we report here that the growth inhibition observed in the mevalonate-producing E. coli strain is due to inhibition of the endogenous type II fatty acid biosynthesis (FAB) pathway by high levels of HMG-CoA (Fig. 1A). This work highlights the unpredictable interactions that can occur in engineering a bacterial cell and illustrates the utility of approaching problems in metabolic engineering by using the tools of systems biology.
Strains, plasmids, and media.The strains and plasmids used in this study are described below and in Table 1 (44, 51).
Strains used in this study
In E. coli DH10B, the arabinose-inducible araBAD promoter (PBAD) system suffers from all-or-none induction (42), in which subsaturating concentrations of arabinose give rise to subpopulations of cells that are fully induced and uninduced. DP10 is an E. coli DH10B derivative containing chromosomal araE, encoding the low-affinity, high-capacity arabinose transport protein, under the control of a constitutive promoter, PCP8, and a deletion of the genes encoding the secondary arabinose transporter, i.e., araFGH. The modifications allow regulatable control of PBAD in a homogeneous population of cells (41).
The low-copy-number plasmid pBAD33MevT contains the previously constructed MevT operon under regulated control of the arabinose-inducible PBAD (44). The operon consists of the genes atoB from E. coli, encoding acetoacetyl-CoA thiolase, and ERG13 and tHMG1 from Saccharomyces cerevisiae, encoding HMG-CoA synthase and truncated HMG-CoA reductase 1, respectively, whose expression allows the in vivo conversion of acetyl-CoA to mevalonate.
In order to investigate the source of toxicity in cells expressing the MevT operon, an inactive-pathway operon was constructed by replacing the catalytic cysteine of S. cerevisiae HMGS (at amino acid position 159), encoded by pBAD33MevT, with an alanine (51). Expression of the modified operon produced active acetoacetyl-CoA thiolase, active HMG-CoA reductase, and full-length, catalytically inactive HMG-CoA synthase. The resulting plasmid was pMevT(C159A) (51).
Medium components and chemicals were purchased from Sigma-Aldrich (St. Louis, MO) and Fisher Scientific (Pittsburgh, PA). For propagation of engineered E. coli strains, Luria broth with Miller's modification (Sigma) was used with appropriate antibiotics for plasmid selection, and 0.06% glucose was used for the repression of the PBAD promoter system. Studies to characterize the cell physiology, metabolite levels, and mRNA transcript levels of the engineered strains were performed using defined C medium (30) supplemented with 3.4% glycerol, all individual amino acids per the work of Neidhardt et al. (46), 4.5 μg/ml thiamine-HCl, micronutrients, and 50 μg/ml (each) chloramphenicol and carbenicillin.
Fatty acids for medium supplementation were purchased from Sigma and employed in defined medium at a concentration of 100 μg/ml in the presence of 400 μg/ml Brij detergent (Sigma).
Growth, metabolite, and mRNA transcript analysis of engineered cells.Cell growth, metabolite concentrations, and mRNA transcript levels of E. coli DP10 expressing genes encoding the active and inactive MevT operons were assayed. Starter cultures of E. coli DP10 harboring pBAD33MevT and pBAD18 (empty vector) or pMevT(C159A) and pBAD18 were inoculated from single colonies and incubated overnight at 37°C in defined C medium supplemented with 0.06% glucose (to repress PBAD) and antibiotics. Overnight starter cultures were diluted to an optical density at 600 nm (OD600) of 0.05 in fresh medium, incubated at 37°C with continuous shaking, and induced with the addition of 1.33 mM (0.02%) arabinose at an OD600 of ∼0.2 to 0.3. During experiments involving metabolite or transcript level analysis, the engineered strains were incubated in baffled shake flasks. Samples were taken at multiple time points during the course of the experiments to assay for cell growth, mevalonate production, transcripts, and intracellular metabolites. Samples for microarray analysis were taken preinduction and during logarithmic growth to capture the early response dynamics of the host to the HMG-CoA toxicity. The OD600 of shake flask samples was measured using a UV spectrophotometer (Beckman), and values for dry cell weight (DCW) were calculated using an equivalence factor of 1 g DCW/liter to an OD600 of 2.5 (4). Experiments were repeated in duplicate to confirm trends in growth and metabolite concentrations.
To screen for fatty acids that alleviate HMG-CoA-associated growth inhibition, cultures were grown in 96-well microtiter plates at 37°C, using defined C medium as described above. The optical density was measured using a microtiter plate reader (SpectraMax; Molecular Devices).
Transcript analysis sample preparation.Biomass designated for transcript analysis was snap-frozen in liquid nitrogen immediately and stored at −80°C until analysis. RNAs were extracted from the cell samples by use of an RNeasy Midi kit (Qiagen). Using 40-μg aliquots of extracted RNA from each sample point, prelabeled cDNA was synthesized using random-primed reverse transcription in a 40-μl volume containing 12.5 μg primers (Invitrogen, Carlsbad, CA), 1× reverse transcriptase buffer (Invitrogen, Carlsbad, CA), 0.01 mM dithiothreitol (Invitrogen, Carlsbad, CA), 1 unit/μl Superase-In (Ambion, Austin, TX), a 0.5 mM concentration (each) of dATP, dCTP, and dGTP (Invitrogen, Carlsbad, CA), 0.1 mM dTTP (Invitrogen, Carlsbad, CA), 0.4 mM amino-allyl-dUTP (Ambion, Austin, TX), and 10 units/μl Superscript II (Invitrogen, Carlsbad, CA), following the enzyme manufacturer's instructions. The cDNA was base hydrolyzed in 100 mM NaOH-10 mM EDTA at 65°C for 10 minutes and then neutralized in HEPES, pH 7.0, at a final concentration of 500 mM. The Tris remaining in the cDNA suspension was removed by three buffer exchange spins using Micron YM-30 columns (Millipore) and eluted in a final volume of 15 μl of water. The cDNA was then labeled using either Alexa 555 or Alexa 647 (Invitrogen, Carlsbad, CA) following the manufacturer's protocol.
Microarray hybridization.Glass microarrays printed with full-length double-stranded DNA prepared by PCR or 70-mer oligonucleotides (Operon) designed to probe every open reading frame of E. coli MG1655 were hybridized in a Tecan hybridization station with ∼6 to 10 μg of labeled cDNA per channel of detection. The hybridization program included a prehybridization step (5× SSC [1× SSC is 0.15 M NaCl plus 0.015 M sodium citrate]-0.2% sodium dodecyl sulfate [SDS]-1% bovine serum albumin, 42°C, 60 min), a 15-hour hybridization step (Ambion Hyb solution 3, 40°C, medium agitation), two low-stringency washes (1× SSC-0.2% SDS, 42°C, 2 min [each]), two high-stringency washes (0.1× SSC-0.2% SDS, 25°C, 2 min [each]), and two final washes (0.1× SSC, 25°C, 2 min [each]). Following hybridization, the slides were scanned with an Axon 4500 scanner.
Transcriptional profile data analysis.The raw scans were globally normalized using Genepix software and then exported to SNOMAD (9) for loess normalization to correct for any hybridization artifacts and to generate local Z scores (see reference 52 for a review of local Z score use in microarray analysis). The local Z scores generated by SNOMAD as well as by serial analysis for microarray (SAM) (60) software were used as guides to determine biologically significant gene expression changes among the replicate hybridization data sets. This list of significant genes was then mined using hierarchal clustering (13) (Cluster 3.0) to determine a base set of clusters in each data set. Once a base set of clusters was chosen, k-means clustering was also used to search the data set for temporal patterns in gene expression (Cluster 3.0).
GC-MS quantification of mevalonate.The mevalonate (mevalonic acid) concentration in cultures of engineered E. coli was determined by gas chromatography-mass spectrometry (GC-MS) analysis as described previously (51). Briefly, culture aliquots were acidified with HCl in glass GC vials to convert mevalonate to mevalonic acid lactone. The acidified cultures were extracted with an equal volume of ethyl acetate containing (−)-trans-caryophyllene (internal standard), and the organic layer was diluted into fresh ethyl acetate prior to GC-MS analysis. Ethyl acetate extracts were analyzed using GC-MS by scanning for only ions m/z 71 and 58, corresponding to mevalonic acid lactone, and ions m/z 189 and 204, corresponding to (−)-trans-caryophyllene. The retention time, mass spectrum, and concentration of extracted mevalonic acid lactone were confirmed using commercial dl-mevalonic acid lactone (Sigma).
Intracellular metabolite extraction and analysis.The concentrations of intracellular acyl-CoAs and adenylate pool were determined by liquid chromatography-MS (LC-MS) analysis of trichloroacetic acid (TCA) culture extracts as described previously (51). To simultaneously and rapidly quench cellular metabolism, isolate E. coli cells from growth medium, and extract metabolites, cells were centrifuged through a layer of silicone oil into a denser solution of TCA. TCA extracts, neutralized with tri-n-octylamine, were analyzed using an Agilent 1100 series LC-MS employing electrospray ionization and operating in positive mode. The following selected ions corresponding to the protonated molecular ion of each metabolite were monitored: for ATP, m/z 508; for ADP, m/z 428; for AMP, m/z 348; for CoA, m/z 768; for acetyl-CoA, m/z 810; for propionyl-CoA, m/z 824; for crotonyl-CoA, m/z 836; for acetoacetyl-CoA, m/z 852; for malonyl-CoA, m/z 854; for succinyl-CoA, m/z 868; for methylmalonyl-CoA, m/z 868; and for HMG-CoA, m/z 912. Retention times, mass spectra, and concentrations of extracted metabolites were confirmed using commercial standards (Sigma). The adenylate energy charge of each strain was calculated from the adenylate pool measurement as defined by Atkinson (3), as follows: $$mathtex$$\[\mathrm{Energy\ charge}{=}\frac{[\mathrm{ATP}]{+}\frac{1}{2}[\mathrm{ADP}]}{[\mathrm{ATP}]{+}[\mathrm{ADP}]{+}[\mathrm{AMP}]}\]$$mathtex$$
Analysis of cellular fatty acid composition.The fatty acid composition and the fatty acid fraction of engineered cells were determined by fatty acid methyl ester (FAME) analysis of lyophilized cell pellets (55). Sample aliquots of the bacterial cultures were centrifuged to pellet the cells, and the biomass was snap-frozen in liquid nitrogen. The frozen pellets were then lyophilized for 24 h at −80°C under vacuum, using a freeze drier. FAME analysis of lyophilized cell pellets was performed by Microbial ID (Newark, DE).
Transcriptomic and metabolic analyses were performed to characterize the HMG-CoA-induced toxicity in the growth-inhibited, mevalonate-producing strain (E. coli DP10 harboring pBAD33MevT). Since the metabolic burden of producing heterologous protein is a well-documented phenomenon (32, 39, 41, 48), we previously designed a control in which the ERG13 gene had a point mutation in its active site and whose expression resulted in a full-length, catalytically inactive protein (54). Thus, this inactive-pathway control [E. coli DP10 harboring pMevT(C159A) and pBAD18] (Fig. 1B) grew under the full burden of heterologous protein production yet suffered no HMG-CoA-associated growth defect since this intermediate could not be synthesized by the mutated ERG13 protein. Because of this, the inactive-pathway strain, E. coli DP10 harboring pMevT(C159A) and pBAD18, provided an ideal control for two-color microarray analysis of the mevalonate-producing, active-pathway strain, E. coli DP10 harboring pBAD33MevT and pBAD18 (Fig. 1B).
Malonyl-CoA accumulates during HMG-CoA-mediated growth inhibition.The growth and acyl-CoA profiles of the active-pathway (mevalonate-producing) strain and the inactive-pathway control strain were analyzed by LC-MS to monitor metabolite changes that correlate with HMG-CoA growth inhibition (Fig. 2). In comparison to the inactive-pathway control, growth of E. coli expressing MevT from pBAD33MevT was inhibited (Fig. 2A), and the growth inhibition correlated directly with the accumulation of HMG-CoA (Fig. 2D), as previously reported (51). This pathway intermediate accumulated to ∼125 nmol/g DCW in the active-pathway strain during log-phase growth and did not accumulate in the inactive-pathway control strain (Fig. 2D).
Growth and metabolite time course profiles for inactive-pathway control strain [E. coli DP10 harboring pMevT(C159A) and pBAD18] (□) and the growth-inhibited, mevalonate-producing strain (E. coli DP10 harboring pBAD33MevT and pBAD18) (▴). Arabinose induction was done at 0.1 g DCW/liter, and the first sample for metabolite extraction was taken at 0.5 h postinduction. Shown are cell growth, in g DCW/liter (A), mevalonate concentration (B), intracellular malonyl-CoA (C), intracellular HMG-CoA (D), intracellular free CoA (E), intracellular acetyl-CoA (F), intracellular acetoacetyl-CoA (G), and energy charge (H).
The most interesting trend noted in the acyl-CoA concentration profiles was the significant diversion of the total acyl-CoA pool to malonyl-CoA during HMG-CoA-associated growth inhibition (Fig. 2C). Malonyl-CoA is a key substrate for the initiation of fatty acid elongation and is typically maintained at relatively low levels in E. coli by the coordinated regulation of long-chain acyl-ACP demand and FAB (<100 nmol/g DCW in the inactive-pathway control strain). Normally, when demand for long-chain acyl-ACP decreases, unused malonyl-CoA is recycled to acetyl-CoA by KAS1 (fabB) and malonyl-CoA-ACP transacylase (encoded by fabD) (28). Yet the growth-inhibited strain converted ∼34% of its total acyl-CoA pool to malonyl-CoA, which represented a >4-fold increase in the intracellular concentration of this fatty acid precursor compared to that in the inactive-pathway control.
Acetyl-CoA, an important central metabolite and precursor to mevalonate, was observed to be twofold lower in the active-pathway strain (Fig. 2F). This decrease in acetyl-CoA concentration following arabinose induction of the MevT genes can be explained partially by diversion of carbon into the engineered pathway. The anomalous accumulation of malonyl-CoA most likely played a role in the lower concentration of acetyl-CoA observed in the active-pathway strain as well, suggesting that the normal regulation of fatty acid metabolite levels was impaired by high concentrations of intracellular HMG-CoA. The accumulation of malonyl-CoA in E. coli DP10 containing pBAD33MevT may also account for the slightly lower level of acetoacetyl-CoA (Fig. 2G) than that in E. coli DP10 harboring pMevT(C159A). While it has been reported that low free CoA levels can inhibit protein production and growth (34), the free CoA concentration in the mevalonate-producing strain was similar to that in the inactive-pathway control strain (Fig. 2E).
The remaining acyl-CoAs that were tracked (propionyl-CoA and succinyl-CoA) did not vary significantly among strains. The adenylate energy charges of all cultures were >0.80 during early logarithmic-phase growth, indicating rapid quenching of cellular metabolism and negligible degradation of adenylates or acyl-CoAs during extraction from the cell (8).
FAB is altered by HMG-CoA accumulation.DNA microarray analysis was performed on the growth-inhibited, mevalonate-producing strain and the inactive-pathway control strain to study the early transcriptional response to HMG-CoA toxicity. RNA was isolated from biomass sampled from each culture just prior to induction with arabinose and at 1 and 3 hours postinduction; the RNA was used to synthesize labeled cDNA. In order to fully elucidate the transcriptional response to HMG-CoA accumulation, the following three separate expression profiles (with technical replicates) were generated by two-color hybridization of the cDNA to DNA microarrays: profile A, the 1- and 3-hour-postinduction transcript profile for the inactive-pathway control [E. coli DP10 containing pMevT(C159A) and pBAD18] relative to a preinduction profile; profile B, the mevalonate-producing strain (E. coli DP10 containing pBAD33MevT and pBAD18) 1- and 3-hour-postinduction transcript profile relative to a preinduction profile; and profile C, the mevalonate-producing strain's 1- and 3-hour-postinduction transcript profile relative to the inactive-pathway strain's 1- and 3-hour-postinduction transcript profile.
There was a consistent up-regulation observed in the expression of the β-ketoacyl-ACP synthase I (encoded by fabB), both over the growth time course of the active-pathway strain (postinduction compared to preinduction) (data not shown) and in the cross-strain comparison (active-pathway strain versus inactive-pathway control strain) (Fig. 3). There was also an up-regulation in transcription of malonyl-CoA-ACP transacylase (encoded by fabD) observed in the ∼3-h postinduction sample, with a smaller up-regulation of fabH. Both fabD and fabH share a common promoter, though there can be extensive posttranscriptional processing of these transcripts (10). FabD is the only enzyme in E. coli that interacts with malonyl-CoA directly, and this occurs during the transfer of the malonyl moiety from CoA to the acyl carrier protein (ACP) (10). Additionally, there were several genes involved in the initial steps of FAB whose expression was up-regulated in mevalonate-producing E. coli, including those encoding biotin biosynthetic enzymes and acetyl-CoA carboxylase (Fig. 3).
Transcript profiles of the initial steps of type II FAB in Escherichia coli. Malonyl-CoA is synthesized from acetyl-CoA by the action of acetyl-CoA carboxylase, a heterotetramer composed of subunits encoded by accABCD. The malonate moiety is transferred from CoA to ACP by the action of malonyl-CoA:ACP transacylase (FabD). See the work of Magnuson et al. for a full review of the E. coli FAB pathway (43). Also shown (inset) are the expression values and Z scores (in parentheses) for FAB genes that exhibited a biologically significant up-regulation in the mevalonate-producing strain (E. coli DP10 containing pBAD33MevT and pBAD18) relative to the inactive-pathway control strain [E. coli DP10 containing pMevT(C159A) and pBAD18] in the microarray analysis. Also shown are the expression profiles for the CFA synthase gene (cfa) as well as two genes in the biotin synthesis pathway, the gene encoding 7,8-diaminopelargonic acid synthase (bioA) and one encoding biotin synthase (bioB).
The alteration in expression of FAB genes and the fourfold increase in malonyl-CoA levels observed in the mevalonate-producing strain indicated that HMG-CoA accumulation altered fatty acid anabolism. Thus, FAME analysis was performed to determine if there were detectable changes in the fatty acid profile that correlated with HMG-CoA accumulation. Indeed, an enrichment of unsaturated fatty acids (UFA) and a decrease in the percentage of saturated fatty acids (SFA) were observed in the membrane lipids of the active-pathway strain compared to the inactive-pathway strain (Fig. 4B and C). Additionally, there was a marked decrease in the amount of fatty acid as a percentage of DCW in the growth-inhibited culture (Fig. 4A).
Cellular fatty acid compositions of the mevalonate-producing strain (E. coli DP10 containing pBAD33MevT and pBAD18) and the inactive-pathway strain [E. coli DP10 containing pMevT(C159A) and pBAD18] obtained by FAME analysis. Open symbols in each graph represent the inactive-pathway strain, and solid symbols represent the mevalonate-producing strain. (A) Total fatty acids as a percentage of DCW. (B) UFA as a percentage of total fatty acids in each strain. ▴ and ▵, 16-carbon UFA (includes 16-carbon CFA); ▪ and □, 18-carbon UFA (includes 18-carbon CFA); and • and ○, CFA (both 16-carbon and 18-carbon CFA). (C) SFA as a percentage of total fatty acids. ▴ and ▵, 16-carbon SFA; ▪ and □, 14-carbon SFA; and • and ○, 12-carbon SFA.
A dramatic up-regulation in the expression of the cyclopropane fatty acid (CFA) synthase gene (cfa) was also observed in the active-pathway strain (Fig. 3). While there has been extensive effort to elucidate the role of CFA in bacteria, there are few clues as to what physiological role they serve (21). Expression of cfa is known to increase upon entry into the stationary phase, and it has been reported that the synthesis of CFA provides E. coli with a method of altering membrane fluidity and/or integrity when normal fatty acid or phospholipid biosynthesis is impeded (21). Again, the fatty acid profiles of membrane lipids correlated with the microarray data quite well, as there was a significant accumulation of CFA observed in the HMG-CoA-stressed cells (Fig. 4B). The high percentage of CFA observed during the log-phase growth of the active-pathway E. coli strain indicated a high degree of stress and a more severe perturbation of FAB than was seen in the inactive-pathway control strain.
The accumulation of HMG-CoA induces a cascade of stress responses.E. coli has evolved several stress response regulons that allow the organism to adapt rapidly to environmental changes. The transcriptional modulation of these regulons (box plots in Fig. 5) provided insight into the toxicity associated with HMG-CoA accumulation in the mevalonate-producing strain. These box plots show the regulation of groups of related genes as a whole based upon EcoCyc (40) and gene ontology classification.
(A) Box plots of transcript expression Z scores for the genes in the σ32 heat shock regulon, oxidative stress-regulated genes (OxyR and SoxS oxidative stress regulons), and osmotically induced operons at 1 hour and 3 hours postinduction. The box plots include the Z scores for all genes detected in the respective functional category based upon EcoCyc (40) annotation and gene ontology classification. The local Z score generated from the SNOMAD analysis represents a change value weighted to account for the confidence in the data, and any gene with a Z score with an absolute value of >1.96 was considered differentially expressed (9). The box plots show the values of the greatest down-regulation, lower quartile (Q1), median, upper quartile (Q3), and largest up-regulation observed for each profile for the regulon or gene group indicated. There were three expression profiles generated in this experiment, as follows: profile A, the 1- and 3-hour-postinduction transcript profile for the inactive-pathway control [E. coli DP10 containing pMevT(C159A) and pBAD18] relative to a preinduction profile; profile B, the mevalonate-producing strain (E. coli DP10 containing pBAD33MevT and pBAD18) 1- and 3-hour-postinduction transcript profile relative to a preinduction profile; and profile C, the mevalonate-producing strain's 1- and 3-hour-postinduction transcript profile relative to the inactive-pathway strain's 1- and 3-hour-postinduction transcript profile. A strong activation of the oxidative and osmotic stress regulons was observed in the active-pathway strain but not in the inactive-pathway control strain. The heat shock regulon was activated early in the active-pathway strain, but overall expression of the regulon was lowered at the 3-hour time point. The heat shock regulon remained highly activated in the inactive-pathway control strain. (B) Time course of expression of genes encoding the 30S and 50S ribosomal proteins as well as the ribosome modulation factor (rmf) observed in the cross-strain microarray analysis (mevalonate-producing strain versus the inactive-pathway control strain). The log2(expression ratio) is defined such that positive values (green) represent up-regulation and negative values (red) represent down-regulation in the mevalonate-producing strain compared to the inactive-pathway control. There was a significant down-regulation of ribosomal protein genes in E. coli DP10 harboring pBAD33MevT (active-pathway strain) compared to their expression in E. coli DP10 harboring pMevT(C159A) (the inactive-pathway control strain).
As shown in the box plots of Z scores (Fig. 5A; Table 2), there was a significant up-regulation of genes involved in osmoregulation in the active-pathway strain, including those encoding trehalose biosynthetic enzymes (otsAB operon), an osmoprotectant/proton symporter (proP), betaine biosynthetic proteins (bet operon), and osmotically inducible genes whose products have not been annotated fully (osmC, osmY, and osmE). No similar osmotic stress response was observed in the inactive-pathway control strain, and indeed, when the two strain's mRNA profiles were compared directly by cross-strain DNA microarray hybridization, the expression levels of the osmoregulatory genes were significantly higher in the growth-inhibited, mevalonate-producing strain than in the inactive-pathway control strain. Of those genes that had biologically significant changes in expression (Table 2), up-regulation of the osmotic response in the active-pathway strain was as high as fivefold for some genes at 3 hours postinduction.
Transcript expression ratios of genes with significant differential expression in the heat shock, osmotic stress, and oxidative stress responsesa
The active-pathway strain also exhibited a significant increase in expression of oxidative stress-associated genes 3 hours after induction (box plot of SoxS and OxyR-regulated genes in Fig. 5A). The up-regulated genes in the HMG-CoA-stressed cells, over the individual time course or in the strain-to-strain comparison, included most of the OxyR regulon, including dps (encoding a DNA binding protein), the suf operon (involved in iron-sulfur cluster repair), grxA (encoding glutaredoxin 1), trxC (encoding thioredoxin 2), gor (encoding glutathione reductase), katG (encoding peroxidase), and ahpC (encoding alkylhydroperoxide reductase) (Table 2). The up-regulation of this stress response regulon is indicative of increased hydrogen peroxide production in the mevalonate-producing strain, especially at the later time point. Again, this trend was not observed in the inactive-pathway control strain; therefore, the H2O2 response appears to be specific to the accumulation of HMG-CoA.
The expression of the mevalonate pathway also elicited a moderate heat shock response consistent with heterologous protein production. As shown in the time course expression profile box plots of σ32-regulated genes in cells harboring the active pathway (Fig. 5A), expression of many members of the heat shock regulon was induced at 1 h postinduction, but then expression of most of these genes decreased at 3 h postinduction. The heat shock regulon genes most highly expressed in the active-pathway strain were those encoding the chaperones ClpB, DnaK, GroEL, and GroES as well as the inclusion body proteins IbpA and IbpB. In contrast, the inactive-pathway control continued to express genes in the heat shock regulon at high levels 3 hours after induction, and the same trend was observed when the two strain's transcriptional profiles were compared directly.
Coincident with expression of the heat shock regulon, there was a strong, two- to fivefold down-regulation of the genes encoding the ribosomal proteins observed in the active-pathway strain at 1 and 3 hours postinduction (time course clustering) (Fig. 5B). A strong heat shock response in E. coli is known to result in a down-regulation of the ribosomal protein genes (12, 53), yet the inactive-pathway strain, which had a much stronger up-regulation of the heat shock genes (Fig. 5A, profile A), did not exhibit a similar down-regulation of the ribosomal genes. Interestingly, the decreased transcription of the ribosomal protein genes in the active-pathway strain correlated with the fourfold up-regulation of the gene encoding the ribosome modulation factor, rmf. This gene, whose product dimerizes the 70S ribosome and reduces translational capacity, is generally expressed during the transition to stationary phase or under conditions of stress (33).
Thus, the two strains both exhibited an activation of the heat shock regulon, but each had a quite different response dynamic. There was a strong expression of this regulon in the active-pathway strain 1 hour after induction, but that response decayed by the third hour. The inactive-pathway control exhibited a steadily increasing induction of the heat shock genes. Heterologous protein production often increases the transcription of heat shock genes encoding molecular chaperones and proteases (12). Thus, the strong down-regulation of the ribosomal proteins, likely part of a coordinated response in the HMG-CoA-stressed cells, limited heterologous protein synthesis and, in turn, led to the general down-regulation of the heat shock regulon observed for the time course of cells harboring pBAD33MevT. Conversely, since the inactive-pathway strain was not stressed by HMG-CoA accumulation, it was able to maintain a higher rate of protein synthesis, which induced increased transcription of heat shock regulon genes, as observed at the later time point. This could explain the observation previously made that HMG-CoA reductase activity is higher in the inactive-pathway strain (51).
Medium supplementation tests.Based on the differential fatty acid auxotrophies reported in the literature for E. coli strains with various mutations of FAB genes, we designed a panel of medium supplementation tests to screen for fatty acids that could relieve the HMG-CoA-associated growth defect. E. coli strain DP10 was transformed with either pBAD33MevT (active pathway), pMevT(C159A) (inactive-pathway control), or pBAD33 (empty vector control) and grown in a defined medium (C medium) in a 96-well plate growth assay. The growth medium was supplemented with 100 μg/ml of oleic acid (cis-Δ9-18:1), palmitoleic acid (cis-Δ9-16:1), palmitic acid (16:0), a combination (16:0 plus cis-Δ9-16:1), or no supplement at all. Both control strains grew well in all medium formulations tested (Fig. 6).
Effect of fatty acid supplementation on growth of mevalonate-producing and control strains of E. coli DP10. (A) Growth of the active-pathway strain [E. coli DP10(pBAD33MevT)] in a defined medium supplemented with 100 μg/ml of oleic acid (cis-Δ9-18:1) (⋄), palmitoleic acid (cis-Δ9-16:1) (□), palmitic acid (16:0) (▵), a combination of fatty acids (16:0 plus cis-Δ9-16:1) (+), or no fatty acid supplement (○) in a 96-well incubator/plate reader monitored continuously at OD600. The growth of the inactive-pathway strain [E. coli DP10 carrying pMevT(C159A)] in nonsupplemented medium (▪) is shown as well. Growth profiles of the inactive-pathway control strain and an empty vector control [E. coli DP10(pBAD33)] were equivalent in medium supplemented with the various fatty acids. (B) Shake flask growth of E. coli DP10(pBAD33MevT) with (•) and without (○) a palmitic acid (16:0) supplement compared to that of the inactive-pathway control [E. coli DP10 carrying pMevT(C159A)] (▪).
The growth inhibition associated with the expression of the active-pathway plasmid was alleviated by 16:0 SFA and 18:1 UFA supplementation and exaggerated by 16:1 UFA supplementation (Fig. 6A). The cultures supplemented with the combination of palmitic and palmitoleic acid grew slightly slower than the cultures with no supplement. Since the growth dynamics in a 96-well plate may not be completely predictive of the physiology of a fully aerobic culture, we grew the active-pathway and inactive-pathway strains in baffled shake flasks in defined medium (C medium), with and without the palmitic acid supplement, and induced expression of the mevalonate pathway genes. Again, the growth inhibition associated with expression of MevT from pBAD33MevT was relieved by the presence of the 16:0 fatty acid (Fig. 6B).
Engineering the heterologous mevalonate pathway into E. coli exposed the host's metabolic network to a biochemical intermediate that the organism had not evolved to counteract. Given the complexity of the metabolic network and the extensive number of interactions that occur in the E. coli cytosol, it is not surprising that a deleterious interaction would arise. While it can be relatively simple to determine that an engineered synthetic biochemical pathway is not functioning in the heterologous host, it is often a far more challenging task to determine exactly what is causing the problem. Using microarray and metabolite analysis of just such a design problem, we have demonstrated that the growth inhibition associated with HMG-CoA accumulation in E. coli DP10 harboring pBAD33MevT is due to inhibition of one or more enzymes involved in the elongation or priming steps of FAB.
It has previously been reported that there is significant conversion of the acyl-CoA pool to malonyl-CoA in E. coli when the initial steps of FAB are inhibited by chemicals, such as the antibiotics thiolactomycin (28, 29) and cerrulenin (17, 29), or genetic methods, such as disruption of protein function (58) and mutation (36). E. coli coordinates FAB with lipid requirements (and hence growth), and the accumulation of long-chain acyl-ACPs is a key signal of slowing growth, which results in the coordinated down-regulation of fabB, fabA, and accAB (27, 28, 35, 36) as well as the inhibition of enzymatic activity (27). During this down-regulation of FAB, excess malonyl-CoA is recycled back to the acetyl-CoA pool through the actions of KAS1 (fabB) and malonyl-CoA-ACP transacylase (fabD) (28). Thus, the impressive accumulation of malonyl-CoA observed in the mevalonate-producing cells is an indication that this complex regulation has been disrupted.
Supplementation of the growth medium with certain fatty acids abolished the HMG-CoA-induced growth inhibition in the active-pathway strain by removing the need for fatty acid anabolism and confirmed the hypothesis that FAB was inhibited, either directly or indirectly, by HMG-CoA accumulation. The results from the entire panel of fatty acid supplements offer insight as to where HMG-CoA may inhibit the FAB pathway. E. coli requires both SFA and UFA to maintain proper membrane fluidity. Harder et al. observed that temperature-sensitive fabB mutants were auxotrophic for trans-UFA or a combination of SFA and cis-UFA at the nonpermissive temperature (25), while fabD mutants grew at the nonpermissive temperature only when the medium was supplemented with 16:0 SFA or 18:1 UFA but not 16:1 UFA (26). Additionally, when this fabD mutant was grown in nonsupplemented medium at a temperature that only partially inhibited growth, the membrane lipids were enriched for 16:1 SFAs. Since the supplementation of UFA or a combination of SFA and UFA failed to relieve the HMG-CoA-associated toxicity, FabB does not appear to be the target of HMG-CoA inhibition in the FAB pathway. Instead, our results suggest that FabD was the target, since inhibited growth of the active-pathway strain in nonsupplemented medium enriched UFA and the addition of palmitic acid (16:0) (and, to a lesser extent, oleic acid [cis-Δ9-18:1]) markedly improved the growth of this strain.
FabD is the only enzyme in E. coli that interacts with malonyl-CoA directly, and this occurs during the transfer of the malonyl moiety from CoA to ACP (10). This reaction is critical to FAB since it provides the two-carbon units needed to extend fatty acids. The reaction mechanism is mediated by an active-site serine that attacks the thioester carbonyl and releases free CoA, leaving a malonyl-FabD complex. Next, the malonyl-FabD ester carbonyl is attacked by the phosphopantetheinyl thiol of ACP, which ultimately yields malonyl-ACP (47). HMG-CoA is structurally very similar to malonyl-CoA and may interfere with substrate binding to the active site. Supporting this hypothesis is the evidence that another acyl-CoA, acetyl-CoA, has been shown to be a weak inhibitor of malonyl-CoA-ACP transacylase activity (38, 47).
The up-regulation of accBC, fabB, fabD, and fabH observed in the growth-inhibited, mevalonate-producing strain is consistent with the hypothesis that HMG-CoA inhibits FAB. This inhibition reduced the availability of long-chain acyl-ACPs, which was sensed by the heterologous host as a decoupling of the rates of FAB and growth. The resulting up-regulation of FAB genes was the host cell's attempt to maintain growth by up-regulating the flux of carbon into fatty acid anabolism. A similar transcriptional response to chemical inhibition of FAB has been documented for Mycobacterium tuberculosis, where exposure to thiolactomycin elicited an up-regulation of the genes encoding β-ketoacyl-ACP synthase I (fabB), malonyl-CoA-ACP transacylase (fabD), and alkyl hydroperoxide reductase C (ahpC) (7), just as observed in the mevalonate-producing strain.
The up-regulation of FAB genes allowed the heterologous host to overcome the most deleterious effect of HMG-CoA, a complete cessation of growth, which is similar in principle to the resistance to thiolactomycin that E. coli gains when fabB is expressed on a multicopy plasmid (11). Yet while the up-regulation of the FAB system was sufficient to allow slow growth in the presence of high levels of cytosolic HMG-CoA, there were consequences to the cell. Jackowski and Rock observed that inhibition of the initial steps of FAB increased the UFA/SFA ratio in the membrane (34) because the inhibition reduced the availability of fatty acids, which, in turn, induced expression of the β-ketoacyl-ACP synthase I (fabB). This enzyme has multiple roles in E. coli's FAB pathway, including both SFA synthesis and the diversion of saturated long-chain acyl-ACPs into the desaturation pathway that produces UFA. Indeed, the increase in FAB gene expression observed in the HMG-CoA-inhibited cells resulted in a coordinate enrichment of UFA in the cell membrane and an overall decrease in fatty acids as a percentage of DCW.
The altered UFA/SFA ratio in the membrane that resulted from HMG-CoA inhibition of fatty acid anabolism in the mevalonate-producing strain could explain the cascade of osmotic, oxidative, and heat shock stress responses observed in that strain's transcriptional profile. Recent studies of the osmosensor ProP in E. coli have discovered that membrane composition is a key signal governing the long-term response to alterations in medium osmolality (59). The oxidative stress response in the active-pathway strain was also an indication of membrane-associated stress, since H2O2 is mostly a by-product of the respiratory electron transport chain in E. coli (20), and both degradation of membrane integrity and osmotic stress have been reported to induce transcription of H2O2 defense genes (14, 57). Finally, the coordinated heat shock response to H2O2 is well established (15, 16), and the sum of all these stress responses appeared ultimately to result in a down-regulation of the translational machinery in the mevalonate-producing strain. Thus, the HMG-CoA-induced changes in membrane composition resulted in a coordinated modulation of many stress response regulons, which may be the ultimate cause for the slow growth observed in HMG-CoA-stressed cells.
In summary, the model for the growth inhibition observed in E. coli expressing the MevT operon is one where the cell's type II fatty acid anabolism was impeded, most likely by inhibition of FabD activity, which in turn limited the availability of long-chain acyl-ACPs. This reduced availability increased the transcription of genes in the initial steps of fatty acid synthesis, which overcame the HMG-CoA inhibition enough to allow slow growth. As a consequence of low levels of acyl-ACP and increased expression of FabB activity, however, the membrane lipids were enriched in UFA. This in turn altered the membrane structural properties and induced the transcription of genes associated with osmotic, oxidative, and heat shock stress. The sum of these responses ultimately limited the growth rate in the mevalonate-producing culture. It should be noted that a similar deleterious interaction does not appear to occur during accumulation of HMG-CoA in S. cerevisiae (56), which has a nondisassociated, type I FAB pathway, and interestingly, the type II pathway antibiotic thiolactomycin is not active against the S. cerevisiae type I FAB pathway. This report highlights the unexpected interactions that can occur when a novel, heterologous biochemical pathway is engineered into a host organism as well as the utility of a systems biology approach to the design problems that inevitably arise in metabolic engineering.
This research was funded by the Bill and Melinda Gates Foundation (through a grant from The Institute of OneWorld Health).
We acknowledge Jack Newman for his assistance.
Received 6 December 2007.
Aldor, I. S., and J. D. Keasling. 2003. Process design for microbial plastic factories: metabolic engineering of polyhydroxyalkanoates. Curr. Opin. Biotechnol. 14:475-483.
Anonymous. 2006. Guidelines for the treatment of malaria. World Health Organization, Geneva, Switzerland.
Atkinson, D. E. 1968. The energy charge of the adenylate pool as a regulatory parameter. Interaction with feedback modifiers. Biochemistry 7:4030-4034.
Bailey, J. E., and D. F. Ollis. 1986. Biochemical engineering fundamentals, 2nd ed. McGraw Hill, New York, NY.
Barbirato, F., J. P. Grivet, P. Soucaille, and A. Bories. 1996. 3-Hydroxypropionaldehyde, an inhibitory metabolite of glycerol fermentation to 1,3-propanediol by enterobacterial species. Appl. Environ. Microbiol. 62:1448-1451.
Berry, A., T. C. Dodge, M. Pepsin, and W. Weyler. 2002. Application of metabolic engineering to improve both the production and use of biotech indigo. J. Ind. Microbiol. Biotechnol. 28:127-133.
Betts, J. C., A. McLaren, M. G. Lennon, F. M. Kelly, P. T. Lukey, S. J. Blakemore, and K. Duncan. 2003. Signature gene expression profiles discriminate between isoniazid-, thiolactomycin-, and triclosan-treated Mycobacterium tuberculosis. Antimicrob. Agents Chemother. 47:2903-2913.
Chapman, A. G., L. Fall, and D. E. Atkinson. 1971. Adenylate energy charge in Escherichia coli during growth and starvation. J. Bacteriol. 108:1072-1086.
Colantuoni, C., G. Henry, S. Zeger, and J. Pevsner. 2002. SNOMAD (standardization and normalization of microarray data): web-accessible gene expression data analysis. Bioinformatics 18:1540-1541.
Cronan, J. E., and C. O. Rock. September 1996, posting date. Biosynthesis of membrane lipids. In R. Curtiss III et al. (ed.), EcoSal—Escherichia coli and Salmonella: cellular and molecular biology. ASM Press, Washington, DC. http://www.ecosal.org.
de Mendoza, D., A. Klages Ulrich, and J. E. Cronan, Jr. 1983. Thermal regulation of membrane fluidity in Escherichia coli. Effects of overproduction of beta-ketoacyl-acyl carrier protein synthase I. J. Biol. Chem. 258:2098-2101.
Dong, H., L. Nilsson, and C. G. Kurland. 1995. Gratuitous overexpression of genes in Escherichia coli leads to growth inhibition and ribosome destruction. J. Bacteriol. 177:1497-1504.
Eisen, M. B., P. T. Spellman, P. O. Brown, and D. Botstein. 1998. Cluster analysis and display of genome-wide expression patterns. Proc. Natl. Acad. Sci. USA 95:14863-14868.
Farr, S. B., and T. Kogoma. 1991. Oxidative stress responses in Escherichia coli and Salmonella typhimurium. Microbiol. Rev. 55:561-585.
Fredriksson, A., M. Ballesteros, S. Dukan, and T. Nystrom. 2005. Defense against protein carbonylation by DnaK/DnaJ and proteases of the heat shock regulon. J. Bacteriol. 187:4207-4213.
Fredriksson, A., M. Ballesteros, S. Dukan, and T. Nystrom. 2006. Induction of the heat shock regulon in response to increased mistranslation requires oxidative modification of the malformed proteins. Mol. Microbiol. 59:350-359.
Furukawa, H., J. T. Tsay, S. Jackowski, Y. Takamura, and C. O. Rock. 1993. Thiolactomycin resistance in Escherichia coli is associated with the multidrug resistance efflux pump encoded by emrAB. J. Bacteriol. 175:3723-3729.
Glick, B. R. 1995. Metabolic load and heterologous gene expression. Biotechnol. Adv. 13:247-261.
Goff, S. A., and A. L. Goldberg. 1985. Production of abnormal proteins in E. coli stimulates transcription of lon and other heat shock genes. Cell 41:587-595.
Gonzalez-Flecha, B., and B. Demple. 1995. Metabolic sources of hydrogen peroxide in aerobically growing Escherichia coli. J. Biol. Chem. 270:13681-13687.
Grogan, D. W., and J. E. Cronan, Jr. 1997. Cyclopropane ring formation in membrane lipids of bacteria. Microbiol. Mol. Biol. Rev. 61:429-441.
Guzman, L. M., et al. 1995. Tight regulation, modulation, and high-level expression by vectors containing the arabinose PBAD promoter. J. Bacteriol. 177:4121-4130.
Harcum, S. W., and W. E. Bentley. 1999. Heat-shock and stringent responses have overlapping protease activity in Escherichia coli. Implications for heterologous protein yield. Appl. Biochem. Biotechnol. 80:23-37.
Harcum, S. W., and W. E. Bentley. 1993. Response dynamics of 26-, 34-, 39-, 54-, and 80-kDa proteases in induced cultures of recombinant Escherichia coli. Biotechnol. Bioeng. 42:675-685.
Harder, M. E., I. R. Beacham, J. E. Cronan, Jr., K. Beacham, J. L. Honegger, and D. F. Silbert. 1972. Temperature-sensitive mutants of Escherichia coli requiring saturated and unsaturated fatty acids for growth: isolation and properties. Proc. Natl. Acad. Sci. USA 69:3105-3109.
Harder, M. E., R. C. Ladenson, S. D. Schimmel, and D. F. Silbert. 1974. Mutants of Escherichia coli with temperature-sensitive malonyl coenzyme A-acyl carrier protein transacylase. J. Biol. Chem. 249:7468-7475.
Heath, R. J., and C. O. Rock. 1996. Regulation of fatty acid elongation and initiation by acyl-acyl carrier protein in Escherichia coli. J. Biol. Chem. 271:1833-1836.
Heath, R. J., and C. O. Rock. 1995. Regulation of malonyl-CoA metabolism by acyl-acyl carrier protein and beta-ketoacyl-acyl carrier protein synthases in Escherichia coli. J. Biol. Chem. 270:15531-15538.
Heath, R. J., S. W. White, and C. O. Rock. 2002. Inhibitors of fatty acid synthesis as antimicrobial chemotherapeutics. Appl. Microbiol. Biotechnol. 58:695-703.
Helmstetter, C., S. Cooper, O. Pierucci, and E. Revelas. 1968. On the bacterial life sequence. Cold Spring Harbor Symp. Quant. Biol. 33:809-822.
Hoffmann, F., and U. Rinas. 2000. Kinetics of heat-shock response and inclusion body formation during temperature-induced production of basic fibroblast growth factor in high-cell-density cultures of recombinant Escherichia coli. Biotechnol. Prog. 16:1000-1007.
Hoffmann, F., and U. Rinas. 2004. Stress induced by recombinant protein production in Escherichia coli. Adv. Biochem. Eng. Biotechnol. 89:73-92.
Izutsu, K., A. Wada, and C. Wada. 2001. Expression of ribosome modulation factor (RMF) in Escherichia coli requires ppGpp. Genes Cells 6:665-676.
Jackowski, S., and C. O. Rock. 1986. Consequences of reduced intracellular coenzyme A content in Escherichia coli. J. Bacteriol. 166:866-871.
James, E. S., and J. E. Cronan. 2004. Expression of two Escherichia coli acetyl-CoA carboxylase subunits is autoregulated. J. Biol. Chem. 279:2520-2527.
Jiang, P., and J. E. Cronan, Jr. 1994. Inhibition of fatty acid synthesis in Escherichia coli in the absence of phospholipid synthesis and release of inhibition by thioesterase action. J. Bacteriol. 176:2814-2821.
Jones, K. L., S. W. Kim, and J. D. Keasling. 2000. Low-copy plasmids can perform as well as or better than high-copy plasmids for metabolic engineering of bacteria. Metab. Eng. 2:328-338.
Joshi, V. C., and S. J. Wakil. 1971. Studies on the mechanism of fatty acid synthesis. XXVI. Purification and properties of malonyl-coenzyme A-acyl carrier protein transacylase of Escherichia coli. Arch. Biochem. Biophys. 143:493-505.
Kanemori, M., H. Mori, and T. Yura. 1994. Induction of heat shock proteins by abnormal proteins results from stabilization and not increased synthesis of sigma 32 in Escherichia coli. J. Bacteriol. 176:5648-5653.
Karp, P. D., M. Riley, M. Saier, I. T. Paulsen, J. Collado-Vides, S. M. Paley, A. Pellegrini-Toole, C. Bonavides, and S. Gama-Castro. 2002. The EcoCyc database. Nucleic Acids Res. 30:56-58.
Khlebnikov, A., K. A. Datsenko, T. Skaug, B. L. Wanner, and J. D. Keasling. 2001. Homogeneous expression of the P(BAD) promoter in Escherichia coli by constitutive expression of the low-affinity high-capacity AraE transporter. Microbiology 147:3241-3247.
Khlebnikov, A., O. Risa, T. Skaug, T. A. Carrier, and J. D. Keasling. 2000. Regulatable arabinose-inducible gene expression system with consistent control in all cells of a culture. J. Bacteriol. 182:7029-7034.
Magnuson, K., S. Jackowski, C. O. Rock, and J. E. Cronan, Jr. 1993. Regulation of fatty acid biosynthesis in Escherichia coli. Microbiol. Rev. 57:522-542.
Martin, V. J. J., D. J. Pitera, S. T. Withers, J. D. Newman, and J. D. Keasling. 2003. Engineering a mevalonate pathway in Escherichia coli for production of terpenoids. 21:796-802.
Nakamura, C. E., and G. M. Whited. 2003. Metabolic engineering for the microbial production of 1,3-propanediol. Curr. Opin. Biotechnol. 14:454-459.
Neidhardt, F. C., P. L. Bloch, and D. F. Smith. 1974. Culture medium for enterobacteria. J. Bacteriol. 119:736-747.
Oefner, C., H. Schulz, A. D'Arcy, and G. E. Dale. 2006. Mapping the active site of Escherichia coli malonyl-CoA-acyl carrier protein transacylase (FabD) by protein crystallography. Acta Crystallogr. D 62:613-618.
Parsell, D. A., and R. T. Sauer. 1989. Induction of a heat shock-like response by unfolded protein in Escherichia coli: dependence on protein level not protein degradation. Genes Dev. 3:1226-1232.
Peiru, S., H. G. Menzella, E. Rodriguez, J. Carney, and H. Gramajo. 2005. Production of the potent antibacterial polyketide erythromycin C in Escherichia coli. Appl. Environ. Microbiol. 71:2539-2547.
Pfeifer, B. A., S. J. Admiraal, H. Gramajo, D. E. Cane, and C. Khosla. 2001. Biosynthesis of complex polyketides in a metabolically engineered strain of E. coli. Science 291:1790-1792.
Pitera, D. J., C. J. Paddon, J. D. Newman, and J. D. Keasling. 2007. Balancing a heterologous mevalonate pathway for improved isoprenoid production in Escherichia coli. Metab. Eng. 9:193-207.
Quackenbush, J. 2002. Microarray data normalization and transformation. Nat. Genet. 32:496-501.
Rinas, U. 1996. Synthesis rates of cellular proteins involved in translation and protein folding are strongly altered in response to overproduction of basic fibroblast growth factor by recombinant Escherichia coli. Biotechnol. Prog. 12:196-200.
Rokosz, L. L., D. A. Boulton, E. A. Butkiewicz, G. Sanyal, M. A. Cueto, P. A. Lachance, and J. D. Hermes. 1994. Human cytoplasmic 3-hydroxy-3-methylglutaryl coenzyme A synthase: expression, purification, and characterization of recombinant wild-type and Cys129 mutant enzymes. Arch. Biochem. Biophys. 312:1-13.
Sasser, M. 1990. Identification of bacteria by gas chromatography of cellular fatty acids. Technical note 101. Microbial ID, Newark, DE.
Seker, T., K. Moller, and J. Nielson. 2005. Analysis of acyl-CoA ester intermediates of the mevalonate pathway in Saccharomyces cerevisiae. Appl. Microbiol. Biotechnol. 67:119-124.
Smirnova, G. V., N. G. Muzyka, and O. N. Oktyabrsky. 2000. The role of antioxidant enzymes in response of Escherichia coli to osmotic upshift. FEMS Microbiol. Lett. 186:209-213.
Subrahmanyam, S., and J. E. Cronan, Jr. 1998. Overproduction of a functional fatty acid biosynthetic enzyme blocks fatty acid synthesis in Escherichia coli. J. Bacteriol. 180:4596-4602.
Tsatskis, Y., J. Khambati, M. Dobson, M. Bogdanov, W. Dowhan, and J. M. Wood. 2005. The osmotic activation of transporter ProP is tuned by both its C-terminal coiled-coil and osmotically induced changes in phospholipid composition. J. Biol. Chem. 280:41387-41394.
Tusher, V. G., R. Tibshirani, and G. Chu. 2001. Significance analysis of microarrays applied to the ionizing radiation response. Proc. Natl. Acad. Sci. USA 98:5116-5121.
Watts, K. T., B. N. Mijts, and C. Schmidt-Dannert. 2005. Current and emerging approaches for natural product biosynthesis in microbial cells. Adv. Synth. Catal. 347:927-940.
Zhu, M. M., P. D. Lawman, and D. C. Cameron. 2002. Improving 1,3-propanediol production from glycerol in a metabolically engineered Escherichia coli by reducing accumulation of sn-glycerol-3-phosphate. Biotechnol. Prog. 18:694-699.
Zhu, M. M., F. A. Skraly, and D. C. Cameron. 2001. Accumulation of methylglyoxal in anaerobically grown Escherichia coli and its detoxification by expression of the Pseudomonas putida glyoxalase I gene. Metab. Eng. 3:218-225.
Applied and Environmental Microbiology May 2008, 74 (10) 3229-3241; DOI: 10.1128/AEM.02750-07
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How can I prove that the four-momentum of relativity is conserved like in classical mechanics?
The four-momentum in special relativity is defined as: $$p = \bigg(\frac{E}{c},\ p_x,\ p_y,\ p_z\bigg)$$ where $$p_x = \gamma (mv_x)$$ $$p_y = \gamma (mv_y)$$ $$p_z = \gamma (mv_z)$$
When solving problems, it is assumed that the individual components of this four-vector at any time will have a constant value. In other words, we assume that this quantity is conserved. How can we prove this fact?
special-relativity momentum conservation-laws
AnonymousAnonymous
Consider the action of a free relativistic particle of mass $m$, given by,
$$S = -m\int dt \, \sqrt{1-\dot x^2}$$
and we can find the Euler-Lagrange equations imply,
$$\frac{\partial}{\partial t}\frac{\partial L}{\partial \dot x} = \frac{\partial}{\partial t} \frac{m\dot x}{\sqrt{1-\dot x^2}} = 0$$
since $\partial L/\partial x = 0$. Note that we can identify $\partial L/\partial \dot x = p = \gamma m\dot x$. Thus, the Euler-Lagrange equations are the statement that the relativistic momenta are conserved. The Hamiltonian is given by a Legendre transform,
$$H = p\dot x - L = \sqrt{m^2 + p^2}$$
and one can see that,
$$\frac{\partial H}{\partial t} = \frac{p \dot p}{\sqrt{m^2+p^2}} = 0$$
since $\dot p =0$ by the Euler-Lagrange equations implying conservation of momentum. We thus have that the energy and relativistic momenta are conserved and so $p^\mu$ itself is conserved.
Both of these conservation laws are a consequence of the fact the system is invariant under translations in time, and the action does not depend on the position of the particle, since only its derivative $\dot x$ appears in the action. By Noether's theorem, this implies energy and momentum conservation.
JamalSJamalS
$\begingroup$ Is there any way to show the conservation law without the use of Lagrangian Mechanics? I am not yet familiar with it. $\endgroup$
$\begingroup$ Playing devil's advocate here: how do you know it's the right action without invoking that it gives the right answer? $\endgroup$
$\begingroup$ @ZeroTheHero It's analogous to the Nambu-Goto action for the relativistic string that sweeps out a worldsheet, except for a particle it sweeps out a wordline. Moreover, the action written in a different form, is equivalent to the one that gives rise to the geodesic equations in general relativity, that is, $\sqrt{g_{\mu\nu}\dot x^\mu \dot x^\nu}$. $\endgroup$
– JamalS
$\begingroup$ @JamalS I don't want to belabour the point too much because I know you have the right action. My observation is that the starting point of this is the experimental observation that the 4-momentum as given in the OP's question is conserved. Whatever action you come up with has to include this conservation law as a consequence. You can come up with all the elegant arguments in the world as to why this or that action should be right, if it is doesn't include conservation of 4-momentum, it's not going anywhere. $\endgroup$
$\begingroup$ @ZeroTheHero I think an argument can be made analogous to bubbles, in that they minimise their surface area with respect to the volume they enclose. Since the action is a one-dimensional version of such an action, maybe there is some way to make a similar argument, that isn't self-referential with respect to the conservation of momentum. $\endgroup$
If a Lagrangian has the symmetry to be translationally invariant, by Noether's theorem, you can show the corresponding conserved quantities are energy and momentum.
gingras.olgingras.ol
There is no a priory way of showing that 4-momentum as defined is conserved. It is however a reasonable guess and, more importantly, it is experimentally verified that this definition leads to conservation of 4-momentum.
I can't for the life of me immediately think of another reasonable definition - it's a 4-vector, it reduces to the usual conservation in the limit of small velocities, etc - but if this definition had failed the experimental test someone would have come up with another definition compatible with experiment.
$\begingroup$ But JamalS has provided a justification for it using the techniques of Lagrangian mechanics. What is wrong with that? $\endgroup$
$\begingroup$ @Anonymous see my comment to Jamal's answer. (Not suggesting JamalS is wrong in anyway.) $\endgroup$
$\begingroup$ So, it's just an experimental fact and not a consequence of the postulates? $\endgroup$
$\begingroup$ @Anonymous As per my last comment to JamalS, AFAIK it is primarily an experimental fact. The form of the 4-vector is dictated by the requirement of properly transforming under Lorentz - that's theoretical. Of course as mentioned above it is immediately included in any theory, and - I want to emphasize this - I don't know of any other reasonable alternative that would compete with the current form. Basically what theory shows is that, if conservation of 4- momentum as defined in your question holds in one inertial frame, it must hold in any other Lorentz-connected frame. $\endgroup$
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Sound propagation at low pressure
I am curious about the properties of sound at low atmospheric pressure. Not the speed of sound, I want to know how lower pressure will affect the distance sound will carry and the frequency range. For example, I know that sound does exist on mars at around 600pa pressure, but travels only a very short distance. Are there any mathematical models for this?
I ask because I have constructed a vacuum chamber for various experiments with a pump rated down to 5pa. At maximum vacuum(I have no way to tell how close to 5pa it is except that water at 0C boils, so below 600pa) sound can be heard through the chamber but only lower frequencies. Above 7khz it is silent, while I can hear 16khz with no vacuum. I wonder if the low pressure is filtering out the high frequency, or is all direct sound transmission lost and the low frequencies are being physically transmitted through the chamber floor?
pressure acoustics vacuum
platatomiplatatomi
$\begingroup$ You didn't show your setup or how you mounted noise-generating device (e.g., bell, audio speaker) in your vacuum chamber, but I would suspect that the low frequencies are being coupled through parts of the vacuum chamber rather than going through the rarefied air in the vacuum chamber. I say this because I don't believe that there is any reason why there should be a strong frequency dependence to the transmission of sound through rarefied air. $\endgroup$
$\begingroup$ Thanks for your answer. The chamber is still a diy work in progress, so I am using sheetmetal for the floor of the chamber through which the vacuum fitting attaches. A jar sits on top of the sheet with wires to a speaker running under the edge, inside the chamber. I am currently using a thick putty to temporarily seal the edge of the jar to the floor. I think the metal floor is conducting and amplifying the sound, I can feel the vibration of the bass in the sheetmetal. $\endgroup$
– platatomi
$\begingroup$ It's probably vibrations getting through whatever solid mounts are holding the speaker. You might give some thought as to how to minimize that as much as possible by using thin wires, sound dampers, etc. $\endgroup$
$\begingroup$ I did try suspending the speaker by wire and wrapping it in foam to isolate the vibrations, but I can still hear the lower frequencies, albeit very faintly. Somewhere I read that the theoretical pressure limit for sound transmission is when the mean free path exceeds the wavelength of the sound. I have no idea if that is true or not. That would explain why only the higher frequencies drop out. But you believe that at pressures possibly under 100pa that no discernable sound propagation should occur? $\endgroup$
$\begingroup$ Hard to say without hearing what your system sounds like as it is being pumped down. There's a system similar to your setup at the Exploratorium (exs.exploratorium.edu/exhibits/no-sound-through-empty-space) I recall that the sound intensity dropped off dramatically as it was pumped down to roughing-pump type vacuums, which seems similar to your vacuum. $\endgroup$
The gory details of this are found in the answer at https://physics.stackexchange.com/a/266046/59023.
Not the speed of sound, I want to know how lower pressure will affect the distance sound will carry and the frequency range. For example, I know that sound does exist on mars at around 600pa pressure, but travels only a very short distance. Are there any mathematical models for this?
Yes, the thing you are looking for is called acoustic impedance, which decreases with decreasing ambient pressure. You may think that a decreasing impedance would allow a sound wave to propagate further, but the reference sound intensity, $I_{o}$, depends upon the characteristic acoustic impedance, $z_{o}$, as: $$ I_{o} = P_{o}^{2}/z_{o} \tag{0} $$ where $P_{o}$ is a constant reference pressure here associated with the hearing threshold, i.e., ~20 $\mu$Pa at 1000 Hz (it's not flat across frequency, but adding the frequency dependence is not necessary to illustrate the main point). The characteristic acoustic impedance is defined as: $$ z_{o} = \rho \ C_{s} \tag{1} $$ where $\rho$ is the mass density and $C_{s}$ is the speed of sound.
The point where such a sound wave would experience strong damping is where the collisional mean free path becomes too large to support the oscillations, i.e., this would occur when the average time between collisions becomes comparable to the wave frequency. Thus, the oscillations would have no restoring force and would damp out.
In weakly damped systems, the intensity of sound decreases as $I\left( r \right) \propto r^{-2}$ while sound pressure decreases as $P\left( r \right) \propto r^{-1}$. If we look at a rough estimate of the atmospheric pressure as a function of altitude in Earth's atmosphere, we reach ~600 Pa by ~43 km.
Using the table in the answer at https://physics.stackexchange.com/a/266046/59023 for 40 km altitude, the magnitudes of $I_{o}$ and $z_{o}$ are ~1.155 x 10-10 W m-2 and ~3.462 Pa s m-1, respectively (at sea level and STP, these satisfy $z_{o}$ ~ 428 Pa s m-1 and $I_{o}$ ~ 9.346 x 10-13 W m-2).
Suppose we start with a sound intensity level of $L_{o}$ = 100 dB and we know that the intensity of the source is given as: $$ I_{src}\left( h \right) = I_{o}\left( h \right) 10^{L_{o}/10} \tag{2} $$ where $h$ is the altitude. So a 100 dB source at sea level would start with $I_{src}\left( 0 \ km \right)$ ~ 9.346 x 10-3 W m-2. To maintain the same intensity at ~40 km, the source intensity would have to increase to $I_{src}\left( 40 \ km \right)$ ~ 1.155 x 10+0 W m-2, i.e., increase by a factor of ~124.
The sound level intensity at a distance $r$ from the source is given by: $$ L_{r}\left( h, r \right) = L_{i,src}\left( h \right) + 20 \ \log_{10} \left( \frac{ 1 }{ r } \right) \tag{3} $$ where a 1 m normalizing distance is used and the source sound level intensity relative to sea level is defined as: $$ L_{i,src}\left( h \right) = 10 \ \log_{10} \left( \frac{ I_{src}\left( 0 \ km \right) }{ I_{o}\left( h \right) } \right) \tag{4} $$ You can see that $L_{i,src}\left( 0 \ km \right)$ = 100 dB, as we defined and so $L_{i,src}\left( 40 \ km \right)$ = 79.1 dB.
Note that sound pressure is related to sound level intensity through: $$ L_{p}\left( r \right) = 20 \ \log_{10} \left( \frac{ P\left( r \right) }{ P_{o} } \right) \tag{5} $$ so $L_{p}\left( r \right)$ = 100 dB corresponds to $P\left( r \right)$ = 2 Pa for $P_{o}$ ~ 20 $\mu$Pa at 1000 Hz and $L_{p}\left( r \right)$ = 79.1 dB corresponds to $P\left( r \right)$ ~ 0.18 Pa. Equation 5 shows that the sound level intensity is defined to be zero at the threshold of hearing, but if we approximate $L_{p}\left( r \right)$ ~ 0.001 dB to be the boundary of hearing then we can estimate how far away from a 100 dB one would need to be to reach this level for an atmospheric pressure of 600 Pa.
For the ~40 km altitude we used before, $L_{r}\left( 40 \ km, r \right)$ goes to ~0.001 dB at $r$ ~ 9 km (~5.6 miles). Note that 100 dB is really loud. A typical subway train arriving at a platform only generates ~90 dB of intensity. A chainsaw at ~1 m is about ~110 dB. Typical breathing is about ~10 dB. So to reduce a 100 dB sound to the equivalent of breathing in an atmospheric pressure equivalent to 40 km altitude above Earth, one would need to be ~2.8 km (~1.8 miles) away.
I wonder if the low pressure is filtering out the high frequency, or is all direct sound transmission lost and the low frequencies are being physically transmitted through the chamber floor?
The ambient pressure reaches ~5 Pa in Earth's atmosphere at an altitude of ~84 km. So if you follow the above steps and use the values in the table for 80 km found at https://physics.stackexchange.com/a/266046/59023, you should find your answer or at least a good enough approximation to figure out what can and cannot be occurring.
honeste_viverehoneste_vivere
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Opposites Attract: A Review of Basic Magnetic Theories
June 08, 2015 by Editorial Team
Electric machineries are based on the basic principles of electromechanical conversion. They use either the electrostatic or the electromagnetic principle. This technical article deals with the magnetic circuit theory for the conversion of one form of energy to another.
Recommended Level
This technical article deals with the magnetic circuit theory for the conversion of one form of energy to another. A static device such as a transformer converts the electrical energy to electrical energy while rotating devices such as a DC machine, induction machine, or synchronous machine convert the mechanical or electrical energy into electrical or mechanical energy. Actuators, solenoids, and relays are also based on this conversion process. This conversion process happens in a magnetic material inside these machines. The magnetic material provides the high flux density which can provide high torque and high machine output per unit volume of the machines. This article is dedicated to the properties of these magnetic materials. We will see the basic methodology for the analysis of these machines by using their magnetic circuits.
A Review on Basic Magnetics
If we use a permanent magnet or let electric current flow through a coil, magnetic field is produced. The direction of magnetic field can be found out using the Right-hand rule which says that if the conductor is held in the right hand in such a way that the thumb indicates the direction of current, then the fingertips will indicate the direction of the magnetic field.
The basic laws related to magnetics are given below.
Faraday's Law
The EMF (or voltage) produced around a closed-loop coil is directly proportional to the rate of change of a magnetic field (time-variant) going through or out of that loop.
$$EMF ∝ \frac{∂\Phi}{∂t}$$
Figure 1. Magnetic Field (Varying with Time)
Lenz's Law
According to this law, the direction of the electromagnetically-induced current is such that its magnetic field opposes direction of the original magnetic field that created the induced current. This is shown in the figure below.
Figure 2. Direction for the Induced Current
As a result, the basic equation of Faraday's law of electromagnetic induction will have a negative sign.
$$EMF=-\frac{∂\Phi}{∂t}$$
Ampere's Law
This law is based on the discovery of the compass needles used for the detection of direction. We know that a current-carrying conductor produces a magnetic field. Lines of magnetic field form a closed path around the wire. The magnitude of the magnetic field density, B, is same on circular paths. B is directly proportional to the current and inversely proportional to the distance of a point on the closed path from the wire.
For a vector B and where dS is small element on the circular path,
$$\mathbf{B}\cdot d\boldsymbol{S} = B\,dS$$
$$B=\frac{µ_{0}I}{2πr}$$
where the value of B is constant around the closed path. Here, µ0 is the permeability of air.
Sum of the products for all such dS elements is given as,
$$\oint B\,dS=B\,\oint\,dS=\frac{µ_{0}I}{2πr}\,\oint dS$$
Consider a circular path, then
$$\oint dS = 2πr$$
We now have,
$$\oint B\,dS=µ_{0}I$$
The Ampere's circuital law is that line integral of $$\mathbf{B} \cdot d\mathbf{S}$$ around any closed path is $$µ_{0}I$$. Here, $$I$$ is the total continuous current passing through any surface bounded by the closed path.
In terms of magnetic field intensity, this terms are reduced to:
$$\oint \mathbf{H}\cdot d\mathbf{l}=I_{Enclosed\,by\,path}$$
Parameters and Terminologies in Magnetics
Now, the basic terminologies are given for the construction of magnetic circuit. Later, we will see the process to form the magnetic equivalent circuit for a given machine.
Magnetic Flux Intensity (H)
If V is the potential of any point, then the electric current will produce a magnetic field of intensity H = -∇V.
The EMF in an electric circuit is analogous to the ampere-turns in a magnetic circuit. This gives the relationship between the current and field intensity.
The relationship between the current and field intensity is given by the Ampere's circuital law mentioned.
$$\oint \mathbf{H}\cdot d\mathbf{l}=\sum{i}$$
$$\mathbf{H}$$ = magnetic field intensity at any point on the closed path of any shape
$$d\mathbf{I}$$= incremental length at the chosen point
Let the angle between the H and dI is Ө. Then
$$\oint H dL\,cosӨ=\sum{i}$$
But for a circular shape, the value of Ө = 0°. Thus, for a circular shape of radius r we have
$$H\, (2πr) = i_{T}$$
Where iT = i1 + i2 - i3 for the Figure 1. For a coil of N turns carrying i current in one conductor, iT = Ni.
Figure 3. Image to Illustrate Ampere Circuital Law
Magnetic Flux Density (B)
It is the flux per unit area. The flux in a magnetic circuit is analogous to the current in electric circuit. It is related to the flux density by the surface integral as shown by the following equation.
$$\Phi=\int_{S}\mathbf{B}\cdot d\mathbf{S}$$
Here, $$\Phi$$ is the flux expressed in Wb (Weber) measured for the surface area S.The unit of B is Wb/m2 or Tesla.
This is analogous to the electric resistance in the electric circuit but it is not necessarily a loss component in the magnetic circuit. The equation for the reluctance is given below which uses Ohm's law and replaces the equivalent magnetic-circuit variables.
$$Reluctance,\,R=\frac{Magnetomotive\,Force}{Flux}$$
$$\Rightarrow R=\frac{NI}{\Phi} \text{ (At/Wb)}$$
Permeance (ᵱ)
It is the inverse of reluctance. This is used to portray the geometrical characteristics for magnetic field.
$$ᵱ=\frac{1}{R}$$
The entire magnetic flux through a magnet does not entirely pass through the low-reluctance path of the core; instead part of the flux also goes to the high-reluctance path of air or leaks out from the core.
As the current passes through the path of least resistance, the magnetic flux also has the ability of leaking out to the surrounding air. There is no magnetic insulator available to eliminate them, but magnetic shielding using DC or AC at low frequency can reduce it to some level. In the case of coupled circuits consisting of two or more coupled circuits having more than one winding, the leakage flux links to one coil without interlinking others.
Fringing
This term is used to illustrate the deviations of the flux lines in an air gap of magnetic machine. Fringing is more significant in air medium than to iron. Fringing increases the effective area of the gap. It is proportional to the length of the air gap.
Absolute and Relative Permeability (µ0, µr )
Permeability tells about the capability of the magnetic substance to favor the making of the magnetic field within itself. Absolute permeability is the ratio of the magnetic field density to the magnetic field intensity in a given medium given by
$$µ=\frac{B}{H}$$
Thus, absolute permeability of a material is given by the slope of the curve obtained between flux density and flux intensity for a particular value of the flux intensity.
The permeability changes with the change in flux intensity and change of the material as shown in the figure below. The different materials require the different values of current to establish a particular level of flux density.
Figure 4. B-H Curve for Magnetic Materials
The flux density B increases linearly when the value of the magnetic field intensity is low. However, as the value of the intensity is increased, flux density increases in a non-linear manner showing the effect of saturation. Hence, the reluctance of the magnetic path is based on the value of the flux density as shown in figure below.
Figure 5. B-H Magnetization Curve
Relative permeability is a dimensionless quantity given by a ratio of a particular magnetic substance to that of a permeability of an air.
$$µ_{r}=\frac{µ}{µ_{0}}$$
Where µ0 is the permeability of air = 4 π X 10-7 Henry/meter.
Table 1. Relative Permeability of Few Materials
A coil is usually wound on the magnetic core to generate the flux. This coil may be represented by the ideal element known as inductance represented by symbol L shown below. Inductance is the flux linkage of the coil per ampere of the current flowing through it.
Figure 6. (a) Basic Magnetic Circuit (b) Equivalent Inductance for a Coil
$$L=\frac{N\Phi}{i}=\frac{N(BA)}{i}=Nµ\frac{HA}{i}=Nµ\frac{HA}{\frac{Hl}{N}}=\frac{{N}^{2}}{\frac{l}{µA}}=\frac{{N}^{2}}{R}$$
Note that inductance is proportional to the square of the number of turns on the coil.
Before starting the construction of the equivalent magnetic circuit, let us revise the basic analogy between the magnetic and electric circuit shown below.
Table 2. Electric and Magnetic Circuits Analogy
Equivalent Magnetic Circuit
The usefulness of the equivalent magnetic circuit is to find out the proper size of magnetic parts of an electric device during the design process, i.e. in finding out the parameters such as inductance and in finding out the air gap flux density for the calculation of power and torque.
The flux density in the core increases with the presence of ferromagnetic material or current-carrying coils. This in turn affects the inductance of the coil.
Although the magnetic field is a distributed parameter phenomenon, we can use the lumped parameter analysis for a definite class of magnetic material as done in the electric circuit analysis. However, the accuracy and precision for such analysis is less than the electric circuit analysis.
Consider a simple magnetic circuit having a ring-shaped magnetic core known as toroid as shown in figure below.
Figure 7. A Toroid
The coil is wrapped around the entire circumference and is carrying the current i through a coil making N turns.
Let the leakage flux is nil as it is mostly confined within the core material.
From Ampere's circuital law, we have,
$$\oint \mathbf{H}\cdot d\mathbf{l}=Ni$$
$$\Rightarrow H \cdot l=Ni=magnetomotive\,force = F \text{ (At)}$$
$$(l=2\,π\,r)$$
$$B = µ\, H \Rightarrow B=µ\frac{Ni}{l} \text{ (Tesla)}$$
As there is no leakage flux, the flux covering the cross section of the toroid is given by
$$\Phi=\oint \mathbf{B}\cdot d\mathbf{A}=B\,A\Rightarrow µ\frac{Ni}{l}A=\frac{Ni}{\frac{l}{µA}}=\frac{Ni}{R}$$
Here, $$\Re$$ is the reluctance of magnetic path given by
$$R=\frac{l}{µA}=\frac{1}{ᵱ}$$
The magnetic equivalent circuit for a toroid can be represented as shown below, which is basically derived from the analogous electric circuit. In this example, we have considered the circular core, but it can be of another form also such as in rectangular form.
Figure 8. (a) Equivalent Magnetic Circuit for a Toroid (b) Equivalent Electric circuit
Equivalent Magnetic Circuit for a Core with Multiple Excitations
Consider the magnetic device shown in Fig. 9 consisting of three coils for excitation carrying the currents i1, i2, i3.
Consider the mean length of this magnetic circuit is L. The coils have turns ratio N1, N2, N3.The first two coils are producing the magnetic fluxes $$\Phi_{1}$$, $$\Phi_{2}$$ in the same direction while the direction of the current in the third coil is in such a way that it produces the flux $$\Phi_{3}$$ in the opposite direction.
Figure 9. Magnetic Device having Multiple Excitations
According to Ampere's circuital law, integral of the magnetic field intensity around any closed path is equal to the total algebraic sum of the electric current in that path.
$$\int_{a}^{b}H\,dl=Ni=F=R\Phi=\frac{l}{µA}\Phi$$
Magnetic flux through surface S is given by
$$\Phi=\int_{S}\mathbf{B} \cdot d\mathbf{S}$$
If there is no saturation, then the value of the magnetic field intensity H varies linearly with the change in the magnetic field density B.
The net magnetomotive force (mmf) is given by
$$\oint H\,dl=\int_{a}^{b}H_{K}dl\,+\,\int_{b}^{c}H_{K}dl\,+\,\int_{c}^{d}H_{K}dl\,+\,\int_{d}^{a}H_{K}dl$$
$$\Rightarrow \oint H\,dl=H_{K}L_{ab}\,+\,H_{K}L_{bc}\,+\,H_{K}L_{cd}\,+\,H_{K}L_{da}=H_{K}L=\Phi R=F=N_{1}I_{1}\,+\,N_{2}I_{2}\,-\,N_{3}I_{3}$$
Thus, the algebraic sum of magnetic potential around the closed path is zero. (Analogous of the KVL in electric circuit)
Thus, the magnetic circuit representation will be:
Figure 10. Equivalent Circuit Representation for the Multiple Excitation System
Equivalent Magnetic Circuit with an Air Gap
In an electric machine, input and output of a magnetic system is isolated from the air gap. Practically, the same flux is required in the magnetic core and the air gap. Thus, an air will require more mmf than the core due to high reluctance. If the value of flux density is high, the core will exhibit saturation. However, the air gap will not get saturated as B-H curve for the air medium is linear (µ is constant for an air medium).
Consider the magnetic structure with a single coil and an air gap having mean length L as shown below.
Figure 11. Magnetic Core having an Air Gap
$$mmf,\;F = Ni$$
Let the core have the reluctance RC and air gap have the reluctance Rg which is given by the following equations as follows:
$$R_{C}=\frac{l_{c}}{µ_{c}A_{C}}$$ and $$R_{g}=\frac{l_{g}}{µ_{0}A_{g}}$$
$$Flux,\, \Phi=\frac{Ni}{R_{C}+R_{g}}$$
$$mmf, \;Ni = H_{C}\,l_{c}+H_{g}\,l_{g}$$
Consider that the air gap is of small size (usual). Thus, fringing effect can be ignored. Also, assume there is no saturation. Then,
$$Ni = H_{C}\,l_{c} + H_{g}\,l_{g} = H_{K}L$$
$$A_{g} = A_{C}$$
The value of flux density will be same both in its core and the air gap given by the ratio of flux to the cross-section area of the core. The equivalent circuit in this case is shown below.
Figure 12. Equivalent Circuit for Magnetic Core with an Air Gap
magnets magnetic flux inductance magnetic circuit
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Does an adjoint of an internal Hom functor of a prounital closed category define a tensor product?
A closed category is a category equipped with internal Hom functors along with a unit object. Now this answer shows that if $C$ is a closed category whose internal Hom functor has a left adjoint, then this left adjoint defines a tensor product on $C$, i.e. it makes $C$ into a monoidal category. But I'm wondering if something more general is true.
A prounital closed category is like a closed category, except that the requirements regarding a unit object are dropped. (I don't think this is standard terminology.) My question is, if $C$ is a prounital closed category whose internal Hom functor has a left adjoint, then does this left adjoint define some kind of tensor product on $C$? Now it presumably wouldn't make $C$ monoidal, since you no longer have a unit object. But would it at least make $C$ into a "semigroupal category", i.e. a category with a tensor product which satisfies all the properties of a monoidal category except the requirements regarding a unit object?
If not, does anyone know of a counterexample?
Browse other questions tagged category-theory examples-counterexamples adjoint-operators monoidal-categories hom-functor or ask your own question.
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Home » MAA Publications » Periodicals » Convergence » Mathematical Treasure: Johannes Scheubel's 1551 Algebra
Mathematical Treasure: Johannes Scheubel's 1551 Algebra
Sidney J. Kolpas (Delaware County Community College)
Dedicated to Dr. Barnabas Hughes: Professor, Mentor, Friend
Above: Title page of Scheubel's Algebrae Compendiosa from the collection of Dr. Sid Kolpas. The full translation of the title from the Latin is Concise Algebra easily described, which brings forth the great wonders of arithmetic. The printer's colophon (logo) is "The fat hen" (Pingui Gallina). The book was printed in Paris in 1551. "Cum Privilegio" means with privilege or permission to print from the king, in this case King Henry II of France.
Algebrae Compendiosa
Selected Examples for the Classroom
Using Algebrae Compendiosa in the Classroom
Who Was the Printer Cavellat?
Johannes Scheubel was born on August 13, 1494, in Kirchheim unter Teck, Germany, and died on February 20, 1570, in Tübingen, Germany, where he is buried. According to his biographer Mary Day (1926), nothing is known about his family since the church register in Kirchheim unter Teck only went back to 1558. Scheubel attended the Latin School of Kirchheim unter Teck, continuing his studies at the University of Liberal Arts in Vienna. He began teaching in 1532 in Leipzig, and then was appointed Professor at the University of Tübingen, where he became "Magister" in 1540. According to Day, he became "Docent of Mathematics" in 1544, teaching arithmetic and geometry. At his death, his mathematical instruments, manuscripts, and library were left to the University of Tübingen. Scheubel's publications include:
De Numeris et Diversis Rationibus seu regulis computationum opusculum a Joannes Scheubelo compositum, Lipsiae (Leipzig), 1545. This was an arithmetic book of five chapters.
Compendium Arithmeticae Artis per Johannem Scheubeliu adornatum et conscriptum, Basel, 1549.
Euclidis Megarensis, Philosophi et Mathematici excellentissimi, sex libri priores, de Geometricis principiis, Graeci et Latini, una cum demonstrationibus propositionum. Algebrae porro regulae, propter numerorum … his libris praemissae sunt …, Basel, 1550. This is Scheubel's Euclid, and is considered his masterpiece. It includes the first publication of Scheubel's Algebrae Compendiosa as its first 76 pages. The author owns a copy of this work.
Algebrae compendiosa facili'sque descriptio, qua depromuntur magna Arithmetices miracula, Paris, 1551. First published as a preface to Scheubel's Euclid. The author owns a copy of this work.
According to Barnabas Hughes in his article, "The Private Library of Johann Scheubel, Sixteenth-Century Mathematician," Scheubel was "an intellectually heavy scholar, balanced and poised, a man of dignity." His library included works on algebra, architecture, arithmetic, the astrolabe, astrology, astronomy, biology, the compass, cosmography, geometry, Greek and Latin literature, medicine, music, natural philosophy, optics, tables, trigonometry, and weights. In particular, he owned a copy of Copernicus's De revolutionibus. He kept abreast of publications in his area of specialty, geometry and arithmetic, which included the works of Cardano, Stifel, Peter Ramus, Christoph Rudolph, and Peter Apian; he was current in the mathematical thinking of the time.
Scheubel was considered by David Eugene Smith (1860-1944) to be an under-rated figure. In his Rara Arithmetica, Smith stated (pp. 235-236):
While Scheubel is not much appreciated to-day, he was really ahead of his time. He tried to banish the expression 'rule of three' and to substitute 'rule of proportion.' His explanation of square root is in some respects the best of the century, and he dismisses with mere mention the 'duplatio' and 'meditatio' of his contemporaries. He extracts various roots as far as the 24th, finding the binomial coefficients by means of the Pascal triangle a century before Pascal made the device famous.
Scheubel's Algebrae Compendiosa is an algebra that does not fully use modern notation. According to Florian Cajori (1859-1930), the book contains the first appearance of the + and – symbols in France (1928, p. 151). N means x0 or 1, Ra., short for Radix, means x1 or x; Pri., short for Primus, means x2; Secun., short for Secundus, means x3; etc. (refer to the table below); that is, xn is indicated by an abbreviation of the Latin name for n-1, the number of multiplications starting with x to obtain xn. Thus, 8N means 8x0 or 8 (N indicating a constant number), 8 Ra. means 8x, 8 Pri. means 8x2, etc.
The table of powers above appears in Algebrae Compendiosa at page (or folio) 6 recto (or 6r).
According to Cajori, mathematical encyclopaedist and textbook author Charles Hutton (1737-1823) felt that Algebrae Compendiosa was "most beautifully printed, and is a very clear though succinct treatise; and both in the form and matter much resembles a modern printed book" (quoted in Cajori 1928, p. 151). The 2010 advertisement of Martayan Lan Rare Books of an original copy of the book, offered at $2,250, indicated that the book was of "pocket format," a specialty of the printer, Gulielmo Cavellat. Cavellet favored the octavo size because it made his books easier to carry and less expensive for university students. (For more information, see Who Was the Printer Cavellat?)
The pages of Algebrae Compendiosa are numbered only on the front side, which was common at the time. In the following images from the book, "r" will denote recto or the front of the page and "v" will denote verso or the back of the page.
Above: Scheubel defined his addition symbol and subtraction symbol on page 1r of his Algebrae Compendiosa. As noted above, this was the first appearance of the + and – symbols in France (Cajori 1928, p. 151).
Below: Scheubel's discussion of powers continued on page 1 verso, or 1v.
On pages 3-9 of Algebrae Compendiosa, Scheubel discussed operations on integers, done exactly as they are taught today. For example, Scheubel stated that when multiplying, if the signs are the same the product is positive, and if the signs are different the product is negative. He then used these rules for operations on integers in the examples in the book.
Algebrae Compendiosa contains many interesting algebra problems and word problems requiring algebraic solutions. There follow some selected examples that can be used in the classroom.
Example: Subtracting binomials
From Algebrae Compendiosa (page 4v)
We would write (8x4 + 7x) – (5x4 – 4x) = 3x4 + 11x. Notice Scheubel's facility with negatives: 7x – (–4x) = 11x.
Example: Multiplying binomials
From Algebrae Compendiosa (page 7r)
The goal here is to compute (8x2 – 9)(8x2 – 9), which is done by computing (8x2 – 9)8x2 = 64x4 – 72x2 and (8x2 – 9)9 = 72x2 – 81 separately, followed by "subtraction of the products": (64x4 – 72x2) – (72x2 – 81) = 64x4 – 144x2 + 81.
Example: Dividing powers
From Algebrae Compendiosa (page 11r)
We might write: \(\frac{2}{3x}\div\frac{8x}{9x^2}=\frac{3}{4}\) or \(\frac{8x}{9x^2}\div\frac{2}{3x}=1\frac{1}{3}.\)
Examples: Solving quadratic equations
Above: Here on page 20r of Algebrae Compendiosa, Scheubel solved the equation 1x2 + 12 = 8x by completing the square. We (but not Scheubel) would first rewrite the equation as 1x2 – 8x = –12. Scheubel's instructions should then sound familiar:
Square 8/2 = 4 to get 16, then add 16 to –12 to get 4. (Literally, 16 minus 12 makes 4.)
Now subtract \({\sqrt 4}=2\) from 4 and add 2 to 4 (that is, compute 4 ± 2) to get the roots 2 and 6 of the equation. Here, "radix quadrata" means "square root."
Scheubel then checked his solutions 2 and 6 by substituting them into his equation, 1x2 + 12 = 8x, and making sure in each case that the two sides were equal to the same number (not shown).
A few pages later, on page 24v of Algebrae Compendiosa, Scheubel solved the quartic (degree 4) equation
9x4 +5x2 = 294
by completing the square. He first rewrote the equation as
1x4 + 5/9 x2 = 294/9,
indicating only the new coefficients 5/9 and 249/9. From here, his method was to square one half of 5/9, or 5/18, to get 25/324, then add 25/324 to 294/9 to get 10609/324. Now \( {\sqrt{\frac{10609}{324}}}={\frac{103}{18}} \) and \[{\frac{103}{18}}-\frac{5}{18}=\frac{98}{18}=\frac{49}{9},\] so that \(x^2=\frac{49}{9}\) and \(x=\frac{7}{3}=2\frac{1}{3}.\) Note that negative square roots were twice ignored.
The equals sign
As can be seen above, Scheubel did not use our equals sign, instead using the Latin word "aequales," meaning "equals." Robert Recorde (1510-1558) would introduce the equals sign just six years later in his Whetstone of Witte (1557).
Barnabas Hughes, in his article, "Robert Recorde and the first published equation" (1993), concurred with Mary Day that Algebrae Compendiosa was Recorde's main source for his Whetstone of Witte. Day, in her 1926 biography of Scheubel, stated that "Recorde must have had Scheubel's Algebra before him when he wrote his book." She showed that many of Recorde's examples and word problems are similar to Scheubel's.
Title page of Robert Recorde's Whetstone of Witte (1557) (Courtesy of Columbia University Libraries)
First appearance of the equals sign ====== in Recorde's Whetstone of Witte (Courtesy of Columbia University Libraries)
Modern translation of third sentence in the paragraph just above the equations: And to avoid the tedious repetition of these words: "is equal to": I will set (as I do often in work use) a pair of parallels, or Gemowe lines, of one length (thus: =), because no two things can be more equal.
See Mathematical Treasure: Robert Recorde's Whetstone of Witte for more images of and information about this text.
Examples: Square roots
Scheubel used "ra." preceding a number to indicate radix or square root; thus, "ra. 7" indicates the square root of 7. Sometimes Scheubel used the modern square root symbol, as in \({\sqrt 7}.\) Moreover, he occasionally used "radix binomii" for the square root of a binomial. For nested square roots, he used "ra.col.," or radix collecti, for the square root of a collection of terms, as in \[{\rm{Ra.col.}}\,32 + {\sqrt{1020}} = {\sqrt{32+\sqrt{1020}}}.\]
Above: Note that the desired form of \[\left(8+\sqrt{28}\right) + \left(4+\sqrt{7}\right)\] was not \(12+3\sqrt{7},\) but rather \(12+\sqrt{63}.\)
Below: The product \[\left({23+\sqrt{448}}\right) \left({4-\sqrt{7}}\right)\] was simplified "only" to \[92-\sqrt{3136}-\sqrt{3703}+\sqrt{7168}.\]
Examples: Higher roots
From Algebra Compendiosa (page 30r)
"Ra. cu." indicates cube root. The problem above, then, is to compute the sum, \({\sqrt[3]{24}}+{\sqrt[3]{81}},\) simplifying the result to the single cube root, \({\sqrt[3]{375}}.\)
Scheubel's method, in modern notation, is to rewrite \({\sqrt[3]{24}}+{\sqrt[3]{81}},\) as \[{\sqrt[3]{3}}\left({\sqrt[3]{8}}+{\sqrt[3]{27}}\right).\] The expression in parentheses simplifies to \(2+3\) or \(5,\) which is then cubed to get \(125.\) That is, \[{\sqrt[3]{3}}\left({\sqrt[3]{8}}+{\sqrt[3]{27}}\right) = {\sqrt[3]{3}}\left({2+3}\right) = {\sqrt[3]{3}}\left({5}\right)={\sqrt[3]{3}}\left({\sqrt[3]{125}}\right)={\sqrt[3]{375}}.\]
From Algebrae Compendiosa: Both examples appear on page 32r.
In the examples above, "ra. ra." indicates fourth root (square root of square root), also indicated by the modified radical sign that appears just before 162 in both examples above. Scheubel used both notations, as can be seen in the examples. In the example at left, Scheubel used the same method he used to add cubic roots in the preceding example to rewrite the sum \({\sqrt[4]{32}}+{\sqrt[4]{162}}\) as the single fourth root, \({\sqrt[4]{1250}}.\) In the example at right, he used the same method to subtract \({\sqrt[4]{32}}\) from \({\sqrt[4]{162}}\) – that is, to compute \({\sqrt[4]{162}}-{\sqrt[4]{32}}\) – and to simplify the difference to the single fourth root, \({\sqrt[4]{2}}.\)
The claim here is that \[{\sqrt[4]{\frac{36}{5}}}\cdot {\sqrt[4]{\frac{4}{5}}} = {\sqrt{\frac{12}{5}}}.\]
Examples: Continuous ratios
Divide 40 into three numbers so that the parts are in a continuous ratio of 1 2/3 to 1.
Answer: 7 17/49, 12 12/49, 20 20/49.
Work: The first is 1 ra. or 1x, the second 1 2/3 x, and the third 2 7/9 x. Their sum, 5 4/9 ra., therefore equals 40 N. We would write 5 4/9 x = 40 or 49/9 x = 40, giving the first as x = 360/49 = 7 17/49, the second as 1 2/3 x = 600/49 = 12 12/49, and the third as 2 7/9 x = 1000/49 = 20 20/49.
From Algebrae Compendiosa (page 16v): King, Princes, Counts, and Soldiers
There is a king, and under him princes, counts, and soldiers. The ratio of counts to princes is twice the number of princes, and the ratio of soldiers to counts is four times the number of princes. One two-hundredth of the number of soldiers is equal to nine times the number of princes. Find the number of princes, counts, and soldiers.
Answer: 15 princes, 450 counts, 27000 soldiers.
Work: By the first hypothesis, let the number of princes \(= x,\) counts \(= 2x^2,\) and soldiers \(= 8x^3.\) By the second hypothesis, \(\frac{1}{200}\cdot8x^3=9x,\) or, as in the last line shown above, \(\frac{1}{25}x^3=9x.\) Solving for \(x\) \(\left(x>0\right)\) gives \(x = 15.\)
So, the number of princes \(= x = 15,\) counts \(= 2x^2 = 450,\) and soldiers \(= 8x^3 = 27000.\)
Example: An investment problem
Three tradesmen invest 170 gold coins (aureos) in a partnership. The first wants his money returned in 3 months, the second in 6 months, and the third in 8 months. In a certain amount of time, this money bears interest amounting to 375 gold coins (aurei). The first man receives for his part, including his share and interest, 75 coins; the second 200 coins; and the third the remaining coins. How much was each man's share, or how much money was collected from each? The result: 60 aurei from the first, 80 from the second, and 30 from the third, as announced at the top of the next page.
From Algebrae Compendiosa (page 18v)
Scheubel's equation was 1 ra. + 1 1/3 ra. + ½ ra. = 170, or 2 5/6 ra. = 170, yielding 1 ra. = 60 and therefore 1 1/3 ra. = 80 and ½ ra. = 30 aurei as the initial investments. How did he get this equation from the given data? Perhaps he obtained it through proportional reasoning like the following:
The first person got 75 aurei for 3 months, so for 6 months the second person would have received 150 at the same rate. But he received 200, so he must have invested \(\frac{200}{150}\) or \(\frac{4}{3}\) as much as the first person. The third person invested for 8 months, and at the same rate as the first person would have received \(\frac{8}{3}(75)=200.\) But he received only 100, so he must have invested one-half as much as the first person. If the first person invested 1 ra., then the second and third invested, respectively, 4/3 ra. and ½ ra. Since the three investments total 170 aurei, Scheubel's equation was
1 ra. + 1 1/3 ra. + ½ ra. = 170.
Example: A military problem
A general has several thousand soldiers under his command. When he attempts to arrange his army in the largest possible square, there are 284 too many soldiers. If he then tries to arrange them in a square with one man more on a side than before, he lacks 25 soldiers. How many soldiers does he have?
Answer: 24 thousand soldiers.
At the top of the next page, Scheubel wrote the total number of soldiers in two different ways:
the number in a square of side length 1 ra., or \(x,\) plus 284 left over, or 1 pri. + 284 N, or \(x^2 + 284,\) and
the number in a square of side length 1 ra. + 1 N, or \(x+1,\) minus the 25 the general lacked to make this square; that is, \((x+1)^2 – 25.\) But Scheubel wrote this as 1 pri. + 2 ra. + 1 N – 25 N, or \(x^2+2x+1-25.\) He then simplified to 1 pri. + 2 ra. – 24 N, or \(x^2+2x-24.\)
Scheubel obtained his answer by setting the two expressions equal to one another to form the equation
1 pri. + 284 N aequales 1 pri. + 2 ra. – 24 N,
or \[x^2+284=x^2+2x-24,\] where each side of the equation gives the total number of soldiers. He simplified this equation to \(308=2x,\) and then to \(x=154.\) Remembering that the total number of soldiers is given by \(x^2+284\) and also by \(x^2+2x-24,\) he could then compute the number of soldiers using one expression and then check his answer using the other expression, obtaining his answer of 24000 soldiers.
Robert Recorde's Whetstone of Witte has the exact same problem, adding evidence that he had Algebrae Compendiosa in front of him when he wrote his book.
From Whetstone of Witte (pages 249-250)
I have used sample problems from Scheubel's Algebra in Elementary, Intermediate, and College Algebra classes to show students how the problems they're studying were written and solved in 1551. Students find the problems fun to work, they gain an appreciation for our modern algebraic notation, and they learn some mathematics history.
Another use in the classroom would be in a History of Mathematics course as part of a larger discussion about the evolution of mathematical notation, and/or to give students the experience of solving problems as they were written and solved in the sixteenth century. In a History of Mathematics or History of Science class, the instructor might also discuss what was happening in the world in 1551 to provide historical context to the discussion of Scheubel's Algebra.
A complete copy of Scheubel's Euclid, which contains Algebrae Compendiosa, can be found on the e-rara website at http://www.e-rara.ch/bau_1/content/titleinfo/1688789.
Algebrae Compendiosa is also available as a print-on-demand book for as low as $10.20 from ReInk (or Reink) Books. See http://www.abebooks.com/servlet/SearchResults?sts=t&tn=algebrae+compendiosa.
A PDF copy of Algebrae Compendiosa can be downloaded from: https://archive.org/details/bub_gb_yyJhlkJwjcQC
Instructors can use the examples in this article or select their own examples from electronic copies or the print-on-demand copy.
According to Joëlle Ducos of the University of Paris–Sorbonne, Guillaume Cavellat was born between 1520 and 1525, and died in 1576. His print shop/book store, established in 1547, could be found at the "sign of the fat hen" (also his colophon) near Cambrai College, University of Paris. Cavellat associated with astronomers and mathematicians, convincing them to use the books he printed. His book production and sales depended largely on the university programs. He chose an octavo format because it was easy for students to carry and was relatively inexpensive. When Cavellat died, his widow Denise Girault, who was the daughter of a bookseller, carried on the business. Back to Algebrae Compendiosa
Cajori, Florian. A History of Mathematical Notations. The Open Court Publishing Company, La Salle, Illinois, 1928. See especially pages 147-151.
Day, Mary S. Scheubel As An Algebraist. Teachers College, Columbia University, NYC, 1926.
Ducos, Joëlle. "Editer des livres scientifiques au XVIe siècle: Guillaume Cavellat" ("Editor of scientific books in the sixteenth century: Guillaume Cavellat"). Uranie: http://uranie.msha.fr/expositions/editer-des-livres-scientifiques-au-xvie-siecle-guillaume-cavellat/
Hughes, Barnabas B. "Johann Scheubel's Revision of Jordanus de Nemore's De numeris datis: An Analysis of an Unpublished Manuscript." Isis, Vol. 63, No. 217, Balding and Mansell, London, June 1972.
Hughes, Barnabas B. "The Private Library of Johann Scheubel, Sixteenth-Century Mathematician." Viator: Medieval and Renaissance Studies. University of California Press, Berkeley, Vol. 3, 1972.
Hughes, Barnabas B. "Robert Recorde and the first published equation." Vestigia Mathematica. Amsterdam, 1993, pp. 163-171.
Martayan Lan Rare Books: http://www.martayanlan.com/cgi-bin/searchresults.cgi?item=371&start=30&map_or_book_id=1&cat=29&catalog=all&ke
Scheubel, Johann. Algebra Compendiosa. Gulielmum Caullet, Paris, 1551.
Smith, David Eugene. Rara Arithmetica. Ginn and Company, Boston, 1908. See especially pages 233 and 235-236.
Index to Mathematical Treasures
Sidney J. Kolpas (Delaware County Community College), "Mathematical Treasure: Johannes Scheubel's 1551 Algebra," Convergence (August 2016)
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\begin{document}
\title{Baseband control of superconducting qubits with shared microwave drives}
\author{Peng Zhao} \email{[email protected]} \affiliation{Beijing Academy of Quantum Information Sciences, Beijing 100193, China} \author{Ruixia Wang} \email{[email protected]} \affiliation{Beijing Academy of Quantum Information Sciences, Beijing 100193, China} \author{Meng-Jun Hu}
\affiliation{Beijing Academy of Quantum Information Sciences, Beijing 100193, China} \author{Teng Ma}
\affiliation{Beijing Academy of Quantum Information Sciences, Beijing 100193, China} \author{Peng Xu}
\affiliation{Institute of Quantum Information and Technology, Nanjing University of Posts and Telecommunications, Nanjing, Jiangsu 210003, China} \author{Yirong Jin}
\affiliation{Beijing Academy of Quantum Information Sciences, Beijing 100193, China} \author{Haifeng Yu} \email{[email protected]} \affiliation{Beijing Academy of Quantum Information Sciences, Beijing 100193, China}
\date{\today}
\begin{abstract} Accurate control of qubits is the central requirement for building functional quantum processors. For the current superconducting quantum processor, high-fidelity control of qubits is mainly based on independently calibrated microwave pulses, which could differ from each other in frequencies, amplitudes, and phases. With this control strategy, the needed physical resource could be challenging, especially when scaling up to large-scale quantum processors is considered. Inspired by Kane's proposal for spin-based quantum computing, here, we explore theoretically the possibility of baseband flux control of superconducting qubits with only shared and always-on microwave drives. In our strategy, qubits are by default far detuned from the drive during system idle periods, qubit readout and baseband flux-controlled two-qubit gates can thus be realized with minimal impacts from the always-on drive. By contrast, during working periods, qubits are tuned on resonance with the drive and single-qubit gates can be realized. Therefore, universal qubit control can be achieved with only baseband flux pulses and always-on shared microwave drives. We apply this strategy to the qubit architecture where tunable qubits are coupled via a tunable coupler, and the analysis shows that high-fidelity qubit control is possible. Besides, the baseband control strategy needs fewer physical resources, such as control electronics and cooling power in cryogenic systems, than that of microwave control. More importantly, the flexibility of baseband flux control could be employed for addressing the non-uniformity issue of superconducting qubits, potentially allowing the realization of multiplexing and cross-bar technologies and thus controlling large numbers of qubits with fewer control lines. We thus expect that baseband control with shared microwave drives can help build large-scale superconducting quantum processors.
\end{abstract}
\maketitle
\section{Introduction}\label{SecI}
For quantum processors built with superconducting qubits, both the control accuracy and the qubit number have shown steady improvement over the past two decades \cite{Kjaergaard2020}. Notably, quantum gate operations, which are generally implemented by using microwave or baseband flux pulses \cite{Krantz2019}, with errors reaching the fault-tolerant thresholds have been achieved in quantum processors with several tens of qubits \cite{Arute2019,Zhu2022,Acharya2022,Kim2021}. Nevertheless, it is known that fulfilling the full promises of quantum computing requires the implementation of fault-tolerant schemes, which will need the high-fidelity control of millions of qubits \cite{Fowler2012,Gidney2021}. In such large-scale superconducting quantum processors, the needed physical resource, such as control electronics and cooling power in cryogenic systems, could be the most challenging obstacle for achieving accurate control of qubits, let alone solving the wiring problem \cite{Frankea2019,Reilly2019,Martinis2020} and the device yield problem \cite{Hertzberg2021,Kreikebaum2020}.
In superconducting quantum processors each qubit has typically a different set of parameters: frequency, anharmonicity, coupling efficiency and signal attenuations in control lines. Thus, in current small-scale quantum processors, to ensure accurate qubit control, each qubit should have its dedicated control pulse with different parameter settings \cite{Kelly2018,Klimov2020}. This means that the microwave control pulses could differ from each other in their amplitudes, frequencies, and phases, while for the baseband flux pulse, their amplitudes could be different. Moreover, these control pulses are generated at room temperature, and then delivered to qubits in the cryogenic system through a series of attenuators and filters for suppressing harmful noises, such as thermal noise \cite{Krantz2019,Chen2018,Krinner2019}. Considering these general arguments, the physical resource for realizing qubit control could be highly related to the type of employed control. To be more specific, the microwave control and its signal synthesis are more complicated and expensive than that of the baseband flux control, for which only a single digital-to-analog converter (DAC) per qubit is needed \cite{Krantz2019,Chen2018,Krinner2019}. Moreover, given the limited available cooling power in cryogenic systems, the microwave control lines generally need the attenuation of $60\,\rm dB$ (about $20\,\rm dB$ at the mixing chamber plate (MXC), for which the available cooling power is smallest), leading to heating loads larger than that of the baseband flux lines (about $20\,\rm dB$, need no attenuation at the MXC stage) \cite{Arute2019,Chen2018,Krinner2019}. Additionally, in large-scale quantum processors, microwave control requires higher-density control lines, making it challenging to suppress the microwave crosstalk \cite{Wenner2011,Rosenberg2019,Huang2021M} and thus to achieve high-fidelity qubit control. By contrast, with baseband flux control, there exists only a single control parameter, potentially allowing the application of multiplexing technologies and cross-bar technologies to address the challenges, e.g., the wiring problem, toward large-scale quantum computing \cite{Hill2015,Vandersypen2017,Veldhorst2017,Li2018}.
Given the above discussion, when scaling up to large-scale quantum processors, implementing baseband flux control could make requirements less stringent than that of microwave control. However, currently, microwave control is generally the essential one for implementing qubit addressing and single-qubit gate operations \cite{Motzoi2009,Chen2016,Krantz2019}, and even for two-qubit gates, such as cross-resonance gates \cite{Chow2011}. In this work, we explore theoretically the possibility of developing the baseband flux control of frequency-tunable qubits with the help of always-on shared microwave drives. It should be noted that previous works on studying baseband control of superconducting qubits mainly focus on low-frequency qubits, e.g., the composite qubit \cite{Campbell2020} and heavy-fluxonium qubit \cite{Zhang2021}, here we focus on the transmon qubits \cite{Koch2007} that have been widely used in current superconducting quantum processors. The basic idea of our control strategy is sketched in Fig.~\ref{fig1}(a), where two qubits are coupled via a coupler and the always-on microwave drive (XY line) is shared by both qubits (in principle, can be extended to multi-qubit cases), the qubit control and the single-qubit addressing can be realized only through the flux (Z) control lines. Our work is motivated by Kane's proposal for realizing spin-based quantum computing \cite{Kane1998}, where the spin qubit is by default off-resonance with the global always-on microwave magnetic field (i.e., at the idle point) and single-qubit gate operations are realized by tuning the spin qubit on-resonance with the field (i.e., at the working point) \cite{Wolfowicz2014,Laucht2015}, as shown in Figs.~\ref{fig1}(b) and~\ref{fig1}(c). Due to the always-on shared drive, the computational states are the basis states of the microwave-dressed qubit \cite{Liu2006,Zhao2022,Wei2022}, and accordingly, in the present work, all the qubit control are analyzed based on this microwave-dressed basis. As an example application of this baseband control strategy, in a system comprising two frequency-tunable transmon qubits coupled via a tunable coupler \cite{Yan2018}, we study the feasibility of this strategy for achieving high-fidelity gate operations. By theoretical analysis, we will show that:
(i) To implement single-qubit gate operations, especially, $\sqrt{X}$ gates, the baseband Z(flux)-control provides great flexibility in the gate tune-up procedure. This flexibility could be used to relieve stringent requirements on qubit frequency, drive strength, and gate time for implementing single-qubit gates, and thus can even compensate for the non-uniformity of qubit parameters, potentially allowing to perform multiplexed control of qubits.
(ii) Since the transmon qubit has a weak anharmonicity, in the traditional microwave control setup, leakage during gate operations can be suppressed by using the derivative removal by adiabatic gate (DRAG) scheme \cite{Motzoi2009}. In our setup, while the DRAG scheme can no longer be directly utilized, we show that by using a modified fast-adiabatic scheme, the leakage can also be suppressed heavily.
(iii) While the always-on microwave drive is detuned from the qubits, it can induce ac-Stark frequency shifts on the qubits \cite{Tuorila2010,Schneider2018,Liu2006,Zhao2022,Wei2022}. Consequently, any fluctuations in the drive amplitude will cause qubit dephasing. By numerical simulation, we study the effect of the amplitude-dependent noise on the qubit and show that the fluctuation-induced dephasing can be eliminated by tuning the qubit away from the drive. Similarly, by numerical simulation of qubit readout dynamics, the impacts of the always-on drive on the readout fidelity can also be neglected safely when the drive detuning is far larger than the drive strength.
(iv) In the qubit architecture with tunable coupling, we show that with the baseband control strategy and the modified fast-adiabatic scheme, high-fidelity single-qubit gates are achievable. We also outline the leading error mechanisms that should be considered carefully when applying baseband control in large-scale quantum systems. Additionally, we further show that with the always-on microwave drive, baseband-controlled two-qubit CZ gates can still be achieved with high gate fidelity and short gate length.
The rest of the paper is organized as follows. In Sec.~\ref{SecII}, we provide an overview of the baseband control scheme. In Sec.~\ref{SecIII}, we consider an example application of the baseband control strategy for achieving high-fidelity single- and two-qubit gates in a qubit architecture with tunable coupling. In Sec.~\ref{SecIV}, we will provide discussions of the challenges and opportunities for realizing the baseband control strategy in superconducting quantum processors. Finally, we provide a summary of our work in Sec.~\ref{SecV}.
\begin{figure}
\caption{Baseband flux control of transmon qubits with a shared always-on microwave drive.
(a) Sketch of a baseband flux controlled two frequency-tunable transmon qubits ($Q_{0}$ and $Q_{1}$) with dedicated Z lines. The two qubits are coupled via a coupler $Q_{c}$, which could be a tunable coupler. Through a shared XY line, the two qubits are driven simultaneously by a global and always-on microwave drive with the frequency $\omega_{d}$ and the constant amplitude $\Omega_{d}$. Baseband flux pulses are delivered to the qubits and coupler through their dedicated Z lines. (b) At the idle point, the qubit is far detuned from the drive, i.e., the drive detuning, $|\Delta_{d}|=|\omega_{01}-\omega_{d}|\gg\Omega_{d}$. Left: Bloch vector in the rotating frame with respect to the drive (here, confined to qubit subspace spanned by the lowest two-energy levels of the transmon qubit). Due to the always-on drive, the Bloch vector (dashed red arrow) at the idle point is tilted toward the X-axis. We thus choose the logical computational states to be the dressed eigenstates defined by the tilted Bloch vector. Right: Energy level diagram of the qubit at the idle point. Here, $\alpha_{q}$ denotes the qubit anharmonicity. (c) When operating the system at the working point, where the qubit is on-resonance with the drive, single-qubit operations can be implemented. Left: Bloch vector (solid red arrow) at the working point. Since the initial Bloch vector is slightly tilted, a small detuning $\delta_{d}$ between the qubit and the drive is needed for enabling complete Rabi oscillations. Right: Energy level diagram of the qubit at the working point.}
\label{fig1}
\end{figure}
\section{Overview of the baseband control setup}\label{SecII}
Here, we provide an overview of the baseband control setup schematically illustrated in Fig.~\ref{fig1}(a). In our setup, frequency-tunable transmon qubits are driven simultaneously by a single always-on global drive with a constant amplitude and each qubit has its dedicated flux control lines, i.e., Z lines. Same to Kane's proposal \cite{Kane1998}, single-qubit addressing or single-qubit gate operations can be implemented by tuning the qubits on-resonance with the global drive, as shown in Fig.~\ref{fig1}(c). By contrast, when biasing at the idle point, as shown in Fig.~\ref{fig1}(b), the qubit is far detuned from the drive, thus in principle, qubit readout and baseband-controlled two-qubit gates can be realized with minimal impacts from the always-on drive. To evaluate the feasibility of the control scheme, in the following, we first consider implementing universal control of an ideal two-level system, which is subjected to an always-on drive, using only Z-control. Next, we consider a more practical case of transmon qubits, which has a weak qubit anharmonicity, making qubits particularly susceptible to leakage during gate operations. We will show that with a fast-adiabatic scheme \cite{Martinis2014b}, Z-controlled single-qubit gate operations can be achieved with fast speed and low leakage. Finally, by biasing the qubit at the idle point, we further study the impact of the always-on drive on the qubit dephasing and qubit readout.
\subsection{Z-control of an ideal two-level system}\label{SecIIA}
For a baseband controlled two-level system (TLS) subjected to an always-on global drive, the system Hamiltonian is (hereinafter, we set $\hbar=1$) \begin{equation} \begin{aligned}\label{eq1} H_{lab}=\frac{\omega_{q}}{2}\sigma_{z}+\Omega_{d}\cos(\omega_{d}t)\sigma_{x} \end{aligned} \end{equation} where $\omega_{q}$ is the bare qubit frequency and can change according to the Z control pulse, $\omega_{d}$ and $\Omega_{d}$ are the frequency and the amplitude of the drive, respectively. Moving into the rotating frame with respect to the global drive and after applying the rotating wave approximation (RWA), the Hamiltonian reads \begin{equation} \begin{aligned}\label{eq2} H_{rot}=\frac{\Delta_{d}}{2}\sigma_{z}+\frac{\Omega_{d}}{2}\sigma_{x} \end{aligned} \end{equation} where $\Delta_{d}=\omega_{q}-\omega_{d}$ denotes the drive detuning. Note that unless otherwise stated, the RWA is used throughout this work.
At the idle point, the drive detuning is far large than the drive strength, thus the Bloch vector is slightly tilted towards the X-axis, as shown in Fig.~\ref{fig1}(b). Generally, the dressed eigenstates defined by this tilted Bloch vector are chosen to be the logical computational states. The tilted angle and the dressed states can be quantitatively obtained by diagonalization of the Hamiltonian in Eq.~(\ref{eq2}), giving rise to \begin{equation} \begin{aligned}\label{eq3} H_{diag}=\frac{\Delta}{2}Z,\,{\rm with}\,Z\equiv\cos{\theta}\sigma_{z}+\sin{\theta}\sigma_{x}, \end{aligned} \end{equation}
where $\theta=\arctan(\Omega_{d}/\Delta_{d})$ is the tilted angle and $\Delta=\sqrt{\Delta_{d}^{2}+\Omega_{d}^{2}}$. From the above discussions, when $|\Delta_{d}|\gg \Omega_{d}$, one can neglect the angle, as well as the difference between the bare states and the dressed states.
As shown in Eq.~(\ref{eq3}), by biasing the qubit at the idle point, the Z rotations can be easily realized by choosing suitable delay times $\tau$ between Z pulses, i.e., \begin{equation} \begin{aligned}\label{eq4} U_{z}=e^{-i\frac{\Delta \tau}{2}Z}. \end{aligned} \end{equation} Note here that compared with the traditional microwave control, Virtual-Z (VZ) gate scheme \cite{McKay2017} is not suitable for the present baseband control. However, similar to the VZ gate, besides time delay, here, no actual control pulses are needed for implementing Z rotations.
Generally, as shown in Fig.~\ref{fig1}(c), by tuning the qubit on-resonance with the global drive, single-qubit X rotations can be achieved. However, we note that since the initial Bloch vector is slightly tilted, as shown in Fig.~\ref{fig1}(b), a small drive detuning $\delta_{d}=|\Omega_{d}^{2}/\Delta_{d}|$ is needed for enabling ideal X rotations with respect to the initial Bloch vector defined by Eq.~(\ref{eq3}). Thus, according to Z control pulses, X rotations can be realized by tuning the qubit from the idle point to the working point with a small overshoot \cite{Barends2019}. This fact is further illustrated by the results shown in Fig.~\ref{fig2}. By initializing the qubit in state $|0\rangle$ and using square pulses (results with cosine-decorated square pulses can be found in Appendix~\ref{A}), Figure~\ref{fig2}(a) shows populations $P_{1}$, i.e., the population in state $|1\rangle$ at the end of the applied pulse, versus the drive detuning $\delta_{d}$ and the pulse length. Here, the drive amplitude is $10\,\rm MHz$ and the detuning at the idle point is $-100\,\rm MHz$. Similarly, given a fixed pulse length of $50\,\rm ns$, Figure~\ref{fig2}(b) shows $P_{1}$ versus $\delta_{d}$ and $\Omega_{d}$. The optimal parameters for X rotations are indicated by the red stars. Indeed, we find that a small frequency overshoot is needed for X rotations.
\begin{figure}
\caption{Flexibility of $\sqrt{X}$ rotations. (a) Population in state $|1\rangle$ (i.e., $P_{1}$ at the end of the pulse) versus the drive detuning and the pulse length of square pulses with the qubit prepared in state $|0\rangle$. Here, the drive strength is $10\,\rm MHz$ and the drive detuning at the idle point is $-100\,\rm MHz$. The red star indicates the optimal parameter set for implementing X rotations, while the dashed and dotted lines indicate the available parameter sets for implementing $\sqrt{X}$ rotations based on numerical simulations and analytical expression in Eq.~(\ref{eq5}), respectively. (b) same as in (a), instead showing $P_{1}$ versus the drive detuning and drive amplitude with the fixed gate length of $50\,\rm ns$.}
\label{fig2}
\end{figure}
In the present work, note that choosing $\sqrt{X}$ gates as the native gates could simplify the tune-up procedure of single-qubit gate operations. This is because:
(i) arbitrary single-qubit rotations can be generated by two $\sqrt{X}$ gates and three Z gates \cite{McKay2017}, i.e., $Z_{\phi1}-\sqrt{X}-Z_{\phi2}-\sqrt{X}-Z_{\phi3}$, with $Z_{\phi}\equiv\exp[-i\phi Z/2]$;
(ii) compared with the native X gate, the implementation of $\sqrt{X}$ gates does not pose stringent requirements on the on-resonance condition, i.e., even the qubit is slightly off-resonance with the drive, $\sqrt{X}$ gates can still be achieved. This can be captured by the analytical expression of Rabi oscillations for the two-level system initialized in state $|0\rangle$, i.e., Rabi's formula \begin{equation} \begin{aligned}\label{eq5} P_{1}(t)=\frac{\Omega_{d}^{2}}{\Omega_{d}^{2}+\Delta_{d}^2}\sin^2\left[\frac{t}{2}\sqrt{\Omega_{d}^{2}+\Delta_{d}^2}\right]. \end{aligned} \end{equation} From Eq.~(\ref{eq5}), implementing $\sqrt{X}$ gates requires $P_{1}=1/2$ at the end of the applied pulse, giving rise to the relations among the pulse length, the drive detuning $\Delta_{d}$, and the drive amplitude $\Omega_{d}$, as illustrated by the dotted lines of Fig.~\ref{fig2}. Accordingly, the results based on numerical simulations are also presented, as indicated by the dashed line of Fig.~\ref{fig2}. Note here that the derivation of the analytical equation ignores the slight tilt at the idle point, and this explains the discrepancy between the analytical and numerical results. Both the analytical and numerical results show that compared to X rotations, the available parameter ranges of $\sqrt{X}$ rotations can provide great flexibility in its tune-up procedure.
Generally, due to the flexibility of $\sqrt{X}$ rotations, for tuning-up $\sqrt{X}$ gates, the above-mentioned overshoot can be ignored. In the next subsection, we will show that following this way, given a fixed drive detuning, $\sqrt{X}$ gates can be realized by only optimizing the ramp times of control pulses, as suggested by Fig.~\ref{fig2}(a). Meanwhile, in large-scale quantum systems with multiplexed control, this flexibility can be the most encouraging advantage as to mitigate single-qubit gate error due to stray coupling between qubits and to compensate for the non-uniformity of qubit parameters. This will be discussed in detail in Sec.~\ref{SecIV}.
\subsection{Baseband control of qubit with fast-adiabatic ramps}\label{SecIIB}
\begin{figure}
\caption{Leakage out of the computational subspace. (a) Energy spectrum (solid lines) versus the drive detuning in the rotating frame corresponding to the global drive. The dashed lines denote the bare energy levels (i.e., spectrum without the drive). Here, the used system parameters are qubit anharmonicity $\alpha_{q}/2\pi=-250\,\rm MHz$, drive frequency $\omega_{d}/2\pi=6.1\,\rm GHz$, and drive amplitude $\Omega_{d}/2\pi=10\,\rm MHz$. (b) Bloch vector for the leakage space spanned by states $\{|1\rangle,|2\rangle\}$. The strength of the coupling between states $|1\rangle$
and $|2\rangle$ is $\sqrt{2}\Omega_{d}$. At the idle point and in the leakage space, the drive detuning is $\Delta_{d}+\alpha_{q}$, while at the working point, the detuning is $\alpha_{q}$. During single-qubit X rotations, the Bloch vector in this leakage space varies according to the drive detuning. In the present work, to avoid possible leakage during qubit control, the qubit idle frequency is far detuned below the drive frequency.}
\label{fig3}
\end{figure}
In the above discussion, the single-qubit baseband control is discussed for an ideal two-level system. Nevertheless, for practical superconducting qubits, such as the transmon qubit, the weak qubit anharmonicity makes single-qubit gate operations particularly prone to leakage outside the qubit subspace. For one such baseband control transmon qubit, which is driven by an always-on global drive, the system Hamiltonian is (hereafter, transmon qubits are modeled as anharmonic oscillators \cite{Koch2007})
\begin{equation} \begin{aligned}\label{eq6} H_{q}=\omega_{q}a_{q}^{\dagger}a_{q}+\frac{\alpha_{q}}{2}a_{q}^{\dagger}a_{q}^{\dagger}a_{q}a_{q} +\frac{\Omega_{d}}{2}(a_{q}^{\dagger}e^{-i\omega_{d}t}+a_{q}e^{+i\omega_{d}t}). \end{aligned} \end{equation}
Here, $a_{q}\,(a_{q}^{\dagger})$ is the annihilation (creation) operator. Figure~\ref{fig3}(a) shows the energy spectrum of the driven qubit versus the drive detuning with qubit anharmonicity $\alpha_{q}/2\pi=-250\,\rm MHz$, drive frequency $\omega_{d}/2\pi=6.1\,\rm GHz$, and drive amplitude $\Omega_{d}/2\pi=10\,\rm MHz$. One can find that due to the global drive, there exits an off-resonance coupling between states $|2\rangle$ and $|1\rangle$, which can cause leakage to state $|2\rangle$
when performing X rotations, i.e., biasing the qubit from its idle point to the working point. Note that there exist two leakage channels, caused by the $|1\rangle\leftrightarrow|2\rangle$
and $|0\rangle\leftrightarrow|2\rangle$ interactions, as shown in Fig.~\ref{fig3}(a). Since the $|0\rangle\leftrightarrow|2\rangle$
interaction involves second-order processes, generally, the induced leakage error is far smaller than that of the $|1\rangle\leftrightarrow|2\rangle$ interaction. Here, we thus mainly focus on the leakage error caused by the $|1\rangle\leftrightarrow|2\rangle$ interaction. However, we note that in our numerical analysis, all the leakage channels, including the $|0\rangle\leftrightarrow|2\rangle$ interaction, are taken into consideration. While this leakage issue can be addressed by using the DRAG scheme in the traditional microwave control setup, this scheme cannot be directly utilized for the baseband flux control setup, as here only Z control is available.
Additionally, in principle, at the idle point, qubits could be far detuned above or below the frequency of the always-on drive. However, to avoid possible leakage error during qubit control, such as gate operations, qubit initialization, and readout, we prefer to bias the qubit away from the harmful avoid crossing caused by coupling between states $|1\rangle$ and $|2\rangle$, as shown in Fig.~\ref{fig3}(a). In this way, during the baseband-controlled gate operations, the qubit system will not sweep through or operate nearby this harmful avoid crossing, generally allowing the suppression of the leakage to state $|2\rangle$. Considering this fact, hereafter, we consider biasing the qubit below the drive frequency at the idle point. Even in the setting, during Z-controlled single-qubit gate operations, leakage error can still occur due to the non-adiabatic error, as shown in Fig.~\ref{fig3}(b). In the following, we will consider using a fast-adiabatic control scheme for suppressing the leakage further.
\begin{figure}
\caption{Minimizing leakage out of the computational subspace for performing Z-controlled X rotations.
(a) Leakage as a function of the ramp time for the transmon qubit initialized in state $|1\rangle$ with the anharmonicity $\alpha_{q}/2\pi=\{-200,\,-250,\,-300\}\rm MHz$ ( denoted by the solid line, dashed line, and dotted line, respectively). Inset shows the typical fast-adiabatic pulse and the fast-adiabatic flat-top pulse (i.e., square pulse with fast-adiabatic ramps) for controlling the qubit frequency from the idle point ($6.0\,\rm GHz$) to the working point ($6.1\,\rm GHz$). The other system parameters are: the hold time (i.e., the pulse length of the flat part) $t_{h}=20\,\rm ns$, drive frequency $\omega_{d}/2\pi=6.1\,\rm GHz$, and drive amplitude $\Omega_{d}/2\pi=10\,\rm MHz$.
(b) Same as in (a), instead showing the population in states $|0\rangle$ and $|1\rangle$ versus the ramp time of the fast-adiabatic flat-top pulse.}
\label{fig4}
\end{figure}
As shown in Fig.~\ref{fig3}(b), the leakage error occurs when one non-adiabatically varies the driving detuning. Considering that coherence times of superconducting qubits are still limited, our target is to find a good flux control pulse, thus the non-adiabatic error is suppressed while maintaining a fast operation speed. Fortunately, this issue has already been addressed successfully by using a fast-adiabatic scheme introduced in Ref.~\cite{Martinis2014b}. Within the scheme, optimal control pulses can be obtained for minimizing non-adiabatic errors for any pulse longer than the chosen pulse length. However, we note that the original scheme only addresses the non-adiabatic error in the pulse ramps. Thus, here, to address the leakage issue in our setting, we consider using a square control pulse with optimal fast-adiabatic ramps, which is obtained following the fast-adiabatic scheme \cite{Martinis2014b} (see Appendix~\ref{B} for details). The inset of Fig.~\ref{fig4}(a) shows the optimal ramp pulse (solid blue line), which is used for generating our target control pulse with a flat middle part and fast-adiabatic ramps (solid orange line). Hereafter, we refer to this pulse as the fast-adiabatic flat-top pulse.
Here, we turn to evaluate the efficiency of the proposed fast-adiabatic flat-top pulse. We consider that the qubit idle frequency is $6.0\,\rm GHz$ and during the implementation of single-qubit X rotations, the drive detuning $\Delta_{d}$ varies from the idle point at $-100\,\rm MHz$ to the work point at $0\,\rm MHz$ and then coming back, according to the fast-adiabatic flat-top pulse. By initializing the qubit in state $|1\rangle$, Figure~\ref{fig4}(a) shows the population leakage to state $|2\rangle$ as a function of ramp times with the hold time of 20 ns and the qubit anharmonicities $\alpha_{q}/2\pi=\{-200,\,-250,\,-300\}\rm MHz$. For easy comparison, the results for applying only the fast-adiabatic pulse are also presented. One can find that by using the fast-adiabatic flat-top pulse, the leakage can be suppressed below $10^{-6}$ for ramp times longer than 10 ns, and inserting a square pulse in the fast-adiabatic pulse does not change the efficiency of the original fast-adiabatic scheme. In Fig.~\ref{fig4}(b), we also show the populations in $|0\rangle$
and $|1\rangle$ versus the ramp times. Additionally, Appendix~\ref{B} presents further results for different drive strengths.
From the results shown in Fig.~\ref{fig4}, one can find that $\sqrt{X}$ rotations can be realized with the ramp time at about 10 ns, giving rise to the total pulse length of about 30 ns. Meanwhile, same as the case for two-level systems (in Sec.\ref{SecIIA}), here, single-qubit Z rotations can be easily implemented by controlling the time delay between Z pulses. Therefore, we could reasonably expect that with the help of the fast-adiabatic scheme, fast-speed single-qubit operations could be achieved with low leakage errors (we will evaluate the single-qubit gate performance in detail in the following section).
\subsection{Dephasing due to fluctuations in the drive amplitude}\label{SecIIC}
\begin{figure}
\caption{Qubit dephasing due to the fluctuations in the amplitude of the always-on drive. (a) Time evolution of the magnitudes of the averaged off-diagonal matrix element, i.e., $|\langle\rho_{01}(t)\rangle|$, for the qubit initialized in state $(|0\rangle+|1\rangle)/\sqrt{2}$. Here, we assume that the drive amplitudes ($\Omega_{d}/2\pi=10\,\rm MHz$) subject to amplitude-dependent Gaussian noise, i.e., $\textsl{N}(0,\sigma)$, and 2000 realizations of noise are used for obtaining $|\langle\rho_{01}(t)\rangle|$. The dashed lines are exponential fits $[1-\exp(-t/T_{\phi})]/2$, giving rise to the dephasing time $T_{\phi}$. (b) Dephasing time versus the noise variance. For given noise variances, the inset shows the dephasing time versus the drive amplitude. Here, the other parameters used are: $\Delta_{d}/2\pi=-100\,\rm MHz$ and $\alpha_{q}/2\pi=-250\,\rm MHz$.}
\label{fig5}
\end{figure}
Within the introduced baseband control setup, at the idle point, the global always-on drive acts as an off-resonance drive and can induce ac-Stark frequency shifts on the qubits. For the two-level system studied in Sec.\ref{SecIIA}, the shift is given as $\delta\omega=\Delta-\Delta_{d}\approx \Omega_{d}^{2}/(2\Delta_{d})$, while, taking the higher energy levels of the transmon qubit into consideration, the shift is \cite{Schneider2018} \begin{equation} \begin{aligned}\label{eq7} \delta\omega\approx \frac{\alpha_{q}\Omega_{d}^{2}}{2\Delta_{d}(\Delta_{d}+\alpha_{q})}. \end{aligned} \end{equation} From Eq.~(\ref{eq7}), the shift has a quadratic-dependent on the drive amplitude, making the qubit frequency more susceptible to possible amplitude noise. Therefore, fluctuations in the drive amplitude can cause qubit dephasing, which has been recently observed in superconducting qubits \cite{Wei2022}.
Here, to numerically study the amplitude-fluctuation-induced qubit dephasing, we consider an amplitude-dependent noise, i.e., amplitude fluctuations are proportional to the amplitudes. By assuming the drive subject to zero-mean Gaussian noise, i.e., $\textsl{N}(0,\sigma)$, we numerically simulate the time evolution of the off-diagonal matrix element $\rho_{01}(t)$ for the qubit initialized in state $(|0\rangle+|1\rangle)/\sqrt{2}$. After averaging $\rho_{01}(t)$ over 2000 trajectories (i.e., realizations of noise), the magnitudes of the off-diagonal matrix element display a clear exponential decay, as shown in Fig.~\ref{fig5}(a). Here, the evolution time is $10\,\mu s$, the other used parameters are: $\Delta_{d}/2\pi=-100\,\rm MHz$, $\Omega_{d}/2\pi=10\,\rm MHz$, and $\alpha_{q}/2\pi=-250\,\rm MHz$, giving rise to $\delta\omega/2\pi\approx-0.36\,\rm MHz$. By fitting the decay curves to $[1-\exp(-t/T_{\phi})]/2$, Figure~\ref{fig5}(b) shows the dephasing time $T_{\phi}$ versus the noise variance $\sigma$. Here, we also show the results for $\Omega_{d}/2\pi=20\,\rm MHz$. Additionally, in the inset, we further show the dephasing times versus the drive amplitudes.
From the results shown in Fig.~\ref{fig5}(b), and given the typical noise variance of $1\%$, we can conclude that the amplitude-noise induced dephasing can be safely neglected by detuning the qubit far from the drive frequency. Meanwhile, we note that to ensure high-fidelity gate operations within sub-100 ns, the drive amplitude itself should be larger than $10\,\rm MHz$. Finally, we note that besides the qubit dephasing, when the always-on drive is shared by multiple qubits, there are two additional potential issues related to the drive and its fluctuation:
(i) qubit decoherence due to the excess quasiparticles (QPs) \cite{Catelani2011}. Previous works have demonstrated that QPs can be injected into a qubit by applying a high-power microwave pulse resonance with its readout resonator \cite{Vool2014,Wang2014}. However, at the present setting, the strength of the always-on drive is far smaller than that of the former case, we thus expect that the contribution of the always-on drive to the excess QPs is negligible.
(ii) when the always-on shared (global) drive is shared by multiple qubits, fluctuations in the drive can result in errors on multiple qubits. At first glance, this can cause correlated errors. However, as discussed in Ref.~\cite{Fowler2014}, provided the fluctuation is small and quantum error correction is frequent, the fluctuation in the shared drive can result in a correlated probability of error on multiple qubits, but the errors themselves will not be correlated. Additionally, as suggested in Fig.5(b) and Eq.(7), by increasing the qubit-drive detuning, the fluctuation-induced dephasing (error) can be heavily suppressed at the system idle time. And during single-qubit gate operations, with the currently available control electronics, single-qubit gates with gate errors below 0.0001 have been demonstrated \cite{Somoroff2021,Li2023}. Thus, we argue that fluctuations in the shared drive itself can indeed cause errors, but probably a rather small one ($<0.0001$). In this case, the control-noise-induced error may still be handled by the error correction schemes \cite{Fowler2014}.
\subsection{Impact of the always-on drive on the qubit readout}\label{SecIID}
\begin{figure}
\caption{Qubit dispersive readout with the presence of the always-on drive.
(a) Histograms of the integrated readout quadrature for the qubit prepared in states $|0\rangle$ (blue) and $|1\rangle$ (orange). The dashed blue line and dashed orange line denote the Gaussian fits of the histograms for states $|0\rangle$
and $|1\rangle$, respectively. The intersection point of the two fitted distributions gives the state-decision threshold. The inset shows the IQ scatter plot of the integrated readout quadrature. (b) Same as in (a), instead showing the results without the always-on drive. (c) Readout error $1-F$ as a function of the drive strength.}
\label{fig6}
\end{figure}
As mentioned in Sec.~\ref{SecI} and Sec.~\ref{SecIIA}, due to the presence of the always-on drive, in this work, the microwave dressed states are defined as the computational states. Here, since the available control over the always-on drive is limited, the previous method \cite{Zhao2022,Huang2021}, in which the dressed state is first mapped back to the corresponding bare state, and then the traditional dispersive readout is employed for inferring the qubit information \cite{Wallraff2005}, cannot be directly utilized. However, as discussed in Sec.~\ref{SecIIA}, when the drive detuning is far larger than the drive amplitude, i.e., $|\Delta_{d}|\gg\Omega_{d}$, the difference between dressed states and bare states can be neglected. Therefore, we expect that by keeping a large ratio of the drive detuning to the drive amplitude, the qubit information can be directly inferred using the traditional dispersive readout scheme.
To explore the possible impact of the always-on drive on the qubit dispersive readout, we numerically simulate the system dynamics during the dispersive readout. By applying a 250-ns square readout pulse with frequency $\omega$ and amplitude $\Omega$ to the readout resonator with decay rate $\kappa$, the full system dynamics are governed by the Hamiltonian \begin{equation} \begin{aligned}\label{eq8} H_{\rm read}=&H_{q}+\omega_{r}a_{r}^{\dagger}a_{r}+g(a_{q}^{\dagger}a_{r}+a_{q}a_{r}^{\dagger}) \\&+\frac{\Omega}{2}(a_{r}^{\dagger}e^{-i\omega t}+a_{r}e^{+i\omega t}), \end{aligned} \end{equation}
where $H_{q}$ denotes the qubit Hamiltonian given in Eq.~(\ref{eq6}), $\omega_{r}$ is the frequency of the readout resonator, $a_{r}\,(a_{r}^{\dagger})$ is the annihilation (creation) operator of the resonator, and $g$ denotes the strength of the qubit-resonator coupling. In this following, the qubit information is encoded into single quadrature, i.e., $I$-quadrature, by choosing the readout frequency to be $\omega=(\omega_{r0}+\omega_{r1})/2$ \cite{Wallraff2005}. Here, $\omega_{r0}$ and $\omega_{r1}$ denote the dressed resonator frequencies with the qubit in states $|0\rangle$ and $|1\rangle$, respectively. The other system parameters are: $\omega_{q}/2\pi=6.0\,\rm GHz$, $\alpha_{q}/2\pi=-250\,\rm MHz$, $\omega_{d}/2\pi=6.1\,\rm GHz$, $\omega_{r}/2\pi=5.0\,\rm GHz$, $g/2\pi=100\,\rm MHz$, $\kappa/2\pi=5\,\rm MHz$, and $\Omega/2\pi=7\,\rm MHz$.
According to Eq.~(\ref{eq8}), we simulate the system dynamics based on solving the stochastic master equation \cite{Johansson2012}. Then, following Ref.~\cite{Walter2017}, we further caulate the integrated readout quadrature with an optimal weight function (see Appendix~\ref{C} for details). With 5000 repetitions of the simulation for each qubit basis state, i.e., $|0\rangle$ and $|1\rangle$, Figure~\ref{fig6}(a) shows the two histograms of the integrated readout quadrature with the qubit prepared in states $|0\rangle$ and $|1\rangle$, respectively. Here, the drive magnitude is $20\,\rm MHz$. For easy comparison, we also present the result for the global drive is absent, as shown in Fig.~\ref{fig6}(b).
Fitting the histograms to Gaussian functions gives the state-decision threshold at the intersection point of the two fitted distributions. Accordingly, the readout fidelity can be calculated as $F=1-[P(0|1)+P(1|0)]/2$, where $P(0|1)$ ($P(1|0)$) denotes the error probability that the qubit initialized in state $|1\rangle$ ($|0\rangle$) is identified as in state $|0\rangle$ ($|1\rangle$). Accordingly, Figure~\ref{fig6}(c) shows the readout error $1-F$ versus the drive strength. One can find that when increasing the drive amplitude from $0$ to $20\,\rm MHz$, while the error shows an upward trend, the increased error is below $1\%$. Moreover, the upward trend also suggests that by further increasing the ratio $|\Delta_{d}|/\Omega_{d}$, the increased error should be heavily suppressed.
\section{An Application in qubit architectures with tunable coupling}\label{SecIII}
\begin{figure}
\caption{Residual coupling with varying qubit frequency. (a) Left: residual resonance XY coupling versus the qubit frequency and the coupler frequency. Horizontal cut through (Left) denotes the result plotted in (Right), i.e., XY coupling versus the qubit frequency with the coupler frequency fixed at $\omega_{c}/2\pi=11.35\,\rm GHz$. (b) Left: residual ZZ coupling versus the frequencies ($\omega_{0}$ and $\omega_{1}$) of the two qubits with the coupler frequency fixed at $11.35\,\rm GHz$. Horizontal cut through (Left) denotes the result plotted in (Right), i.e., ZZ coupling versus frequency of $Q_{1}$ with the $Q_{0}$'s frequency fixed at $\omega_{0}/2\pi=6.0\,\rm GHz$. Vertical cut through (Left) denotes the result plotted in the inset of (Right), i.e., ZZ coupling versus the frequency of $Q_{0}$ with the $Q_{1}$'s frequency fixed at $\omega_{1}/2\pi=5.9\,\rm GHz$.}
\label{fig7}
\end{figure}
Given the overview of the baseband control scheme, in this section, we will present the application of this scheme in a qubit architecture with tunable coupling. As depicted in Fig.~\ref{fig1}(a), we consider that two frequency-tunable transmon qubit $Q_{0}$ and $Q_{1}$ are coupled via a tunable coupler $Q_{c}$ (i.e., an auxiliary transmon qubit) and both qubits are driven by an always-on global drive. After applying RWA, the system Hamiltonian is given by
\begin{equation} \begin{aligned}\label{eq9} H=&\sum_{j=0,1,c}\big(\omega_{j}a_{j}^{\dagger}a_{j}+\frac{\alpha_{j}}{2}a_{j}^{\dagger}a_{j}^{\dagger}a_{j}a_{j}\big) \\&+\sum_{\substack{k=0,1,c\\j\neq k}}g_{jk}(a_{j}a_{k}^{\dagger}+a_{j}^{\dagger}a_{k}) \\&+\sum_{i=0,1}\frac{\Omega_{d}}{2}(a_{i}^{\dagger}e^{-i\omega_{d}t}+a_{i}e^{+i\omega_{d}t}), \end{aligned} \end{equation} where $\omega_{j}$ and $\alpha_{j}$ are the bare qubit frequency and the qubit anharmonicity of $Q_{j}$, $q_{j}\,(q_{j}^{\dagger})$ is the associated annihilation (creation)
operator, and $g_{jk}$ denotes strength of the coupling between $Q_{j}$ and $Q_{k}$. Hereafter, the system state is denoted by the notation $|Q_{0}Q_{c}Q_{1}\rangle$ and the used system parameters are: the qubit anharmonicity $\alpha_{0}/2\pi=\alpha_{1}/2\pi=-250\,\rm MHz$, the coupler anharmonicity $\alpha_{c}/2\pi=-200\,\rm MHz$, the direct qubit-qubit coupling strength $g_{01}/2\pi=13\,\rm MHz$ (at $\omega_{0}/2\pi=\omega_{1}/2\pi=5.5\,\rm GHz$), the qubit-coupler coupling strength $g_{0c}/2\pi=g_{1c}/2\pi=160\,\rm MHz$ (at $\omega_{0(1)}/2\pi=\omega_{c}/2\pi=5.5\,\rm GHz$), the drive amplitude $\Omega_{d}/2\pi=10\,\rm MHz$, and the drive frequency $\omega_{d}/2\pi=6.1\,\rm GHz$.
Note here that the RWA is used for simplifying numerical simulation (otherwise, given the always-on drive, Floquet methods could be employed here \cite{Huang2021,Shirley1965,Sambe1973,Petrescu2021}). However, in the present two-qubit system with tunable coupling, the non-RWA terms in the original Hamiltonian (see Appendix~\ref{D} for details) can significantly affect the effective coupling between qubits and can shift the bare qubit frequency. Thus, here, considering non-RWA terms while still working within the RWA formalism, we keep second-order corrections from the non-RWA terms and find that with this correction (details on its derivation can be found in Appendix~\ref{D}), the results agree well with the results without applying the RWA. Accordingly, the corrections are taken into consideration throughout the following discussion.
Before going into details of the baseband controlled gate operations, we give a few brief discussions of the tunable coupling architecture. For performing gate operations in multiqubit systems, the key benefit of the introduced tunable coupler is that the inter-qubit coupling strength can be tuned off by biasing the coupler at a certain frequency point, i.e., zero-coupling point. However, the zero-coupling point can change when the qubit is just biased slightly away from its idle point. Fortunately, in the tunable coupling architecture, biasing the qubit slightly away, generally, only causes a small increase in the residual inter-qubit coupling. This can be found in Fig.~\ref{fig7}. Figure~\ref{fig7}(a) shows the strengths of the residual resonance XY coupling versus the qubit frequency and the coupler frequency, while Figure~\ref{fig7}(b) shows the residual ZZ coupling versus the frequencies of the two qubits with the coupler frequency fixed at $11.35\,\rm GHz$. Here, the XY coupling and the ZZ coupling are numerically calculated by the diagonalization of the Hamiltonian Eq.~(\ref{eq9}) in the rotating frame defined by the always-on drive. To be more specific, the XY coupling is extracted as half the energy difference between dressed eigenstates $|10\rangle$ and $|01\rangle$, while the ZZ coupling is $\zeta_{zz}=(E_{11}-E_{10})-(E_{01}-E_{00})$. Here, $E_{ij}$ denotes the energy of dressed eigenstate $|ij\rangle$, which is adiabatically connected to the bare state $|i0j\rangle$ \cite{Ghosh2013}.
According to the above results, in the following discussion, we consider that at the system idle point, the frequency of qubit $Q_{0}$ and qubit $Q_{1}$ are $\omega_{0}/2\pi=6.0\,\rm GHz$ and $\omega_{1}/2\pi=5.9\,\rm GHz$, respectively, and the frequency of coupler $Q_{c}$ is $\omega_{c}/2\pi=11.35\,\rm GHz$. Therefore, the residual ZZ coupling is below $10\,\rm kHz$ at the system idle point. Moreover, during the gate operations based on slightly tuning qubit frequency, such as implementing single-qubit gates by tuning the qubit from its idle point to the working point (e.g., at $6.1\,\rm GHz$), the residual inter-qubit ZZ coupling can always be below $10\,\rm kHz$.
\subsection{Single-qubit gate operation}\label{SecIIIA}
\begin{figure}
\caption{Performing Z-controlled X rotations with the fast-adiabatic flat-top pulse in the two-qubit system with tunable coupling. (a) Leakage as a function of the pulse ramp time for qubit $Q_{0}$ initialized in state $|1\rangle$. Inset shows the population in states $|0\rangle$ and $|1\rangle$ versus the ramp time for the Z-controlled X rotations. The solid lines and dashed lines represent the results with the coupler biased at two different idle points, i.e., $11.35\,\rm GHz$ and $11.25\,\rm GHz$, respectively. Same as in Fig.~\ref{fig4}, here, the hold time of the utilized fast-adiabatic flat-top pulse is fixed at $t_{h}=20\,\rm ns$. (b) Same as in (a), instead showing the results for $Q_{1}$. Additionally, here also shows the population leakage to $Q_{0}$, as indicated by the orange lines.}
\label{fig8}
\end{figure}
Following the scheme introduced in Sec.~\ref{SecIIB}, here, Z-controlled single-qubit gates are realized by using the fast-adiabatic flat-top pulse. Note here that in the present two-qubit system, single-qubit gate operations for one qubit are tuned up and characterized with the other qubit in its ground state $|0\rangle$.
Figure~\ref{fig8} shows the leakage versus the ramp time of the pulse with the qubit initialized in its excited state $|1\rangle$. One can find that while for $Q_{0}$, the result is in line with our theory discussed in Sec.~\ref{SecIIB}, i.e., by increasing the ramp time, the leakage can be further suppressed, the result of $Q_{1}$ seems unreasonable at first glance. However, during gate operations applied to $Q_{1}$, $Q_{1}$ is tuned from its idle point at $5.9\,\rm GHz$ to the working point at about $6.1\,\rm GHz$, according to the fast-adiabatic pulse, while
$Q_{0}$ is fixed at its idle point at $6.0\,\rm GHz$. Therefore, during the pulse ramp, $Q_{1}$ will sweep through a tiny avoided crossing formed by the residual resonance XY coupling between $Q_{0}$ and $Q_{1}$ at $6.0\,\rm GHz$. On contrast, during single-qubit gate operations, $Q_{0}$ will not sweep through $Q_{1}$. As shown in Fig.~\ref{fig7}(a), the strength of the residual XY coupling is about $0.1\,\rm MHz$. Thus, sweeping through this avoided crossing slowly will generally cause more leakage into the nearby qubit $Q_{1}$, as shown in Fig.~\ref{fig8}(b), where the orange solid line denotes the population of $Q_{0}$ in state $|1\rangle$. One can find that for $t_{r}\geq 10\,\rm ns$, the leakage into $Q_{0}$ gives the leading contributions to the total leakage error. These results suggest that there exists a trade-off between gate error resulting from the qubit itself and error from spectator qubits, i.e., suppressing leakage into $|2\rangle$ favors longer gate times, while mitigating the leakage into the $Q_{0}$ favors short gate times. This observation is in agreement with that in previous work \cite{Zhao2022b}.
To address the above issue, one can change the idling coupler frequency, the XY coupling can thus be further suppressed at the resonance point, i.e., $6.0\,\rm GHz$. By biasing the coupler at $11.25\,\rm GHz$, the residual XY coupling is suppressed below $0.01\,\rm MHz$. Accordingly, the leakage into $Q_{0}$ is indeed suppressed heavily, as shown in Fig.~\ref{fig8}(b), where the dashed lines show the results with the coupler biased at $11.25\,\rm GHz$.
Here, we turn to evaluate the gate performance of the baseband controlled single-qubit gates, and use the metric of the state-average gate fidelity \cite{Pedersen2007} in the following discussion (details on the fidelity calculation can also be found in Ref.~\cite{Zhao2022b}). As mentioned in Sec.~\ref{SecIIA}, in the present work, we focus on the implementation of $\sqrt{X}$ gates. From the inset of Figs.~\ref{fig8}(a) and ~\ref{fig8}(b), one can find that for both qubits, $\sqrt{X}$ gates can be realized with a ramp time of about $10\,\rm ns$, giving rise to the total gate time of about $30\,\rm ns$. Moreover, even by biasing the coupler at $11.35\,\rm GHz$, Figure~\ref{fig8} shows that when the ramp time is about $10\,\rm ns$, the leakage error can still be suppressed below $5\times10^{-5}$ for both qubits. This is to be expected, since sweeping through the tiny avoided crossing with fast speed could suppress leakage. By optimizing the ramp times, we find that for both qubits, up to single-qubit Z rotations, $\sqrt{X}$ gates can be achieved with gate fidelity exceeding $99.999\%$ (for $Q_{0}$, the gate fidelity is $99.9998\%$ and the optimal gate time is $30.2\,\rm ns$, while for $Q_{1}$, are the $99.9996\%$ and $29.4\,\rm ns$). As mentioned before, Z gates can be easily realized by choosing suitable time delays between flux pulses. In this way, universal single-qubit gates can be achieved by combining Z gates and $\sqrt{X}$ gates.
\subsection{Two-qubit CZ gate}\label{SecIIIB}
\begin{figure}
\caption{Performing CZ gates with the fast-adiabatic flat-top pulse in the two-qubit system with tunable coupling. During the gate operations, qubit $Q_{0}$ is fixed at its idle point, i.e., $6.0\,\rm GHz$, qubit $Q_{1}$ and coupler $Q_{c}$ are tuned from their idle points ($5.9\,\rm GHz$ and $11.35\,\rm GHz$) to the working points, resulting in a complete population oscillation between states $|101\rangle$ and $|200\rangle$. (a) The cosine-decorated square pulse with the ramp time of $10\,\rm ns$ for implementing CZ gates. Up to single-qubit phase gates, the gate fidelity is $99.82\%$. (b) Time evolution of the qubit state population during the gate operation with the cosine-decorated square pulse. Here, $P_{ij}$ denotes the population in state $|i0j\rangle$ for the two-qubit system initialized in state $|i0j\rangle$. (c) For biasing the coupler, the fast-adiabatic flat-top pulse with a hold time of $10\,\rm ns$ is employed, while for $Q_{1}$, a cosine-decorated square pulse with the ramp time of $6\,\rm ns$ is used. Here, the total pulse length is $30.7\,\rm ns$ and up to single-qubit phase gates, the CZ gate fidelity is $99.94\%$. (d) Time evolution of the qubit state population during the gate operation with the fast-adiabatic pulse, showing that the population swap between qubits is suppressed below $10^{-3}$.}
\label{fig9}
\end{figure}
Having discussed the single-qubit control, we now turn to the two-qubit case. Here, we consider the implementation of CZ gates in the two-qubit system with an always-on drive. During the gate operations, $Q_{0}$ is fixed at its idle point, i.e., $6.0\,\rm GHz$, $Q_{1}$ is tuned from its idle point ($5.9\,\rm GHz$) to the working point, where a complete oscillation between states $|101\rangle$
and $|200\rangle$ can occur. Meanwhile, the coupler is tuned from its idle point at $11.35\,\rm GHz$ to a working point at about $7\,\rm GHz$, giving rise to the CZ coupling strength of $20\,\rm MHz$ (see Appendix~\ref{D}).
Figure~\ref{fig9}(a) shows the typical control pulse, i.e., the pulse with a flat middle part and cosine-shaped ramps (see Appendix~\ref{A} for details), with a pulse length of $30\rm ns$ for the CZ implementation. By numerically optimizing the working points of $Q_{c}$ and $Q_{1}$, the gate fidelity of the implemented CZ gate (up to single-qubit Z phases) is $99.82\%$. After inspecting the qubit dynamics during the gates, one can find that the leading error source is the population swap between two qubits, as shown in Fig.~\ref{fig9}(b). Following the fast-adiabatic scheme discussed in Sec.~\ref{SecIIB} and the previous work \cite{Martinis2014b,Sung2021}, here, the fast-adiabatic flap-top pulses, as shown in Fig.~\ref{fig9}(c), with a hold time of $10\,\rm ns$, is used to suppress the population swap. Accordingly, the population swap is indeed largely suppressed, as shown in Fig.~\ref{fig9}(d), improving the CZ gate fidelity to $99.94\%$ with a gate time of $30.7\,\rm ns$. Additionally, we note that generally, by increasing the gate length, the residual gate error can be further suppressed (see also in Appendix~\ref{E}).
The above results show that although there exists an always-on global drive, high-fidelity two-qubit gates can still be achieved in a short time. This success is mainly based on the fact that during the gate operations, the global drive is far detuned from both qubits and the coupler.
\section{discussion}\label{SecIV}
Given the above theoretical analysis of the implementation of the baseband flux control in tunable coupling architecture, in the following, we will give a few discussions of the challenges and opportunities for realizing the baseband control strategy in large-scale superconducting quantum processors.
\subsection{Practical challenges}\label{SecIVA}
While our theoretical study shows that baseband controlled gate operations can be realized with high fidelity and fast speed, we note that besides the qubit decoherence, there exist several practical experimental issues that will limit the available gate performance:
(i) Flux pulse distortion. Flux pulse distortion has been demonstrated as a critical issue faced by baseband flux-controlled gate operations \cite{Jerger2019,Rol2020,Foxen2019}. Moreover, the above-demonstrated high-fidelity gate operations are achieved by using pulse shaping technologies, thus, the impact of flux pulse distortion can become more prominent in our setting.
(ii) Stray coupling beyond nearest neighbors. Generally, in our setting, the always-on drive is shared by multiple qubits. When performing single-qubit gates in parallel, multi-qubit will be tuned on-resonance with the same drive. This means that any stray coupling between these qubits will cause population swaps among these qubits, as discussed in Sec.~\ref{SecIIIA}, leading to additional gate errors compared to isolated gates. While near-neighbor couplings between qubits can be controlled well in the tunable coupling architecture, parasitic coupling beyond nearest neighbors can still exist due to, such as stray capacitive coupling, in multi-qubit systems \cite{Barends2014,Zajac2021,Yanay2022,Zhao2022b}. This will degrade the efficiency of the baseband control strategy in large-scale quantum processors.
(iii) Defect modes, such as TLSs \cite{Muller2019}. Same as in (ii), when performing single-qubit gates, the working frequencies of multi-qubit are almost limited to a fixed one, i.e., the frequency of the shared drive. This will limit the ability to mitigate the impacts from defect modes by tuning the qubit away from the defects \cite{Klimov2018}.
(iv) Keeping track of the single-qubit phase accumulation. In our setting, the qubit frequency at its idle point is detuned from the always-on drive. Thus, the single-qubit phase will accumulate at the speed of the drive detuning $\Delta$ during the idle time. While the accumulated phase can be employed to realize single-qubit Z gates, on the other hand, when performing gate sequences or quantum circuits, the accumulated phase should be tracked carefully over the whole time domain. Compared with the traditional microwave control, this could complicate the implementation of quantum circuits.
In addition, we note that owing to the great flexibility of Z-controlled $\sqrt{X}$ gates, as discussed in Sec.\ref{SecIIA}, the issues, related to (ii) and (iii), may be addressed. From the results shown in Fig.~\ref{fig2}, we can find that given a fixed drive amplitude or a fixed pulse length, $\sqrt{X}$ gates can be achieved with a small drive detuning, for which its magnitude can even be compared with that of the always-on drive. Thus, when implementing isolated or paralleled single-qubit gates, the working frequencies of qubits can be biased intentionally at different frequency points, thus impacts of sub-MHz stray coupling can be mitigated. Similarly, the defect's impact can be suppressed by biasing qubits away from the leading defect modes.
\begin{figure}
\caption{(a) Multiplexing control of qubit lattices of frequency-tunable superconducting qubits with shared XY and Z lines and local programmable memory. During the parallel gate operations, the local memory can be used to switch on or off the control on the individual qubit, and can provide the static bias for compensating qubit non-uniformity. (b) Network of word lines for digital addressing the local memory.}
\label{fig10}
\end{figure}
\subsection{Opportunities for solving challenges towards large-scale quantum processors}\label{SecIVB}
Currently, in small-scale superconducting quantum processors, each qubit has its dedicated control lines, such as XY lines and flux (Z) lines. Moreover, due to the non-uniformity of qubit parameters, such as qubit frequency and anharmonicity, the coupling efficiency between qubits and control lines, and the signal attenuation and the distortion in control lines, control pulses can differ from qubit to qubit. Thus, generally, microwave control pulses differ with each other in their amplitudes, frequencies, and phases, while for the baseband flux pulse, their amplitudes could be different. When scaling up to large-scale quantum computing, such strategy is not scalable.
Given the recent progress in the pursuit of scalable spin-based quantum computing with multiplexing technologies and crossbar technologies \cite{Hill2015,Vandersypen2017,Veldhorst2017,Li2018}, we may also consider how to utilize these technologies for solving the above-mentioned challenges toward large-scale superconducting quantum processors. One possible example is schematically illustrated in Fig.~\ref{fig10}(a), where both the XY and Z lines are shared by multiple qubits in a square lattice of frequency-tunable qubits. Similarly, in qubit architectures with tunable couplers, the Z-line shared scheme could be employed for controlling tunable couplers, thus enabling the implementation of baseband-controlled two-qubit gates in parallel.
Note that in principle, the shared control scheme should be applicable for driving all qubits with a single continuous microwave source. As a practical application, we expect that the shared scheme could be feasible for tens of superconducting qubits. With this strategy, the number of needed microwave drive lines is, at least, an order of magnitude less than that in the traditional setting. Additionally, to integrate with the widely used flip-chip technology, the shared drive may be applied to multiple qubits through a shared XY line. According to the discussion given in Ref.~\cite{Krinner2019}, we estimate that when the strength of the always-on microwave drive is about 10 MHz ($\sqrt{X}$ gates with a gate time of 30 ns), the required power is about -14 dBm at the room temperature, and through a series of attenuators and filters (giving rise to the total attenuation of 60 dB), the actual power delivered to the qubit chip is about -74 dBm. As in the traditional setting, the average required power per qubit drive line is about -78 dBm (to the qubit chip)~\cite{Krinner2019}, we expect that the heating issue of the present shared control scheme can be addressed.
Compared with the spin qubit, it seems that the superconducting qubit can provide more flexible control over its physical size and qubit parameters \cite{Martinis2020,Barends2014,Zhao2020N,Mamin2022,Zhao2022c,Chow2015}, yet, it can also show prominent non-uniformity. Unfortunately, the success of the multiplexing technologies and crossbar technologies highly hinges on the uniformity of qubit parameters. This can be more prominent for superconducting quantum processors based on individual microwave control. In the context of the implementation of multiplex control of superconducting qubits, baseband flux control may alleviate this issue of non-uniformity. Within our baseband control setup, and applying the multiplexing technologies shown in Fig.~\ref{fig10}(a), there are three main leading challenges from the qubit non-uniformity:
(i) The non-uniform amplitude of the shared microwave drive or (ii) flux pulse felt by qubits (caused by, such as the different coupling efficiency between qubits and the global XY/Z line and the signal attenuation in control lines);
(iii) Independently calibrated parameters of flux pulse, including pulse length and pulse shape, for implementing accurate control on individual qubits.
However, as discussed in Sec.\ref{SecIIA}, for two-level systems, given a fixed pulse length and pulse shape (i.e., square shape, see Appendix~\ref{A} for results with smooth pulses), owing to the flexibility of Z-controlled single-qubit gates (based on $\sqrt{X}$ gates), the available parameter ranges (i.e., the drive amplitude and the drive detuning) can be explored for compensating the non-uniform of drive amplitude. This exciting feature is illustrated in Fig.~\ref{fig2}(b). Furthermore, similar results can also be obtained for superconducting qubits, such as transmon qubits. Figures~\ref{fig11}(a) and~\ref{fig11}(b) present the results for transmon qubits with $50$-$\rm ns$ square pulses and smooth pulses (i.e., cosine-decorated square pulse with a hold time of $50\,\rm ns$ and a ramp time of $10\,\rm ns$), respectively. Here, the qubit anharmonicity is $-250\,\rm MHz$ and the other used parameters are same as in Fig.~\ref{fig2}. In addition, we note that to achieve uniform control pulses, the Z gate scheme based on time-delay and the proposed fast-adiabatic scheme cannot be employed. Here, we can instead use cosine-decorated square pulses for implementing Z gates by tuning the qubit from the idle point, and for suppressing leakage. From Figs.~\ref{fig11}(a) and~\ref{fig11}(b), one can find that by adding cosine-shaped ramps, the leakage error can be suppressed below $10^{-4}$, while for the square pulse the leakage can approach $10^{-3}$. To further suppress the leakage error, one can increase the ramp time or decrease the drive amplitude. However, this will increase gate length and thus cause more decoherence errors.
\begin{figure}
\caption{Flexibility of $\sqrt{X}$ rotations on superconducting transmon qubits.
(a) Population in states $|0\rangle$ (left panel) and $|2\rangle$ (right panel) versus the drive detuning and the drive amplitude with the qubit prepared in state $|1\rangle$. Here, the qubit anharmonicity is $-250\,\rm MHz$, the drive detuning at the idle point is $-100\,\rm MHz$, and the length of the square pulse is $50\,\rm ns$. The dashed lines indicate the available parameter sets for implementing $\sqrt{X}$ rotations. (b) same as in (a), instead showing the case with cosine-decorated square pulses. The ramp time and the hold time are $10\,\rm ns$ and $50\,\rm ns$, respectively.}
\label{fig11}
\end{figure}
Considering the above results, the above three challenges can be effectively overcome by only addressing the non-uniformity issue of flux pulse amplitudes. This non-uniformity issue could be removed by developing on-chip programmable memory, such as the one demonstrated by using Single Flux Quantum (SFQ) logic \cite{Johnson2010,McDermott2010}, which could be used to compensate for the remaining non-uniformity. In this way, combined with the word lines for digital addressing, as shown in Fig.~\ref{fig10}(b), it is possible using only a few global XY and Z lines to achieve parallel control of large numbers of qubits \cite{Hill2015,Vandersypen2017,Veldhorst2017,Li2018}. Nevertheless, given the practical experimental limitations, such as the limited cooling power, the realization of the local programmable memory, which is compatible with superconducting qubits, is still rarely explored \cite{Johnson2010}, and undoubtedly, will be one of the most crucial challenges for implementing multiplexing control technologies. Last, but not least, we must stress that before solving the issue of pulse distortion, the efficiency of the above-discussed scheme could be limited for implementing gate-based quantum computing.
\section{conclusion}\label{SecV}
In this work, we propose and theoretically study the possibility of implementing baseband control of superconducting qubits, which are subjected to an always-on global drive. Our results provide a general understanding and the basic principles of realizing the baseband control scheme for superconducting qubits, such as frequency-tunable transmon qubits. In the qubit architecture with tunable coupling, we show that high-fidelity and fast-speed gate operations are possible by employing this baseband control scheme. Additionally, we have further discussed potential challenges and opportunities for implementing such baseband control strategy toward large-scale superconducting quantum processors.
\begin{acknowledgments} We acknowledge helpful discussions with Zhaohua Yang, Yanwu Gu, and Zhi-Hai Liu. This work was supported by the National Natural Science Foundation of China (Grants No.12204050, No.11890704, and No.11905100), the Beijing Natural Science Foundation (Grant No.Z190012), and the Key-Area Research and Development Program of Guang Dong Province (Grant No. 2018B030326001). P.X. was supported by the National Natural Science Foundation of China (Grant Nos. 12105146, 12175104). \end{acknowledgments}
\emph{Note added.}-- During the preparation of this manuscript, we became aware of a recent related work \cite{Bejanin2022}, which presents the experimental demonstration of baseband-controlled single-qubit gates in superconducting transmon qubits.
\appendix
\section{Flexibility of $\sqrt{X}$ rotations}\label{A}
\begin{figure}
\caption{Same as in Fig.~\ref{fig2}, instead showing results with cosine-decorated square pulses. Here, the hold time is the pulse length of the flat part of the cosine-decorated square pulse, and the ramp time of the pulse is fixed at $10\,\rm ns$. }
\label{fig12}
\end{figure}
In Fig.~\ref{fig2}, we show the dynamics of Z-controlled two-level systems subjected to an always-on drive. Here, as shown in Fig.~\ref{fig12}, we further present the result for the case with cosine-decorated square pulses, i.e., \begin{align} \Delta(t)\equiv \begin{cases} \Delta_Z[1-\cos{(2\pi \frac{t}{t_r}})]/2 \;, &0<t<t_r/2\\ \Delta_Z\;, &t_r/2<t<t_g-t_r/2\\ \Delta_Z[1-\cos{(2\pi \frac{t_g-t}{t_r}})]/2 \;, &t_g-t_r/2<t<t_g \end{cases} \label{eqa1} \end{align}
where, $\Delta_Z$ denotes the peak pulse amplitude, $t_r$ is the ramp time, and $t_g$ represents the total pulse length. One can find that same as that with square pulses, to implement X rotations, a small overshoot is needed, as indicated by the red stars. Accordingly, as shown in Fig.~\ref{fig13}, we also give the results, including both the population in state $|0\rangle$ and the leakage into state $|2\rangle$, for the transmon qubit. Here the qubit anharmonicity is $-250\,\rm MHz$, and the transmon qubit is prepared in state $|1\rangle$.
\begin{figure}
\caption{Same as in Figs.~\ref{fig2}(a) and~\ref{fig12}(a), instead showing the results for the transmon qubit with the anharmonicity of $-250\,\rm MHz$. (a) Population in states $|0\rangle$ (left panel)
and $|2\rangle$ (right panel) versus the drive detuning and the pulse time with the qubit prepared in state $|1\rangle$. Here, the drive detuning at the idle point is $-100\,\rm MHz$, and the drive amplitude is $10\,\rm MHz$. The dashed lines indicate the available parameter sets for implementing $\sqrt{X}$ rotations. (b) same as in (a), instead showing the case with cosine-decorated square pulses. The ramp time is $10\,\rm ns$.}
\label{fig13}
\end{figure}
\section{fast-adiabatic pulse}\label{B}
As discussed in the main text, the fast-adiabatic scheme is employed for designing an optimal control pulse for suppressing leakage \cite{Martinis2014b}. Our strategy is using the Slepian-based method to design an optimal ramp pulse, and then inserting a square pulse into the optimal pulse. In this way, the target fast-adiabatic flat-top pulse is synthesized for implementing low-leakage X rotations. Therefore, here we only focus on finding the optimal ramp pulse. As shown in Fig.~\ref{fig3}(a), the leakage error results from the non-adiabatic evolution in the leakage subspace. Generally, during X rotations i.e., the drive detuning varies from the idle point with the control angle $\theta_{i}=\arctan{[\sqrt{2}\Omega_{d}/(\Delta_{d}+\alpha_{q})]}$ to the working point with the control angle $\theta_{f}=\arctan{[\sqrt{2}\Omega_{d}/\alpha_{q}]}$, and then comes back. Following Ref.~\cite{Martinis2014b}, the ramp pulse with a length of $t_{g}$ can be parameterized in terms of Fourier basis functions, and is given by
\begin{eqnarray} \begin{aligned}\label{eqb1} \theta(t)=\theta_{i}+ \frac{\theta_{f}-\theta_{i}}{2}\sum\limits_{n=1,2,3...}\lambda_{n}\left[1-\cos\frac{2n\pi t}{t_{g}}\right] \end{aligned} \end{eqnarray} with constraints on the coefficients $\Sigma_{n\,\rm{odd}}\,\,\lambda_{n}=1$. Here, to find the optimal pulse, we consider keeping three Fourier terms. Following Ref.~\cite{Martinis2014b}, the three optimized coefficients can be obtained numerically by minimizing the integrated spectral density of the pulse above a chosen frequency.
In the main text, the efficiency of the above-discussed optimal pulse is only evaluated with the drive amplitude of $10\,\rm MHz$. Here, we present more results with different drive strengths. In Figs.~\ref{fig14}(a) and~\ref{fig14}(b), we show the results with $\Omega_{d}/2\pi=15\,\rm MHz$ and $\Omega_{d}/2\pi=20\,\rm MHz$, respectively. Similar to the results shown in Fig.~\ref{fig4}, we find that fast-speed $\sqrt{X}$ rotations can be realized with low leakage.
\begin{figure}
\caption{leakage with varying driving strengths. Same as in Fig.~\ref{fig4}, here shows the qubit population and leakage versus the ramp time for the transmon qubit initialized in state $|1\rangle$ with the anharmonicity $\alpha_{q}/2\pi=-250\,\rm MHz$. (a) The drive amplitude is $\Omega_{d}/2\pi=15\,\rm MHz$ and the hold time is fixed at $10\,\rm ns$. (b) The drive amplitude is $\Omega_{d}/2\pi=20\,\rm MHz$ and the hold time is fixed at $5\,\rm ns$. The other system parameters are the same as in Fig.~\ref{fig4}.}
\label{fig14}
\end{figure}
\section{readout}\label{C}
Here for easy reference, following Ref.~\cite{Walter2017}, we briefly describe the processing of the recording obtained from continuous measurement to infer qubit states. As mentioned in the main text, by encoding the qubit information into single quadrature, i.e., in-phase quadrature $I$, the qubit state can be inferred by recoding $I_{t}=\langle a_{r}^{\dagger}+a_{r}\rangle(t)$ during the continuous measurement. To maximize the readout fidelity with a given measurement time (here is $t_i=250\,\rm ns$), the records of $I_{t}$ are integrated over the measurement time $t_{i}$ with a weight function, giving rise to the integrated readout quadrature value
\begin{equation} \begin{aligned}\label{eqc1} I=\sqrt{k}\int_{0}^{t_i}W_{t}[I_t-\langle I_{t}^{(0)}\rangle]dt \end{aligned} \end{equation} with the weight function \begin{equation} \begin{aligned}\label{eqc2}
W_{t}\varpropto|\langle I_{t}^{(1)}\rangle- \langle I_{t}^{(0)}\rangle|\, {\rm with\,} \int_{0}^{t_i}W_{t}^2 dt=1. \end{aligned} \end{equation}
Here, $\langle I_{t}^{(0)}\rangle$ and $\langle I_{t}^{(1)}\rangle$ represent the expectation values of $I_{t}$ for the qubit prepared in $|0\rangle$ and $|1\rangle$, respectively.
\section{RWA and corrections}\label{D}
\begin{figure}
\caption{Numerically calculated XY coupling strengths of (a) CZ coupling $J_{101-200}$ (the resonance coupling between $|101\rangle$ and $|200\rangle$) and (b) iSWAP coupling $J_{100-001}$ (the resonance coupling between $|100\rangle$ and $|001\rangle$). The blue and orange solid lines show the results without the RWA and the results with the RWA, respectively, while the green dashed lines denote results with the RWA and the corrections.}
\label{fig15}
\end{figure}
For two frequency-tunable transmon qubits ($Q_{0}$ and $Q_{1}$) coupled via a tunable coupler $Q_{c}$, the system Hamiltonian is given by \begin{equation} \begin{aligned}\label{eqd1} H_{0}=&\sum_{j=0,1,c}\big(\omega_{j}a_{j}^{\dagger}a_{j}+\frac{\alpha_{j}}{2}a_{j}^{\dagger}a_{j}^{\dagger}a_{j}a_{j}\big) \\&+\sum_{\substack{k=0,1,c\\j\neq k}}g_{jk}(a_{j}a_{k}^{\dagger}+a_{j}^{\dagger}a_{k}) \\&+\sum_{\substack{k=0,1,c\\j\neq k}}g_{jk}(a_{j}a_{k}+a_{j}^{\dagger}a_{k}^{\dagger}), \end{aligned} \end{equation} After applying the RWA, the non-RWA term, i.e., the terms in the third line of Eq.~(\ref{eqd1}), is omitted, giving rise to \begin{equation} \begin{aligned}\label{eqd2} H_{\rm RWA}=&\sum_{j=0,1,c}\big(\omega_{j}a_{j}^{\dagger}a_{j}+\frac{\alpha_{j}}{2}a_{j}^{\dagger}a_{j}^{\dagger}a_{j}a_{j}\big) \\&+\sum_{\substack{k=0,1,c\\j\neq k}}g_{jk}(a_{j}a_{k}^{\dagger}+a_{j}^{\dagger}a_{k}) \end{aligned} \end{equation}
As mentioned in the main text, the non-RWA term can significantly affect the effective coupling between qubits and shift the bare qubit frequency. This can be found in Fig.~\ref{fig15}, where the numerically calculated XY coupling strengths for iSWAP coupling $J_{100-001}$ (the resonance coupling between $|100\rangle$ and $|001\rangle$) and CZ coupling $J_{101-200}$ (the resonance coupling between $|101\rangle$ and $|200\rangle$) are presented based on Eq.~(\ref{eqd1}) and Eq.~(\ref{eqd2}).
To remove the discrepancy within the RWA formalism, we consider eliminating the non-RWA terms in Eq.~(\ref{eqd1}) by using the unitary transformation \cite{Zueco2009,Yan2018} \begin{equation} \begin{aligned}\label{eqd3} U=\exp{\left[-\sum_{j\neq k}\frac{g_{jk}}{\Sigma_{jk}}(a_{j}a_{k}-a_{j}^{\dagger}a_{k}^{\dagger})\right]} \end{aligned} \end{equation} with $\Sigma_{jk}=\omega_{j}+\omega_{k}$. This gives $H_{\rm RWA_C}=U^{\dag}H_{0}U$. Expanding the above equation and keeping term up to second-order in the small parameters $g_{ic}/\Sigma_{i}$, we have \begin{equation} \begin{aligned}\label{eqd4} H_{\rm RWA_C}\approx&\sum_{j=0,1,c}\big(\tilde{\omega}_{j}a_{j}^{\dagger}a_{j}+\frac{\alpha_{j}}{2}a_{j}^{\dagger}a_{j}^{\dagger}a_{j}a_{j}\big) \\&+\sum_{\substack{i=0,1}}g_{ic}(a_{i}a_{c}^{\dagger}+a_{i}^{\dagger}a_{c}) \\&+\tilde{g}_{01}(a_{0}a_{1}^{\dagger}+a_{0}^{\dagger}a_{1}) \end{aligned} \end{equation} with the renormalized qubit frequency and coupler frequency \begin{equation} \begin{aligned}\label{eqd5} &\tilde{\omega}_{i}=\omega_{i}-\frac{g_{01}^{2}}{\Sigma_{01}}-\frac{g_{ic}^{2}}{\Sigma_{ic}},\, {\rm with}\,i=0,1 \\&\tilde{\omega}_{c}=\omega_{c}-\frac{g_{0c}^{2}}{\Sigma_{0c}}-\frac{g_{1c}^{2}}{\Sigma_{1c}}, \end{aligned} \end{equation} and the effective strength of qubit-qubit coupling \begin{equation} \begin{aligned}\label{eqd6} \tilde{g}_{01}=g_{01}-\frac{g_{0c}g_{1c}}{2}\left(\frac{1}{\Sigma_{0c}}+\frac{1}{\Sigma_{1c}}\right). \end{aligned} \end{equation} Taking the above-obtained corrections into consideration, the results (dashed lines) with the RWA show a great agreement with the results without the RWA, as shown in Fig.~\ref{fig15}. Finally, we note that in our numerical analysis, each transmon qubit is modeled as an anharmonic oscillator truncated with five levels. Additionally, to further reduce the computational expenses, we further project the full system Hamiltonian to a smaller subspace where at most five excitations are permitted. We justify this choice by checking the resulting gate error compared to models with more energy levels or excitations and find that the variation of the gate error is below $10^{-6}$.
\section{CZ gate error}\label{E}
\begin{figure}
\caption{Gate error versus the gate length. The blue and orange lines denote the errors of CZ gates using the fast-adiabatic flat-top pulse and the cosine-decorated square pulse. As in Fig.~\ref{fig9}(a), the ramp time of the cosine-decorated square pulse is $10\,\rm ns$. The hold times of the fast-adiabatic flat-top pulses are $\{0,\,10,\,20,\,30,\,40\}\,{\rm ns}$.}
\label{fig16}
\end{figure}
As mentioned in the main text, generally, increasing the gate length can further suppress the parasitic interaction (i.e., qubit-qubit swap interactions and qubit-coupler swap interactions) induced gate errors. Here, we provide more numerical results on this issue. Figure~\ref{fig16} shows the gate error versus gate length for the discussed two types of pulses, i.e., the fast-adiabatic flat-top pulse and the cosine-decorated square pulse.
\end{document} | arXiv |
The Lewis structure for the chlorate ion is Calculate the formal charge on the chlorine (Cl) atom
In: Chemistry
A) The Lewis structure for the chlorate ion is
Calculate the formal charge on the chlorine (Cl) atom.
Calculate the formal charge on each of the oxygen (O) atoms labeled \(\mathrm{a}, \mathrm{b},\) and \(\mathrm{c}\) in the following Lewis structure.
Express your answers as integers separated by commas.
What are the formal charges on the sulfur (S), carbon (C), and nitrogen (N) atoms, respectively, in the resonance structure that contributes most to the stability of the thiocyanate ion, SCN\%u2212?
The possible resonance structures for the thiocyanate ion, SCN\%u2212, are
A Lewis structure is a diagram that represents the chemical bonding between atoms of molecules and a lone pair of electrons that may exist in the molecule. It is also called a dot structure representing a lone pair and bond pair of electrons in the molecule. Lewis structure is used to draw covalently bonded molecules as well as coordination compounds.
- Lewis's structure represents the bonding and lone pair of electrons in the molecule.
- Valence electrons are the electrons present in the outermost shell of an atom. Dots represent the electron position around the atoms, and lines or dot pairs are used to represent covalent bonds between atoms.
- The Lewis structure is based on the concept of the octet rule. So, the electrons shared in each atom should have 8 electrons in its outer shell.
- The formal charge (F.C) of the atom can be calculated by using the following formula.
$$ \left(\begin{array}{l} \text { Formal } \\ \text { charge (F.C) } \end{array}\right)=\left(\begin{array}{l} \text { no.ofvalence } \\ \text { electronsinatom } \end{array}\right)-\frac{1}{2}\left[\begin{array}{l} \text { no.ofbonding } \\ \text { electrons } \end{array}\right] -\left[\begin{array}{l}\text { no.ofnon }-\text { bonding } \\ \text { electrons }\end{array}\right] $$
The formal charge on the chlorine atom in the chlorate ion is zero.
Number of valance electron for Clis7
Number of bonding electrons for Clis10
Number of non - Bonding electrons for Clis2
Substituting these numbers in formal charge (FC)
\(\mathrm{FC}=7-\left(\frac{1}{2} \times 10\right)-2\)
\(=0\)
Therefore, the formal charge on the chlorine atom is zero in the chlorate ion.
Formal charge on oxygen \((\mathrm{a})=0\)
Formal charge on oxygen \((\mathrm{b})=0\)
Formal charge on oxygen \((\mathrm{c})=-1\)
Part B The charge on each oxygen atom is \(0,0,-1 .\)
Oxygen(a) :
Number of valance electron for \(\mathrm{O}(\mathrm{a})=6\)
Number of bonding electrons for \(\mathrm{O}(\mathrm{a})=4\)
Number of non \(-\) Bonding electrons for \(\mathrm{O}(\mathrm{a})=4\)
Substituting these numbers in formal charge :
\(\mathrm{FC}=6-\left(\frac{1}{2} \times 4\right)-4\)
Oxygen(b) :
Number of valance electron for \(\mathrm{O}(\mathrm{b})=6\)
Number of bonding electrons for \(\mathrm{O}(\mathrm{b})=4\)
Number of non \(-\) Bonding electrons for \(\mathrm{O}(\mathrm{b})=4\)
Oxygen(c)
Number of valance electron for \(\mathrm{O}(\mathrm{c})=6\)
Number of bonding electrons for \(\mathrm{O}(\mathrm{c})=2\)
Number of non \(-\) Bonding electrons for \(\mathrm{O}(\mathrm{c})=6\)
\(=-1\)
Therefore, the formal charge on each oxygen atom is \(0,0,-1 .\)
The most stable structure is structure \(B\) :
For the structure B:
Formal charge on sulfur(S) \(=0\)
Formal charge on carbon \((\mathrm{C})=0\)
Formal charge on nitrogen \((\mathrm{N})=-1\)
Part C The formal charge on sulfur, carbon and nitrogen is 0,0,-1 .
Sulfur(S) :
Number of valance electron for \(\mathrm{S}=6\)
Number of bonding electrons for \(\mathrm{S}=4\)
Number of non - Bonding electrons for \(\mathrm{S}=4\)
\(\operatorname{Carbon}(\mathrm{C})\)
Number of valance electron for \(\mathrm{C}=4\)
Number of bonding electrons for \(\mathrm{C}=8\)
Number of non \(-\) Bonding electrons for \(\mathrm{C}=0\)
Nitrogen \((\mathrm{N}):\)
Number of valance electron for \(\mathrm{N}=5\)
Number of bonding electrons for \(\mathrm{N}=4\)
Number of non \(-\) Bonding electrons for \(\mathrm{N}=4\)
Therefore, the formal charge on sulfur, carbon and nitrogen is 0,0,-1
CH3+ lewis structure
CH3NO2 Lewis structure
Draw a Lewis structure for PO4 3
Molecular geometry of IF2^-1 and lewis structure?
Based on formal charges, draw the most preferred Lewis structure for ClO3-?
Classify each of the following as a Lewis acid or a Lewis base.
the magnitude of the charge of the electron is
What magnitude and sign of charge Q will make the force on charge q zero?
Classify each of the following as a Lewis acid or a Lewis base. CO2 P(CH3)3 H2O | CommonCrawl |
\begin{document}
\title{Minimizers for the thin one-phase free boundary problem}
\begin{abstract} We consider the ``thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in $\mathbb R^{n+1}_+$ plus the area of the positivity set of that function in $\mathbb R^n$. We establish full regularity of the free boundary for dimensions $n \leq 2$, prove almost everywhere regularity of the free boundary in arbitrary dimension and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight.
While our results are typical for the calculus of variations, our approach does not follow the standard one first introduced in \cite{AltCaffarelli}. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments which are less reliant on the underlying PDE.
\end{abstract}
\setcounter{tocdepth}{2} \tableofcontents
\section{Introduction}\label{secIntro}
This article is devoted to the study of the regularity properties of a weighted version of the thin one-phase problem. More precisely we investigate even, nonnegative minimizers of the following functionals: denote $x\in{\mathbb R}^{n+1}$ by $x=(x',y)\in {\mathbb R}^n\times {\mathbb R}$, and for $\beta \in (-1,1)$ we define
\begin{equation}\label{eqLocalizedFunctional}
\mathcal{J}(v,\Omega):= \int_\Omega |y|^\beta |\nabla v|^2\ dx + m(\{v>0\}\cap{\mathbb R}^n\cap\Omega), \end{equation} where $m$ stands for the $n$-dimensional Lebesgue measure. Here, and throughout the paper, the integration is done with respect to the $(n+1)$-dimensional Lebesgue measure unless stated otherwise.
This functional is finite for open sets, $\Omega$, and functions in the weighted Hilbert space,
$$H^1(\beta,\Omega): = \{ v\in L^2(\Omega;|y|^\beta) : \nabla v\in L^2(\Omega;|y|^\beta)\},$$ equipped with the usual weighted norm.
Our main concern is to investigate fine regularity properties of the free boundary of minimizers $v$ of \eqref{eqLocalizedFunctional}, that is the set, $$ F(v):=\partial_{\mathbb R^n} \left \{ v(x,0)>0 \right \} \cap \Omega. $$ Since the free boundary lies on a codimension 1 subspace of the ambient space ${\mathbb R}^{n+1}$, such a problem is called a {\sl thin one-phase free boundary problem}. This type of free boundary problem has been investigated for the first time by Caffarelli, Roquejoffre and the last author in \cite{CaffarelliRoquejoffreSire} in relation with the theory of semi-permeable membranes (see, e.g., \cite{duvautLions}). As we will describe later this is an analogue of the classical one-phase problem (also called the Bernoulli problem) but for the fractional Laplacian.
The Bernoulli problem was first treated in a rigorous mathematical way by Alt and Caffarelli in the seminal paper \cite{AltCaffarelli}: in the Bernoulli problem we consider minimizers of \eqref{eqLocalizedFunctional} where $\beta = 0$ and the second term is replaced by $\mathcal L^{n+1}(\{v>0\}\cap\Omega)$ (where $\mathcal L^{n+1}$ stands for the Lebesgue measure in ${\mathbb R}^{n+1}$). In particular, for the Bernoulli problem, the free boundary fully sits in the ambient space, ${\mathbb R}^{n+1}$. In \cite{AltCaffarelli}, the authors provided a general strategy to attack this type of problem. Out of necessity we needed to modify this blueprint in several substantial ways (see below for a more detailed comparison). For more information on the one-phase problem (and some of its variants) we refer to the book of Caffarelli and Salsa (and references therein) \cite{CaffarelliSalsa}, and to the more recent survey of De Silva, Ferrari and Salsa \cite{DeSilvaFerrariSalsaSurvey}.
As noticed in \cite{CaffarelliRoquejoffreSire}, problem \eqref{eqLocalizedFunctional} is related in a tight way to the {\sl standard } one-phase free boundary problem but with the Dirichlet energy replaced by the Gagliardo semi-norm $[u]_{\dot H^\alpha}$, for $\alpha=\frac{1-\beta}{2} \in (0,1)$. This connection suggests that the thin one-phase problem is actually intrinsically a {\sl nonlocal } problem, though the energy in \eqref{eqLocalizedFunctional} is clearly local. \subsection*{Connection with the fractional one-phase problem} As previously mentioned, the functional $\mathcal{J}$ introduced by Caffarelli, Roquejoffre and the last author in \cite{CaffarelliRoquejoffreSire} is a local version of the following nonlocal free boundary problem: given a function $f\in L^1_{\loc}({\mathbb R}^n)$ with suitable decay at infinity, we can define its fractional Laplacian at $x\in{\mathbb R}^n$ by
$$(-\Delta)^\alpha f(x) = c_{n,\alpha}\, p.v. \int_{{\mathbb R}^n} \frac{f(x)-f(\xi)}{|x-\xi|^{n+2\alpha}}\, d\xi.$$ At the formal level, we are interested in solutions of the free boundary problem \begin{equation}\label{eqEulerLagrangeFrac} \begin{cases} (-\Delta)^\alpha f = 0 & \mbox{in } \Omega \cap \{f>0\},\\ \partial_\nu^\alpha f = A & \mbox{on } \Omega\cap F(f), \end{cases} \end{equation} where $\partial_\nu^\alpha f (x):=\lim_{ \Omega \cap \{f>0\} \ni \xi\to x}\frac{f(\xi)-f(x)}{((\xi-x)\cdot \nu(x))^\alpha}$ and where $f$ satisfies a given ``Dirichlet boundary condition" on the complement of $\Omega$.
As in the case of the classical Laplacian (see \cite{AltCaffarelli}), we are interested in obtaining equation \rf{eqEulerLagrangeFrac} as the Euler-Lagrange equation of a certain functional. Given a locally integrable function $f$, consider its fractional Sobolev energy
$$\left[f\right]_{\dot H^\alpha ({\mathbb R}^n)}:=\iint_{{\mathbb R}^{2n}} \frac{|f(x)-f(\xi)|^2}{|x-\xi|^{n+2\alpha}}\, d\xi\, dx.$$ Since we want to study competitors which vary only in a certain domain $\Omega$, it is natural to consider only the integration region which may suffer variations when changing candidates. Thus, we define the energy \begin{equation}\label{eqNonLocalFunctional}
J(f,\Omega):= c_{n,\alpha} \iint_{{\mathbb R}^{2n}\setminus (\Omega^c)^2} \frac{|f(x)-f(\xi)|^2}{|x-\xi|^{n+2\alpha}}\, d\xi\, dx + m(\{f>0\}\cap\Omega). \end{equation} We say that $f\in L^1_{\loc}$ is a minimizer of $J$ in $\Omega$ if $J(f,\Omega)$ is finite and $ J(f,\Omega)\leq J(g,\Omega)$ for every $g$ satisfying that $f-g\in \dot H^\alpha({\mathbb R}^n)$ and such that $f(x)=g(x)$ for almost every $x\in \Omega^c$. We say that $f$ is a global minimizer if it is a minimizer for every open set $\Omega\subset {\mathbb R}^n$. Note that both terms in \rf{eqNonLocalFunctional} are in competition, since a minimizer of the fractional Sobolev energy in $\Omega$ is $\alpha$-harmonic and, thus, if it is non-negative outside of $\Omega$ it is strictly positive inside of $\Omega$, maximizing the second term.
Consider now the Poisson kernel for fixed $n\in{\mathbb N}$ and $0<\alpha<1$ \begin{equation}\label{eqPoisson}
P_y(\xi):=P_{n,\alpha}(\xi,y)=c_{n,\alpha} \frac{|y|^{2\alpha}}{|(\xi,y)|^{n+2\alpha}} \quad\quad \mbox{ for every }(\xi,y)\in {\mathbb R}^n\times{\mathbb R}. \end{equation} The Poisson extension of $f\in L^1_{\loc}({\mathbb R}^n)$ is given by \begin{equation}\label{eqPoissonExtension} u(x',y):= f * P_y(x')=\int_{{\mathbb R}^n} P_{n,\alpha}(\xi,y) f(x'-\xi)\, d\xi \quad\quad \mbox{ for every }(x',y)\in {\mathbb R}^n\times{\mathbb R}. \end{equation} By \cite{CaffarelliSilvestre}, with a convenient choice of the constant one gets
$$\lim_{y \searrow 0}y^{1-2\alpha} u_y(x',y)=-(-\Delta)^\alpha f(x')$$ in every point where $f$ is regular enough. Moreover, the extension satisfies the localized equation $\nabla\cdot (|y|^\beta \nabla u)=0$ weakly, away from ${\mathbb R}^n\times \{0\}$. The whole point is that local minimizers of \eqref{eqNonLocalFunctional} can be extended via the previous Poisson kernel $P_y$ to (even) minimizers of \eqref{eqLocalizedFunctional} (see the Appendix for a precise statement). Therefore, the thin one-phase problem appears as a ``localization" of the one-phase problem for the fractional Laplacian. Notice that, and this is of major importance for us, this localization technique does not carry over to other types of nonlocal operators besides pure powers of second-order elliptic operators. This is a major drawback of the theory, in the sense that, at the moment, it seems to be impossible to tackle one-phase problems involving more general operators than the fractional Laplacian. The main point is we do not know how to prove any kind of monotonicity for general integral operators.
This connection between the nonlocal analogue of the Bernoulli problem and our thin one-phase problem allows us to simplify several arguments by working in the purely nonlocal setting. However, this underlying nonlocality is also the reason why several results, which came more easily in the setting of \cite{AltCaffarelli}, are non-trivial or substantially harder for us. For example, perturbations of solutions need to take into account long range effects which makes classical, local, perturbation arguments much more difficult.
In the paper \cite{CaffarelliRoquejoffreSire}, the authors proved basic properties of the minimizers for the functional $\mathcal J$ such as optimal regularity, non-degeneracy near the free boundary, and positive densities of phases. Also they provided an argument for $n=2$ showing that Lipschitz free boundaries are $C^1$. A feature of the functional $\mathcal J$ is that the weight $|y|^\beta$ is either degenerate or singular at $\left \{ y=0 \right \}$ (except in the case $\beta=0$). Such weights belong to the Muckenhoupt class $A_2$ and the seminal paper of Fabes, Kenig and Serapioni \cite{FabesKenigSerapioni} investigated regularity issues for elliptic PDEs involving such weights (among other things). After that, \cite{deSilvaSavinSire} proved an $\varepsilon$-regularity result and \cite{Allen} showed the existence of a monotonicity formula for this setting.
In the case $\beta=0$, the problem is still degenerate in the sense that derivatives near the free boundary blow up. The case $\beta=0$ has been thoroughly investigated in the series of papers by De Silva, Savin and Roquejoffre \cite{desilvaRoque,DSSJDE,deSilvaSavinJEMS}.
The main goal of our paper is to provide a full picture of the regularity of the free boundary for any power $\beta \in (-1,1)$, both in terms of measure-theoretic statements and partial (or full) regularity results. From this point of view our contribution is a complement of the paper by De Silva and Savin \cite{deSilvaSavinJEMS} for $\beta=0$. It has to be noticed that the standard approach to regularity of Lipschitz free boundaries as developed by Caffarelli (see the monograph \cite{CaffarelliSalsa}) does not seem to work in our setting.
\subsubsection*{Our approach to regularity}
In \cite{AltCaffarelli} (and many subsequent works), the minimizing property of the solution is used to prove that the distributional Laplacian of that solution is an Ahlfors-regular measure supported on the free boundary. This implies (amongst other things) that the free boundary is a set of (locally) finite perimeter, and thus almost every point on the free boundary has a measure theoretic tangent. One can then work purely with the weak formula (i.e. the analogue of \eqref{eqEulerLagrangeFrac}) to prove a ``flat implies smooth" result which, together with the existence almost everywhere of a measure theoretic tangent, has as a consequence that the free boundary is almost everywhere a smooth graph and the free boundary condition in \eqref{eqEulerLagrangeFrac} holds in a classical sense at the smooth points.
A similar ``flat implies smooth" result exists in our context (this is essentially due to De Silva, Savin and the last author, \cite{deSilvaSavinSire}, see Theorem \ref{theoImprovement} below). However, showing that the free boundary is the boundary of a set of finite perimeter proves to be much more difficult. Due to the nonlocal nature of the problem, $-\mathrm{div}(|y|^\beta \nabla u)$ (considered as a distribution) is not supported on the free boundary. Furthermore, the scaling of this measure does not allow us to conclude that the free boundary has the correct dimension (much less that it is Ahlfors regular).
To prove finite perimeter, we take the following approach inspired by the work of de Silva and Savin: after establishing some preliminaries
we prove crucial compactness results. This, along with a monotonicity formula originally due to Allen \cite{Allen} allows us to run a dimension reduction argument in the vein of Federer or (in the context of free boundary problems) Weiss \cite{WeissMinimum}. With this tool in hand, we show that the set of points at which no blow-up is flat is a set of lower dimension. Locally finite perimeter and regularity for the reduced boundary then follow from a covering argument and some standard techniques.
Here and throughout the paper, we will denote the ball of radius $r$ in ${\mathbb R}^{n+1}$ centered at the origin by $B_r$, and $B_r':=B_r\cap {\mathbb R}^n\times\{0\}$. Moreover, for the definition of ${\mathbf H}^\beta$, see Section \ref{s:preliminaries}. We may then summarize our regularity results in the following theorem.
\begin{theorem}\label{t:mainregularity}[Main Regularity Theorem] Let $u\in {\mathbf H}^\beta(B_1)$ be a (non-negative, even) local minimizer of $\mathcal{J}$ in $B_1 \subset \mathbb R^{n+1}$. Let $B_{1,+}'(u):=\{x=(x',0)\in B_1: u(x)>0\}$, let $F(u)$ be the boundary of $B_{1,+}'(u)$ inside of ${\mathbb R}^n\times \{0\}$ and assume that $0\in F(u)$. Then, \begin{enumerate} \item $B_{1,+}'(u)$ (as a subset of ${\mathbb R}^n\times \{0\}$) is a set of locally finite perimeter in $B_1'$. \item We can write the free boundary as a disjoint union $F(u) = \mathcal R(u) \cup \Sigma(u)$, where $\mathcal R(u)$ is open inside $F(u)$, and for $x \in \mathcal R(u)$ there exists an $r_x > 0$ such that $B(x, r_x) \cap F(u)$ can be written as the graph of a $C^{1,s}$-continuous function. \item Furthermore, the set $\Sigma(u)$ is of Hausdorff dimension $\leq n-3$ (and, therefore, of $\mathcal H^{n-1}$-measure zero). In particular, for $n\leq 2$, $\Sigma(u)$ is empty, and moreover, if $n=3$ then $\Sigma(u)$ is discrete.
\end{enumerate} The constants (implicit in the set of finite perimeter, and the H\"older continuity of the functions whose graph gives the free boundary) depend on $n$ and $\beta$ but not on $\norm{u}_{{\mathbf H}^\beta(B_1)}$. \end{theorem}
As usual $\Sigma(u) \subset F(u)$ is called the \emph{singular set} of the free boundary: the set of points around which $F(u)$ cannot be parameterized as a smooth graph and all the blow-ups will be non-trivial minimal cones, see Theorem \ref{theoImprovement}.
Our second contribution concerns the structure and size of the singular set. It builds on recent major works on quantitative stratification \cite{NaberValtortaRectifiable}, extended to free boundary problems (in particular the one-phase problem) by Edelen and the first author \cite{EdelenEngelstein}.
\begin{theorem}\label{t:singset} Let $u\in {\mathbf H}^\beta(B_1)$ be a (non-negative, even) local minimizer of $\mathcal{J}$ in $B_1$ and $0\in F(u)$. Let $B_{1,+}'(u):=\{x=(x',0)\in B_1: u(x)>0\}$ and $F(u)$ be the boundary of $B_{1,+}'(u)$ inside $B_1'$. Then, there exists a $ k_\alpha^*\geq 3$ such that $\Sigma(u)$ is $(n-k_\alpha^*)$-rectifiable and $$\mathcal{H}^{n-k^*_\alpha}(\Sigma(u)\cap D)\leq C_{n,\alpha, {\rm dist}(D,\partial B_1)} \quad\quad \mbox{for every }D\subset\subset B_1.$$
\end{theorem}
In \cite{DSJ}, De Silva and Jerison constructed a singular minimizer for the Alt-Caffarelli one-phase problem in dimension $7$, giving the dimension bound $k^* \leq 8$ in the previous theorem in this case (see \cite{EdelenEngelstein}). This result is not known for the thin one-phase problem. The reason is that the one-phase problem, seen from the nonlocal point of view involving the fractional Laplacian, is related to the so-called nonlocal minimal surfaces introduced by Caffarelli, Roquejoffre and Savin \cite{CaffarelliRoquejoffreSavin}. Indeed, in \cite{savinValdinoci}, the authors proved that a fractional version of Allen-Cahn equation converges variationally to the standard perimeter functional for $\alpha \geq 1/2$ and to the so-called nonlocal minimal surfaces for $\alpha <1/2$. We can then conjecture the bound $k^*_\alpha \leq 8$ for $\alpha \geq 1/2$ by analogy with the result for the standard one-phase problem but the bound for $\alpha <1/2$ is not clear at all. However, one knows that there is no singular cone in dimension $2$ for nonlocal minimal surfaces \cite{SVcone} and that the Bernstein problem is known for those in dimensions 2 and 3 \cite{figalliValdinoci}.
We would like also to make a last remark about a result which is of purely nonlocal nature. In the case of the one-phase problem, one can show that the distributional Laplacian is a Radon measure along the free boundary. In the case of the thin one-phase free boundary problem, due to the nonlocality of the problem, such a behavior does not happen in the sense that we will show that the fractional Laplacian is an absolutely continuous measure with respect to $n$-dimensional Lebesgue measure with a precise behavior. This phenomenon is of purely nonlocal nature and similar to the fact that the fractional harmonic measure is of trivial nature. More precisely, every minimizer $u$ satisfies $\nabla\cdot (|y|^\beta \nabla u)=0$ weakly, away from ${\mathbb R}^n\cap \{u \leq 0\}$. Thus, equation \rf{eqEulerLagrangeFrac} above can be understood as an Euler-Lagrange equation for the functional $\mathcal{J}$ in the sense that the restriction to ${\mathbb R}^n$ of a given minimizer $u$ in $\Omega\subset {\mathbb R}^{n+1}$ and with asymptotic behavior $u(x,y)=\mathcal{O}(|(x,y)|^\alpha)$ is always a solution to \rf{eqEulerLagrangeFrac} for $A=A(\alpha)$ at ``nice" points of the free boundary.
A brief summary of this paper follows. In Sections \ref{s:Compactness} and \ref{s:monotonicity} we discuss compactness of minimizers and we recall Allen's monotonicity formula to derive some immediate consequences. In Section \ref{s:finperimeter} we show that the positive phase is a set of locally finite perimeter, establishing the first part of Theorem \ref{t:mainregularity} (modulo energy bounds), and we show that the singular set can be identified using the Allen-Weiss density. Section \ref{s:r2} is devoted to deducing full regularity of minimizers in ${\mathbb R}^{2+1}$ concluding the proof of Theorem \ref{t:mainregularity}.
Once we have established the finite perimeter, in Section \ref{s:UniformBounds} we remove the dependence of the estimates on the energy of the minimizer in the previous theorems, using a rather subtle argument which combines results from all the previous sections. A crucial step is to analyze some basic properties of the distributional fractional Laplacian of our minimizer. As stated above this analysis will not be enough to establish that the positivity set of the minimizer is a set of locally finite perimeter. We believe that many of these results may be of independent interest. For example, corresponding results for the classical Bernoulli problem have been used to understand free boundary problems for harmonic measure (see \cite{KenigToroDCDS}).
Finally, Section \ref{s:Rect} is devoted to the proof of Theorem \ref{t:singset}.
\subsubsection*{Notation}
We denote the constants that depend on the dimension $n$, $\alpha$ and perhaps some other fixed parameters which are clear from the context by $C$. Their value may change from an occurrence to another. On the other hand, constants with subscripts as $C_0$ retain their values along the text. For $a,b\geq 0$, we write $a\lesssim b$ if there is $C>0$ such that $a\leq Cb$. We write $a\approx b$ to mean $a\lesssim b\lesssim a$.
Let $u$ be a continuous function in ${\mathbb R}^{n+1}$. Then we write $\Omega_+(u):=\Omega\cap \{u>0\}$, and we denote the zero phase, the positive phase and the free boundary by \begin{align*} &\Omega_0(u):=\{x\in {\mathbb R}^n\times\{0\}: u(x)=0\}^\circ{},\\ &\Omega_+'(u):=\Omega_+\cap ({\mathbb R}^n\times\{0\})=\{x\in {\mathbb R}^n\times\{0\}: u(x)>0\},\mbox{ and}\\ &F(u):=F_\Omega(u)=\partial (\Omega_+(u)\cap {\mathbb R}^n\times\{0\}) \cap \Omega, \end{align*} respectively. Here both the boundary and the interior are taken with respect to the standard topology in ${\mathbb R}^n$. Note that ${\mathbb R}^n\times \{0\}$ is the disjoint union of $\Omega_0(u)$, $\Omega_+'(u)$ and $F(u)$ whenever $u$ is non-negative. We also call $F_{\rm red}(u)=F_{{\rm red},\Omega}(u)$ the points of $F_\Omega(u)$ where the free boundary is expressed locally as a $C^1$ surface. Finally, let $\Sigma(u)=\Sigma_\Omega(u)=F_\Omega(u)\setminus F_{{\rm red},\Omega}(u)$. In general we will write $\Omega':=\Omega\cap({\mathbb R}^n\times\{0\})$.
Throughout the paper we will often fix $\beta \in (-1, 1)$ but then refer to $\alpha\in (0,1)$ or vice versa. These two numbers are always connected by the relationship $\alpha = \frac{1-\beta}{2}$.
\section{Preliminaries}\label{s:preliminaries}
In this section, we provide the known results concerning the problem under consideration. We say that a function $u$ is \emph{even} if it is symmetric with respect to the hyperplane ${\mathbb R}^n\times \{0\}$, that is, $u(x',y)=u(x',-y)$. The function spaces that we will consider are the following $$ {\mathbf H}^\beta (\Omega):= \{u \in H^1(\beta,\Omega) : u\mbox{ is even and non-negative}\}$$ and $$ {\mathbf H}^\beta_{\loc}(\Omega):= \{u\in L^2_{\loc}(\Omega): u \in {\mathbf H}^\beta(B) \mbox{ for every ball } B\subset \subset \Omega\}.$$ We will omit $\Omega$ in the notation when it is clear from the context. \begin{definition}\label{defMinimizer}
We say that a function $u\in {\mathbf H}^\beta_{\loc}(\Omega)$ is a (local) minimizer of $\mathcal{J}$ in a domain $\Omega$ if for every ball $B\subset\subset \Omega$ and for every function $v\in {\mathbf H}^\beta(B)$ such that the traces $v|_{\partial B} \equiv u|_{\partial B}$, the inequality $$\mathcal{J}(u,B)\leq \mathcal{J}(v,B)$$ holds.
\end{definition}
As usual for several free boundary problems, it is a natural question to exhibit a particular (global) solution so that one gets an idea of the qualitative properties of general solutions. Let us consider the following function: for every $x\in {\mathbb R}^n$ let $$f_{n,\alpha}(x):= c_{n,\alpha} (x_n)_+^\alpha ,$$ where $a_+=\max\{0,a\}$. If $n=1$, $f_{1,\alpha}$ is a solution to \rf{eqEulerLagrangeFrac} for a convenient choice of $c_{1,\alpha}$ (see \cite[Theorem 3.1.4]{BucurValdinoci}). In fact one can see that the same is true for $n\geq 1$ using Fubini's Theorem conveniently, with \begin{equation}\label{eqLaplacianTrivial} -(-\Delta)^\alpha f_{n,\alpha}(x) = c_{n,\alpha} (x_n)_-^{-\alpha}, \end{equation} where $a_-=\max\{0,-a\}$.
As a toy question we wonder whether the trivial solutions are minimizers. Indeed, this is the case, as we will see later in Section \ref{secReduction}. \begin{proposition}\label{propoTrivial} Let $n\in{\mathbb N}$ and $0<\alpha<1$. Then the trivial solution $u_{n,\alpha}:=f_{n,\alpha}*P_y$ is a minimizer of $\mathcal{J}$ in every ball $B\subset {\mathbb R}^{n+1}$. \end{proposition}
Next we collect the main properties of minimizers in the unit ball proven in \cite[Theorems 1.1-1.4, Proposition 3.3 and Corollary 3.4]{CaffarelliRoquejoffreSire}. \begin{theorem}\label{theoCRS}
If $u\in {\mathbf H}^\beta(B_1)$ is a minimizer of $\mathcal{J}$ in $B_1$ with $\norm{u}_{\dot {\mathbf H}^\beta(B_1)}:=\norm{\nabla u}_{L^2(B_1,|y|^\beta)} \leq E_0$ and $x_0\in F(u)\cap B_{\frac12}$, then it satisfies \begin{enumerate}[P1:] \item Optimal regularity (see \cite[Theorem 1.1]{CaffarelliRoquejoffreSire}): $\norm{u}_{\dot C^\alpha(B_{1/2})}\leq C{(1+E_0)}$. \item Nondegeneracy (see \cite[Theorem 1.2]{CaffarelliRoquejoffreSire}): $u(x)\geq C {\rm dist}(x,F(u))^\alpha $ for $x\in B_{\frac12}'$. \item Interior corkscrew condition (see \cite[Proposition 3.3]{CaffarelliRoquejoffreSire}): there exists $x_+\in B_r'(x_0)$ so that $B'(x_+, C_0r)\subset \Omega_+'(u)$.
\item Positive density (see \cite[Theorem 1.3]{CaffarelliRoquejoffreSire}): $|\Omega_0\cap B_r'(x_0)| \gtrsim r^n$. \label{itemPositive} \item Blow-ups are minimizers (see \cite[Corollary 3.4]{CaffarelliRoquejoffreSire}): The limit of a blow-up sequence $u_k(x):=\frac{u(x_0+\rho_k x)}{\rho_k^\alpha}$ converging weakly in $H^1(\beta,B_1)$ and uniformly is a global minimizer.\label{itemBlowup} \item Normal behavior at the free boundary (see \cite[Theorem 1.4]{CaffarelliRoquejoffreSire}): the boundary condition in \rf{eqEulerLagrangeFrac} is satisfied at every point on the free boundary with a measure theoretic normal (see \cite{EvansGariepy}) for a prescribed value of $A$.\label{itemNormal} \end{enumerate} All the constants depend on $n$ and $\alpha$; and also on $E_0$ except for the one in \emph{P1}. \end{theorem}
A major tool in the present paper is an $\epsilon-$regularity result, i.e. in the language of free boundaries a statement of the type ``flatness implies smoothness". In \cite{deSilvaSavinSire}, the authors proved such an $\epsilon$-regularity result for viscosity solutions to the overdetermined system associated to minimizers of $\mathcal J$. Here we establish that all local minimizers are in fact viscosity solutions. While this verification may be standard for experts in the field, we include it here for the sake of completeness.
\begin{theorem}[$\epsilon$-regularity]\label{theoImprovement} \label{th:iof} There exists $\epsilon>0$ depending only on $n$, $\alpha$ and $E_0$ such that for every non-negative, even minimizer $u$ of the energy \rf{eqLocalizedFunctional} on a ball $B\subset {\mathbb R}^{n+1}$ with $\norm{u}_{{\mathbf H}^\beta(B)}\leq E_0 r(B)^{\frac n2}$ and \begin{equation}\label{eqImprovementOfFlatness} \{(x,0)\in B: x_n\leq -\epsilon\}\subset B_0(u) \subset \{(x,0)\in B: x_n\leq \epsilon\}, \end{equation} we have that $F(u)\in C^{1,\gamma}_{\rm loc}(\frac12 B)$, with $0<\gamma<1$. \end{theorem} Note that the dependence on $E_0$ will be removed in Section \ref{s:UniformBounds}. \begin{proof}
We say that $u$ is a \emph{viscosity solution} to \begin{equation}\label{eqViscositySol} \begin{cases}
\nabla\cdot (|y|^\beta \nabla u) = 0 & \mbox{in }B_1^+(u),\\ \lim_{t\to 0+} \frac{u(x_0+t\nu (x_0),0)}{t^\alpha} = 1, & \mbox{for } (x_0,0)\in F(u), \end{cases} \end{equation} if \begin{enumerate}[i)] \item $u\in C(B_1)$, $u\geq 0$,
\item $u\in C^{1,1}_{\rm loc}(B_{1,+}(u))$, $u$ is even and it solves $\nabla\cdot (|y|^\beta \nabla u) = 0$ in the viscosity sense, and \item any strict \emph{comparison subsolution} (resp. supersolution) cannot touch from below (resp. from above) at a point $(x_0,0)\in F(u)$. \end{enumerate}
We claim that \begin{equation}\label{eqMinimizersViscosity} \mbox{every non-negative even minimizer is a viscosity solution.} \end{equation}
Conditions (i) and (ii) have been verified in \cite{deSilvaSavinSire,VitaThesis}. To verify our claim it suffices to prove condition (iii) above: that any strict comparison subsolution cannot touch $u$ from below at a point $(x_0, 0) \in F(u)$. The analogous claim for strict comparison supersolutions will follow in the same way.
Let us recall (see, e.g. Definition 2.2 in \cite{deSilvaSavinSire}), that $w\in C(B_1)$ is a strict comparison subsolution (resp. supersolution) to \eqref{eqViscositySol} if \begin{enumerate}[a)] \item $w \geq 0$, \item $w$ is even with respect to $\{y= 0\}$, \item $w\in C^2(\{w > 0\})$,
\item $\mathrm{div}\left(|z|^\beta \nabla w\right) \geq 0$ in $B_1 \backslash \{y = 0\}$,
\item $F(w)$ is locally given by the graph of a $C^2$ function and for any $x_0 \in F(w)$ we may write \begin{equation}\label{e:exp} w(x,y) = aU((x-x_0)\cdot \nu(x_0), y) + o(\|(x-x_0, y)\|^\alpha),\qquad (x,y) \rightarrow (x_0,0).\end{equation} Here $U$ is the extension of the trivial solution (see \cite{deSilvaSavinSire}), $\nu(x_0)$ is the unit normal to $F(w)$ considered as a subset of ${\mathbb R}^n$ pointing into $\{w > 0\}$ and $a \geq 1$. \item Furthermore, either the inequality is strict in d), or $a >1$ in e). \end{enumerate}
So assume that $w \geq u$ where $w$ is a strict comparison subsolution and $u$ is some minimizer and that $w = u$ at $(x_0,0) \in F(u)$. Since $u(x_0, 0) = 0$ it follows that $(x_0, 0) \in F(w)$ and with a harmless rotation we can guarantee that $\nu((x_0,0)) = e_n$. We want to show that $e_n$ is also the measure theoretic unit normal to $F(u)$. Indeed, since $F(w)$ is $C^2$ there must exist a ball $B \subset \{w > 0\}$ which is tangent to $F(w)$ at $(x_0,0)$. It must then be that case that $B \subset \{u > 0\}$ as well. Thus $(x_0,0) \in F(u)$ has a tangent ball from the inside which, by \cite{CaffarelliRoquejoffreSire} Proposition 4.5 implies that $u$ has the asymptotic expansion $$u(x,y) = U((x-x_0)\cdot \nu(x_0), y) + o(\|(x-x_0, y)\|^\alpha),\qquad (x,y) \rightarrow (x_0,0).$$
If $u \geq w$ this implies that $w$ must satisfy the expansion in \eqref{e:exp} with $a = 1$ at the point $x_0$. This, in turn, implies that $\mathrm{div}\left(|z|^\beta \nabla w\right) > 0$ in $B_1 \backslash \{y = 0\}$ (by the definition of a strict subsolution). Furthermore, since $w\in C^2$ where $\{w > 0\}$ we can guarantee that $\mathrm{div}\left(|z|^\beta \nabla w\right) \geq 0$ in all of $B_1 \cap \{w > 0\}$.
Let us return to the ball $B$ which is a subset of $\{u > 0\}$ and $\{w > 0\}$ and for which $(x_0, 0) \in \overline{B}$. We know that $w-u \neq 0$ in $B\setminus\{y=0\}$ (this is because $w$ strictly satisfies the differential inequality in $B$ away from $\{y = 0\}$) and we know that $w-u$ is a subsolution in $B$. Furthermore $(x_0, 0) \in B$ is a strict maximum, so by the Hopf lemma in \cite[Proposition 4.11]{cabreSire} it must be that $$\lim_{t\downarrow 0^+}\frac{({w-u})(x_0+t\nu (x_0),0)}{t^\alpha} > 0.$$ This contradicts the fact $u$ and $w$ both satisfy \eqref{e:exp} at $(x_0,0)$ with $a = 1$. Therefore, $(x_0,0)$ must not have been a touching point and $u$ is indeed a viscosity solution.
Since, $u$ is a viscosity solution, \cite[Theorem 1.1]{deSilvaSavinSire} applies and we get the desired $\varepsilon$-regularity. \end{proof}
\section{Compactness of minimizers}\label{s:Compactness}
In this section we prove important results on the compactness of minimizers. As we mentioned above, our contribution is that convergent sequences of minimizers also converge in the relevant weighted Sobolev spaces strongly rather than just weakly. This will prove essential to the compactness arguments used in the later sections of this paper.
\subsection{Caccioppoli Inequality}\label{secCaccioppoli}
First we want to show that the distribution $\lambda:=\nabla\cdot(|y|^\beta \nabla u)$ is in fact a Radon measure with support in the complement of the positive phase as long as $u$ is a minimizer. In Section \ref{s:UniformBounds} we will come back to this measure to understand its behavior around the free boundary.
\begin{lemma}\label{lemRadonMeasure}
Let $\Omega\subset {\mathbb R}^{n+1}$ be an open set, and let $u\in W^{1,2}_{\loc}(\Omega, |y|^\beta)$ be such that $\nabla\cdot (|y|^\beta \nabla u)=0$ weakly in $\Omega_+(u)$, i.e., for every $\eta\in C^\infty_c(\Omega_+(u))$, \begin{equation}\label{eqBetaHarmonic}
\langle \nabla\cdot(|y|^\beta \nabla u), \eta\rangle := - \int (|y|^\beta \nabla u) \nabla \eta=0. \end{equation}
Then $\lambda:=\nabla\cdot(|y|^\beta \nabla u)$ is a positive Radon measure supported on $\{u=0\}$ and for every $v\in W^{1,2}(\Omega, |y|^\beta) \cap C_{c}(\Omega)$ \begin{equation}\label{eqMeasureDef}
\int v \, d\lambda = -\int |y|^\beta \nabla u\cdot \nabla v. \end{equation} \end{lemma} \begin{proof} Indeed, by \rf{eqBetaHarmonic} the quantity
$$-\int |y|^\beta \nabla u\cdot \nabla \zeta=-\int |y|^\beta \nabla u \cdot \nabla \left(\zeta \max\left\{\min\left\{2-\frac{u}{\varepsilon},1\right\},0\right\}\right) \geq -\int_{\Omega\cap \{0<u<2\varepsilon\}} |y|^\beta |\nabla u| |\nabla \zeta|\xrightarrow{\varepsilon\to0} 0$$ defines a positive functional on positive $\zeta\in C^{0,1}_c(\Omega)$. Moreover, for compact $K\subset \Omega$, consider a Lipschitz function $f_K$ such that $\chi_K\leq f_K\leq \chi_\Omega$. If $\zeta\in C^{0,1}_c(K)$, by the positivity shown above we obtain
$$-\int |y|^\beta \nabla u\cdot \nabla \zeta\leq - \norm{\zeta}_{L^\infty}\int |y|^\beta \nabla u\cdot \nabla f_K\leq C_{K,u} \norm{\zeta}_{L^\infty} $$ and, by Hahn-Banach's theorem, we can extend the functional to a positive functional in $C_c(\Omega)$, that is given by integration against a positive Radon measure by the Riesz representation theorem.
The fact that \rf{eqMeasureDef} holds for all functions in $W^{1,2}(\Omega, |y|^\beta) \cap C_{c}(\Omega)$ follows by a standard density argument. \end{proof}
The Caccioppoli inequality is the first step to proving convergence in a Sobolev sense. It will also be useful when we remove the {\it a priori} dependence of our results on the Sobolev norm of the minimizer.
\begin{lemma}[Caccioppoli Inequality] \label{cacciop}
Let $B \subset {\mathbb R}^{n+1}$ be a ball of radius $r$ centered on ${\mathbb R}^n\times\{0\}$, and let $u \in W^{1,2}(B, |y|^\beta)$ be such that $\nabla\cdot (|y|^\beta \nabla u)=0$ weakly in $B\cap \{u>0\}$. Then
$$\int_{\frac12 B} |y|^\beta |\nabla u|^2 \leq \frac{4}{r^2} \int_{B\setminus \frac12 B} |y|^\beta u^2.$$ \end{lemma} \begin{proof}
Let $\eta$ be a Lipschitz function such that $\chi_{\frac12 B}\leq \eta\leq \chi_B$ and with $|\nabla \eta|\leq \frac{1}{r}$. By Lemma \ref{lemRadonMeasure}
$$0= \int_B u \eta^2 d\lambda = \int_B |y|^\beta \nabla u\cdot \nabla(u \eta^2).$$ By the Leibniz rule
$$\int_B |y|^\beta \eta^2 |\nabla u|^2 =- \int_B |y|^\beta 2 u \eta \nabla u\cdot \nabla\eta,$$ and using H\"older's inequality we get
$$\int_{\frac12 B} |y|^\beta |\nabla u|^2 \leq \int_B |y|^\beta \eta^2 |\nabla u|^2 \leq \int_B |y|^\beta 4 u^2 |\nabla\eta|^2 \leq \frac{4}{r^2} \int_{B\setminus \frac12 B} |y|^\beta u^2.$$ \end{proof}
\begin{lemma}\label{lemComparisons} Let $u \in {\mathbf H}^\beta(B_r)$ be a minimizer of $\mathcal{J}$ in $B_{2r}$ and $0\in F(u)$. Then
$$ r^{-n/2}\norm{ \nabla u}_{L^2(\frac12 B_r;|y|^{\beta})} \leq r^{-\alpha} \norm{u}_{L^\infty (B_r)} \leq \norm{u}_{\dot C^\alpha(B_r)} { \leq C \left(1+r^{-n/2} \norm{ \nabla u}_{L^2( B_{2r};|y|^{\beta})} \right)}.$$ \end{lemma}
\begin{proof} The first inequality is Caccioppoli, the middle estimate is trivial and the last follows from \emph{P1} in Theorem \ref{theoCRS}. \end{proof}
\subsection{Compactness}
In the following lemma we prove the compactness of minimizers in the relevant Sobolev spaces. For convenience, we also detail several compactness results which were either already proven in \cite{CaffarelliRoquejoffreSire} or are standard consequences of the non-degeneracy estimates in Theorem \ref{theoCRS}. Nevertheless, we include full proofs here for the sake of completeness. We note here (as we did above and will do again below) that while we currently need to assume the uniform bound on the H\"older norm of the functions $u_k$ we can get rid of this assumption in the light of the results of Section \ref{s:UniformBounds}.
\begin{lemma}[Compactness results]\label{lemCompactness}
Let $\{u_k\}_{k=1}^\infty\subset {\mathbf H}^\beta_{\loc}(\Omega)$ be a sequence of minimizers in a domain $\Omega\subset {\mathbb R}^{n+1}$ with $\norm{u_k}_{\dot C^\alpha(\Omega)}\leq E_0$ with non-empty free boundary. Then there exists a subsequence converging to some $u_0\in{\mathbf H}^\beta_{\loc}(\Omega)$ such that for every bounded open set $G\subset \subset \Omega$ we have
\begin{enumerate}
\item\label{eka} $u_k\rightarrow u_0$ in $C^{\beta}(G)$ for every $\beta<\alpha$,
\item $u_k\rightarrow u_0$ in $L^p(G)$ for every $p\leq\infty$,
\item $\partial \{u_k>0\}\cap \bar G \rightarrow \partial\{u_0>0\}\cap \bar G$ in the Hausdorff distance,
\item $\chi_{\{u_k>0\}}\rightarrow\chi_{\{u_0>0\}}$ in $L^1(G)$, and
\item $\nabla u_k \to \nabla u_0$ in $L^p(G;|y|^\beta)$ for every $p\leq 2$.
\end{enumerate}
\end{lemma}
\begin{proof}
The first claim follows from uniform H\"older continuity and compact embeddings of H\"older spaces. The claim (2) follows from (1) easily.
We now prove the third claim. Let $\epsilon>0.$ We will first show that for $x \in {\mathbb R}^n$ we have \begin{equation} \label{hausconv1} d(x, F(u_0))>\epsilon \Rightarrow d(x,F(u_k))>\frac\epsilon2 \end{equation} for large $k.$ This implies that $F(u_k)\subset \{x\colon\, d(F(u_0),x)<2\epsilon\}$ for $k$ large enough.
Let $B(x,\epsilon)\subset F(u_0)^c.$ If $u_0$ is positive in $B(x,\epsilon)$ then it is bounded from below by a positive number in $B(x,\epsilon/2).$ In this case $u_k$ are also positive in $B(x,\epsilon/2)$ for large $k$ due to uniform convergence in $G$. Thus $B(x,\epsilon/2)\subset F(u_k)^c$ for large $k.$ If $u\equiv0$ in $B'(x,\epsilon)$ then due to the uniform convergence we know that for $k$ large enough $u_k<C\epsilon^\alpha$ in $B'(x,\epsilon)$, where $C$ is a constant given by \emph{P2} in Theorem \ref{theoCRS} so that $u_k$ has no free boundary points in $B(x,\epsilon/2)$ for all large $k.$ This proves \eqref{hausconv1}.
Next we will show that for all large $k$ \begin{equation} \label{hausconv2} F(u_0)\subset \{x\colon\, d(F(u_k),x)<\epsilon\}. \end{equation}
If this was not true we could find a point $x\in F(u_0)$ and a subsequence of $u_k$ such that $B'(x,\epsilon)\subset F(u_k)^c$ for every $k$ in the subsequence. If the subsequence contains infinitely many $u_k$ such that $u_k\equiv0$ in $B(x,\epsilon)$ then also $u_0\equiv 0$ due to uniform convergence. Otherwise, the sequence contains infinitely many $u_k$ for which $B(x,\epsilon)$ is contained in the positive phase. In this case the non-degeneracy implies that in $B(x,\epsilon/2)$ we have $u_k>C\epsilon^\alpha,$ with $C$ independent of $k.$ Again uniform convergence implies the same lower bound for $u_0,$ which contradicts our choice
$x\in F(u_0).$
To show the fourth claim we notice that $F(u_0)$ has zero Lebesgue measure by the Lebesgue differentiation Theorem and the positive density of the zero phase. Take an open set $V\supset F(u_0)$ with $m(V\cap G )<\epsilon.$ For large $k$ we have $F(u_k)\cup F(u_0)\subset V\cap G$, so $\norm{\chi_{\{u_k>0\}}- \chi_{\{u_0>0\}}}_{L^1(G)}<\epsilon.$
Also the sequence is uniformly bounded in $H^{1,p}(G;|y|^\beta)$ by the Caccioppoli inequality. This implies by compactness \cite[1.31 Theorem]{HeinonenKilpelainenMartio} the
weak convergence of $\nabla u_k$ in $L^p(G;|y|^\beta).$ To obtain strong convergence, use Lemma \ref{lemStronger} below.
\end{proof}
It remains to show that weak convergence implies strong convergence.
\begin{lemma}\label{lemStronger} Any sequence of minimizers $\{u_k\}_{k=0}^\infty$ in $\Omega\subset {\mathbb R}^{n+1}$ with $u_k\rightarrow u_0$ uniformly and
$\nabla u_k\rightharpoonup \nabla u_0$ weakly in $L^2_{\loc}(\Omega,|y|^\beta)$ satisfies that $\nabla u_k\to \nabla u_0$ in $L^2_{\loc}(\Omega,|y|^\beta)$. \end{lemma} \begin{proof} Let $\eta\in C^{0,1}_c(\Omega)$ be a non-negative function. We claim that for every $\varepsilon>0$ there exists $j_0$ so that
$$\int |y|^\beta \eta |\nabla u-\nabla u_j|^2\leq \varepsilon$$ for $j\geq j_0$.
First we isolate the main difficulty
$$\int |y|^\beta \eta |\nabla u_0-\nabla u_j|^2=\int |y|^\beta \eta (\nabla u_0-\nabla u_j)\cdot\nabla u_0 - \int |y|^\beta \eta (\nabla u_0-\nabla u_j)\cdot\nabla u_j.$$ By weak convergence,
$$\left|\int |y|^\beta \eta (\nabla u_0-\nabla u_j)\cdot\nabla u_0\right|\leq \varepsilon/4 $$ for $j$ big enough. Note that this is true even if the $u_j$ are not minimizers. The bound on the second term, however, needs the minimization property.
We observe that \begin{equation}\label{e:usedivergence} \int |y|^\beta \eta (\nabla u_0-\nabla u_j)\cdot\nabla u_j = \underbrace{\int |y|^\beta (\nabla u_0-\nabla u_j)\cdot\nabla (\eta u_j)}_{=:I} - \underbrace{ \int |y|^\beta u_j (\nabla u_0-\nabla u_j)\cdot \nabla \eta}_{=:II}.\end{equation}
To estimate $I$ in \eqref{e:usedivergence}, let $\lambda_j$ be the measures corresponding to $u_j$ from Lemma \ref{lemRadonMeasure}. By \rf{eqMeasureDef} we get that
$$\int |y|^\beta (\nabla u_0-\nabla u_j)\cdot\nabla (\eta u_j) = \int \eta u_j \, d\lambda_0 - \int \eta u_j\, d\lambda_j.$$ Since $\lambda_j$ is supported on $\{u_j = 0\}$ we have that $$\int \eta u_j \, d\lambda_j=0$$ for every $j$ (including $j = 0$ as $u_0$ is also a minimizer to $\mathcal J$, see Corollary 3.4 in \cite{CaffarelliRoquejoffreSire}).
To finish the estimate on $I$ in \eqref{e:usedivergence} we observe that
$$\int \eta u_j \, d\lambda_0=\int \eta (u_j-u_0) \, d\lambda_0\leq \sup_{{\rm supp}\; \eta} |u_j-u_0| \int \eta \, d\lambda_0 .$$
By uniform convergence on compact subsets, for $j$ big enough, $\sup_{{\rm supp}\; \eta} |u_j-u_0|\leq \frac{\epsilon}{4\norm{\eta}_{L^1(\lambda_0)}}$.
We turn towards estimating $II$ in \eqref{e:usedivergence}: \begin{align}\label{eqbreak3}
|II|= \left| \int |y|^\beta u_j (\nabla u_0-\nabla u_j)\cdot \nabla \eta\right|
\nonumber & \leq \left| \int |y|^\beta (\nabla u_0-\nabla u_j)\cdot (u_0 \nabla \eta)\right| \\
& \quad + \sup_{{\rm supp}\; \eta} |u_j-u_0| \norm{\nabla u_0-\nabla u_j}_{L^2(\Omega,|y|^\beta)}\norm{\nabla \eta}_{L^2(\Omega,|y|^\beta)}.
\end{align} The first term goes to zero by weak convergence of $\nabla u_j$ to $\nabla u_0$. The second term satisfies
$$\sup_{{\rm supp} \eta} |u_j-u_0| \norm{\nabla u_0-\nabla u_j}_{L^2({\rm supp} \eta,|y|^\beta)}\norm{\nabla \eta}_{L^2(\Omega, |y|^\beta)}\leq \varepsilon/4$$
for $j$ big enough, by uniform convergence and the uniform bound of $\norm{\nabla u_j}_{L^2({\rm supp}\; \eta,|y|^\beta)}$ derived from the Caccioppoli inequality in Lemma \ref{cacciop} together with uniform convergence. \end{proof}
Lemma \ref{lemCompactness} implies that minimizers converge to minimizers (which was observed in Corollary 3.4 in \cite{CaffarelliRoquejoffreSire}), but also implies the stronger fact that the energy is continuous under this convergence:
\begin{corollary}\label{c:ContinuityofEnergy}
Let $u_k$ be a sequence of minimizers in $\Omega \subset \mathbb R^{n+1}$ with $u_k \rightarrow u_0$ locally uniformly and $\sup_k \|u_k\|_{{\mathbf H}^\beta} < \infty$. Then $u_0$ is also a minimizer to $\mathcal J$ in $\Omega$ and for any $B \subset \subset \Omega$ we have $\mathcal{J}(u_k, B) \rightarrow \mathcal{J}(u_0, B)$. \end{corollary}
\section{Monotonicity formula and some immediate consequences}\label{s:monotonicity}
From \cite{Allen} we have the following monotonicity formula: \begin{theorem}[Monotonicity formula, see { \cite[Theorem 4.3]{Allen}} ]\label{theoMonotonicityMinimum} Let $u\in {\mathbf H}^\beta(B_\delta(x_0))$ be a minimizer in $B_\delta(x_0)$ for the functional $\mathcal{J}$ with $x_0\in F(u)$. Then the function
$$r\mapsto \Psi^u_r(x):=\Psi(r)=\frac{\mathcal{J}(u,B_r(x_0))}{r^{n}} - \frac{\alpha}{ r^{n+1}}\int_{\partial B_r(x_0)} |y|^{\beta} u^2 \, d\mathcal{H}^{n}$$ is defined and nondecreasing in $(0,\delta)$, and for $0<\rho<\sigma<\delta$, it satisfies
$$\Psi(\sigma)-\Psi(\rho) = \int_{B_\sigma(x_0)\setminus B_\rho(x_0)} |y|^{\beta}\frac{2\left|\alpha u (x) -(x-x_0)\cdot \nabla u(x)\right|^2 }{|x_0-x|^{n+2}}dx \geq 0.$$ \end{theorem}
As a consequence, the blow-up limits are cones, in the sense of the following corollary. \begin{corollary}\label{coroWeakLimits} Let $u\in {\mathbf H}^\beta(B_\delta(x_0))$ be a minimizer in $B_\delta(x_0)$ with $x_0=(x_0',0)$. Consider a decreasing sequence $0<\rho_k\xrightarrow{k\to\infty}0$ and the associated rescalings $u_k(x):=\frac{u(x_0+\rho_k x)}{r^\alpha}$. Then the \emph{Allen-Weiss density} $$\Psi^{u}_0(x_0):=\lim_{r\searrow 0} \Psi^{u}_r(x_0)$$
is well defined. Furthermore, for every bounded open set $D\subset{\mathbb R}^{n+1}$ and $k\geq k(D)$ this subsequence $u_{k}$ is bounded in $H^{1,2}(D;|y|^\beta)$ and, passing to a subsequence $u_{k_j}$, converges (in the sense of Lemma \ref{lemCompactness}) to $u_0$ which is a globally defined minimizer of $\mathcal J$ that is homogeneous of degree $\alpha$. \end{corollary} The proof is the same as in \cite[Theorem 2.8]{WeissMinimum}
\begin{remark}[Non-uniqueness of blow-ups] We call the function $u_0$ appearing in Corollary \ref{coroWeakLimits} a \emph{blow-up} of $u$ at $x_0$. A priori, the function $u_0$ may depend on the subsequence $u_{k_j}$. However, a simple scaling argument shows that for all radii $r \geq 0$ and all blow-ups $u_0$ to $u$ at $x_0$ we have $$\Psi^{u_0}_r(0) \equiv \Psi^u_0(x_0).$$ \end{remark}
\subsection{Dimension reduction}\label{secReduction} We use the homogeneity of the blow-ups to obtain dimension estimates on the points in the free boundary for which there exists a non-flat blow-up. This process is known as ``dimension reduction" and has been applied to a variety of situations (see \cite{WeissMinimum} for its application to the Bernoulli problem).
The first lemma shows that blow-up limits of blow-up limits have additional symmetry:
\begin{lemma}\label{lemConeBlowup} Let $u\in {\mathbf H}^\beta_{\loc}({\mathbb R}^{n+1})$ be an $\alpha$-homogeneous minimizer of $\mathcal{J}$ and let $x_0\in F(u)\setminus\{0\}$. Then any blow-up limit $u_0$ at $x_0$ is invariant in the direction of $x_0$, i.e., for every $x\in {\mathbb R}^{n+1}$ and every $\lambda\in{\mathbb R}$, $$u_0(x+\lambda x_0)=u_0(x) .$$ \end{lemma}
\begin{proof} Let $x \in {\mathbb R}^{n+1}$, and consider its decomposition $x=\widetilde{x}+\lambda x_0$ with $\widetilde{x}\in \langle x_0 \rangle^\bot$. We only need to check that \begin{equation}\label{eqTargetIdentity} u_0(x)=u_0(\widetilde{x}). \end{equation}
\begin{figure}
\caption{The distance ${\rm dist}(P_1,P_3)=\mathcal{O}(\rho_k^2)$.}
\label{figTrigonometry}
\end{figure}
Consider a ball $B=B(0,r) \subset {\mathbb R}^{n+1}$ so that $\widetilde{x}, x\in B$. Let $\{\rho_k\}$ be a sequence of radii converging to zero and such that $u_k(x):=\frac{u(x_0+\rho_k x)}{\rho_k^\alpha}$ converges to $u_0$
uniformly on $B_r$. For $k$ big enough, $\norm{u_k-u_0}_{L^\infty(B_r)}<\varepsilon$. Then, \begin{equation}\label{eqPartialStep}
|u_0(x)-u_0(\widetilde{x})|\leq 2\varepsilon + |u_k(x)-u_k(\widetilde{x})|. \end{equation}
To control the last term above, we use the homogeneity of $u$. Writing $P_1:=x_0+\rho_k \widetilde{x}$ and $P_2:=x_0+\rho_k x$ we have $\rho_k^\alpha u_k(\widetilde{x})=u(P_1)$ and $\rho_k^\alpha u_k(x)=u(P_2)$. Let $P_3$ be the intersection between the line through $P_1$ and $x_0$ and the line through the origin and $P_2$ (see Figure \ref{figTrigonometry}). By homogeneity of $u$
$$u(P_2)=u(P_3)\left(\frac{|P_2|}{|P_3|}\right)^\alpha=u(P_3)\left(1 \pm \frac{|P_2-P_3|}{|P_3|}\right)^\alpha.$$ Thus, \begin{align*}
\rho_k^\alpha|u_k(x)-u_k(\widetilde{x})|
& \leq \left|u(P_1) - u(P_3)\left(1 + \mathcal{O}(\rho_k)\right)^\alpha \right|
\leq \left|u(P_1) - u(P_3)\right| + \left|u(P_3)\right|\mathcal{O}(\rho_k). \end{align*}
By Thales' Theorem, $|P_1- P_3|=\frac{|P_1- P_2||P_3-x_0|}{|x_0|}=\mathcal{O}(\rho_k^2)$ and using the $\dot{C}^\alpha$ character of $u$ and the fact that $u(x_0)=0$, we get \begin{align*}
\rho_k^\alpha|u_k(x)-u_k(\widetilde{x})|
& \leq \norm{u}_{\dot{C}^\alpha}\left( \left|P_1- P_3\right|^\alpha + \left|P_3\right|^\alpha \mathcal{O}(\rho_k)\right)
= \mathcal{O}(\rho_k^{2\alpha} )+ \mathcal{O}(\rho_k), \end{align*} and \rf{eqTargetIdentity} follows by \rf{eqPartialStep} since $\rho_k\to 0$. \end{proof}
We then recall that a minimizer with a translational symmetry is actually a minimizer without that symmetry in one dimension less. This is known as ``cone splitting":
\begin{lemma}\label{lemDimensionReduction} Let $u\in {\mathbf H}^\beta_{\loc}({\mathbb R}^{n+1})$ be an $\alpha$-homogeneous minimizer of $\mathcal{J}$ in ${\mathbb R}^{n+1}$ which is invariant in the direction $e_n$. Then $\widetilde{u}(x',y):=u(x',0,y)$ is a minimizer of $\mathcal{J}$ in one dimension less.
\end{lemma}
\begin{proof}
The proof is a slight variation of {\cite[Proof of Lemma 3.2]{WeissMinimum}}.
\end{proof}
Next we provide a non-standard proof of Proposition \ref{propoTrivial}, that is, to show that the trivial solution is a minimizer. We use \emph{P\ref{itemBlowup}} in a sequence of conveniently chosen blow-ups and a dimension reduction argument, based on the following lemma. Note that the proposition could also be proven via a classical dimension reduction argument.
\begin{proof}[Proof of Proposition \ref{propoTrivial}] Consider a non-zero minimizer $u$ with non-empty free boundary (see \cite[Pro\-position 3.2]{CaffarelliRoquejoffreSire} for its existence), choose a free boundary point $x_0\in F(u)$ and consider $u_0$ to be a blow-up weak limit at this point, which exists and is $\alpha$-homogeneous by Lemma \ref{coroWeakLimits}. Then $u_0$ is also a global minimizer by \emph{P\ref{itemBlowup}} and not null by the nondegeneracy condition.
Next we argue by induction: given $0\leq j\leq n-2$ let $u_j$ be an $\alpha$-homogeneous global minimizer different from $0$ such that it is invariant in a $j$-dimensional linear subspace $H_j\subset {\mathbb R}^n$, i.e., for every $v\in H_j$ and every $x'\in{\mathbb R}^{n}$, $$u_j(x',y)=u(x'+v,y).$$
Consider a point $x_j \in F(u_j)\setminus (H_j\times\{0\})$ which exists as long as $j<n-1$ by the interior corkscrew condition and positive density, and let $u_{j+1}$ be a blow-up limit at this point, which is again an $\alpha$-homogeneous global minimizer. We claim that $u_{j+1}$ is invariant in fact in the $(j+1)$-dimensional subspace $H_j+\langle x_j' \rangle$.
Indeed $u_{j+1}$ is invariant in $\langle x_j'\rangle$ by Lemma \ref{lemConeBlowup}. On the other hand, since $u_j$ is invariant in $H_j$, so are the functions in the blow-up sequence and, thus, $u_{j+1}$ is invariant in $H_j$. Thus, for $v\in H_j$, $v_0\in \langle x_j'\rangle$ and $x\in{\mathbb R}^{n+1}$ we get $$u(x+v+x_j')=u(x+v)=u(x),$$ and the claim follows.
Thus, after $n-1$ steps, we obtain $u_{n-1}$ which is an $\alpha$-homogeneous global minimizer invariant in an $(n-1)$-dimensional space $H_{n-1}$, with non-empty free boundary. Thus, $$u_{n-1}(x',0)=C_{n,\alpha}(x'_n)_+^\alpha,$$ where the constant is given by \emph{P\ref{itemNormal}}. The proposition follows by Proposition \ref{propoExtension}. \end{proof}
\subsection{Upper semicontinuity} Next we show that Allen-Weiss' energy at a fixed radius is continuous both with respect to the minimizer and with respect to the point: \begin{lemma}\label{lemContinuity} Let $u_j\in {\mathbf H}^\beta_{\loc}(\Omega)$ be minimizers of $\mathcal{J}$ in $\Omega$ and $u_j \to u_0$ in the sense of Lemma \ref{lemCompactness}. Then, for $x_j\to x_0$ and $r<{\rm dist}(x_0,\partial\Omega)$, $$ \Psi_r^{u_j}(x_j) \xrightarrow{j\to\infty} \Psi_r^{u_0}(x_0).$$ \end{lemma} \begin{proof} Let $\varepsilon>0$. We want to check that for $j$ big enough,
$$\left| \Psi_r^{u_j}(x_j) -\Psi_r^{u_0}(x_0)\right|\leq \varepsilon.$$
We will consider the three terms of the energy separately. For the first term,
$$ \int_{B_r(x_j)}|y|^{\beta} |\nabla u_j|^2- \int_{B_r(x_0)}|y|^{\beta} |\nabla u_0|^2 \leq r^n \varepsilon/3$$
follows from the $L^2$ convergence of the gradients. Indeed, if $\delta_j:=|x_j-x_0|\leq \delta $ for $j$ big enough and $B_{r+\delta}\subset \Omega$, then \begin{align*}
\int_{B_r(x_j)}|y|^{\beta} |\nabla u_j|^2- \int_{B_r(x_0)}|y|^{\beta} |\nabla u_0|^2
& \leq
\int_{B_r(x_j)}|y|^{\beta} \left(|\nabla u_j|^2- |\nabla u_0|^2\right)+ \int_{B_r(x_j)\Delta B_r(x_0)}|y|^{\beta} |\nabla u_0|^2 \\
& \leq \int_{B_{r+\delta}(x_j)}|y|^{\beta} \left(|\nabla u_j|^2- |\nabla u_0|^2\right)+ \int_{(B_{r+\delta_j}\setminus B_{r-\delta_j})(x_0)}|y|^{\beta} |\nabla u_0|^2 \\
&\leq r^n \varepsilon/3.
\end{align*}
For the measure, we estimate
$$\left|\int_{B_r(x_j)'} \chi_{\Omega_+(u_j)} dm - \int_{B_r(x_0)'} \chi_{\Omega_+(u_0)}dm \right| \leq r^n \varepsilon/3$$ for $j$ big enough as a consequence of $\chi_{\Omega_+(u_j)}\to \chi_{\Omega_+(u_0)}$ in $L^1_{\loc}$ as before. The fact that
$$\alpha \left| \int_{\partial B_r(x_j)} u_j^2- \int_{\partial B_r(x_0)} u_0^2 \right| \leq r^{n+1} \varepsilon/3 $$ for $j$ big enough is a straight consequence of the uniform convergence and the continuity of $u_0$. \end{proof}
It is well known that the limit of a decreasing sequence of continuous functions is upper semicontinuous (see {\cite[Theorem 1.8]{DalMaso}}). The monotonicity formula also implies the following result. \begin{lemma}\label{lemUSC} Let $u_j\in {\mathbf H}^\beta_{\loc}(\Omega)$ be minimizers of $\mathcal{J}$ in $\Omega$ and $u_j \xrightarrow{j\to\infty}u_0$ in the sense of Lemma \ref{lemCompactness}, with $x_j\in F(u_j)$ for $j\in {\mathbb N}$. Then, if $x_j\to x_0$ and $r_j\to 0$, $$\limsup_{j} \Psi_0^{u_j}(x_j)\leq \limsup_{j} \Psi_{r_j}^{u_j}(x_j) \leq \Psi_0^u(x_0).$$ \end{lemma} \begin{proof} The first inequality comes from monotonicity.
To see that $$\limsup_{j} \Psi_{r_j}^{u_j}(x_j) \leq \Psi_0^{u_0}(x_0),$$ it is enough to check that for every $r>0$ $$ \limsup_{j} \Psi_{r_j}^{u_j}(x_j) \leq \Psi_r^{u_0}(x_0),$$ or using monotonicity it suffices to show that for every $\varepsilon>0$ and $j$ big enough, $$ \Psi_r^{u_j}(x_j) - \Psi_r^{u_0}(x_0) \leq \varepsilon.$$ But this is true for $j$ big enough because the left-hand side converges to $0$ by the continuity of the energy from Lemma \ref{lemContinuity}. \end{proof}
\section{Measure-theoretic properties}\label{s:finperimeter}
\subsection{Finite perimeter} We will show that $\Omega'_+(u)$ is a set of locally finite perimeter. Then $F_{\rm red}(u)$ will coincide with the measure-theoretic reduced boundary by the $\epsilon$-regularity theorem, see \cite[Sections 4.6 and 4.7]{AltCaffarelli}.
\begin{definition} For every $0<\alpha<1$ we can define
$$k^*_\alpha:=\inf \left\{k\in{\mathbb N}: \exists\mbox{ an $\alpha$-homogeneous minimizer $u\in {\mathbf H}^\beta_{\loc}({\mathbb R}^{k+1})$ such that $\Sigma(u)=\{0\}$}\right\}.$$ \end{definition} Note that, to the best of our knowledge, there is no result showing that $k^*_\alpha$ needs to be finite.
\begin{lemma}\label{lemTrivialIsTheChosen} Let $u$ be an $\alpha$-homogeneous minimizer of $\mathcal{J}$ in ${\mathbb R}^{n+1}$ with $n < k^*_\alpha$. Then $u$ is a rotation of the trivial solution. \end{lemma} See \cite[Section 3]{WeissMinimum} for the proof.
From the positive density properties, we know that $k^*_\alpha\geq 2$. From the homogeneity of the blow-ups we find out that the free boundary in ${\mathbb R}^{1+1}$ is in fact a collection of isolated points. Later in Theorem \ref{FullRegDim2} we will show that in fact $k^*_\alpha\geq 3.$ \begin{lemma}[Isolated singularities] \label{IsolSing} Let $u\in {\mathbf H}^\beta_{\loc}(\Omega)$ for $\Omega\subset {\mathbb R}^{1+1}$ be a minimizer of $\mathcal{J}$ in $\Omega$. Then $F(u)$ has no accumulation points in $\Omega$. \end{lemma} \begin{proof} Arguing by contradiction, we assume that $F(u)$ has an interior accumulation point which, without loss of generality, we assume to be the origin.
Let $(x_k,0)$ be a sequence of singular points converging to $0$ with $x_k >0$. Consider the blow-up rescaling $u_k(x):=\frac{u(x_k x)}{x_k^\alpha}$. Note that $u_k(0,0)=u_k(1,0)=0$. Moreover, by the interior corkscrew condition, there exist $z_k\in (1/2, 3/2)$ such that $u_k|_{B'_c(z_k, 0)}>0$, so $u(z_k,0)\gtrsim C$ by the non-degeneracy condition.
Choosing a subsequence, we may assume that $z_k\to z_0\geq 1/2$, and $u_k\to u_0$ in the sense of Lemma \ref{lemCompactness}. In particular $u_0$ is homogeneous by Corollary \ref{coroWeakLimits}, reaching a contradiction with the fact that $u_0(1,0)=0$ and $u_0(z_0,0)\gtrsim C$. \end{proof}
We will prove the local finiteness of the perimeter of the free boundary adapting a proof of De Silva and Savin in \cite{deSilvaSavinJEMS}. Our proof is essentially the same, but we repeat it for the sake of completeness.
As in \cite{deSilvaSavinJEMS} we say that a set $A\subset {\mathbb R}^n$ satisfies the property (P) if following holds: for every $x\in A$ there exists an $r_x>0$ such that for every $0<r<r_x$, every subset $S$ of $B(x,r)\cap A$ can be covered with a finite number of balls $B(x_i,r_i)$ with $x_i\in S$ such that \begin{equation}
\label{PropP}
\sum_i r_i \leq r^\alpha/2. \end{equation}
\begin{lemma} \label{MinimDim}
If $\mathcal H^t (\Sigma(U))=0$ for some $\alpha>0$ and for every minimal cone $U$ in ${\mathbb R}^{n+1}$ then $\mathcal H^t (\Sigma(u))=0$ for every minimizer $u$ of $\mathcal J$ defined on $\Omega\subset {\mathbb R}^{n+1}$ \end{lemma}
\begin{proof}
We first show that $\Sigma(u)$ satisfies the property (P). If (P) does not hold we find a point $y\in \Sigma(u)$ for (P) is violated for a sequence $r_k\rightarrow 0.$ We consider the blow-up sequence
\begin{equation}
u_{r_k}(x)=r_k^{-\alpha}u(y+r_k x).
\end{equation} By Corollary \ref{c:ContinuityofEnergy} we may assume, by taking a subsequence, that $u_{r_k}$ converges to a minimal cone $U$. By our assumptions we may cover $\Sigma(U)\cap B(0,1)$ with a finite collection of balls $\{B(x_i,\frac{\rho_i}{10})\}_{i=1}^k$ with $$ \sum_i \rho_i^t\leq \frac12. $$
By Lemma \ref{lemCompactness} we know that free boundaries converge in Hausdorff sense and thus the set $F(u_{r_k})\cap B(0,1)\setminus \bigcup_i B(x_i,\rho_i/5)$ is flat for all large $k.$ From Theorem \ref{th:iof} we infer that all singularities must be covered by the same balls, that is, for all $k\geq k_0$ \begin{equation}
\Sigma(u_{r_k})\cap B(0,1)\subset \bigcup_i B(x_i,\rho_i/5). \end{equation} After rescaling we see that $u$ satisfies the condition for property (P) in the ball $B(y,r_k),$ which is a contradiction. Therefore the property (P) holds as claimed.
Consider the set $D_k:=\{y\in \Sigma(u): r_y\geq1/k \}.$ Fix a point $y_0\in D_k.$ By property (P) applied to $r_0=1/k$ we find a finite cover of $D_k\cap B(y_0,r_0)$ with balls $B(y_i,r_i),$ $y_i\in D_k,$ satisfying $$ \sum_i r_i^t\leq r_0^t/2. $$ Similarly, for each ball $B(y_i,r_i)$ in the cover we use the property (P) to find a finite number of balls $B(y_{ij}, r_{ij}),$ $y_{ij}\in D_k,$ which cover $D_k\cap B(y_i,r_i)$ and satisfy $$ \sum_j r_{ij}^t\leq r_i^t/2, $$ and thus $\sum_{i,j} r_{ij}\leq r_0^t/4$. By repeating the argument $N$ times we obtain a cover of $D_k\cap B(y_0,r_0)$ by balls $B(z_l,r_l)$ which satisfies $$ \sum_l r_l^t \leq 2^{-N}r_0^t. $$ This implies that $\mathcal H^t(B(y_0,r_0)\cap D_k)=0$ and thus $\mathcal H^t(D_k)=0.$ By countable additivity we obtain the claim.
\end{proof}
\begin{lemma} \label{MeasReduction}
If $\mathcal H^t (\Sigma(U))=0$ for some $t>0$ and for every minimal cone in ${\mathbb R}^{n+1}$, we then have that $\mathcal H^{t+1} (\Sigma(V))=0$ for every minimal cone $V$ in ${\mathbb R}^{(n+1)+1}.$ \end{lemma} \begin{proof} Without loss of generality we may assume $\Sigma(V)\neq \{0\}.$ Let $x\in \Sigma(V)\setminus \{0\}.$ By Corollary \ref{c:ContinuityofEnergy} the blow-ups at any point of $\Sigma(V)\cap \partial B$ converge to a minimal cone in dimension $(n+1)+1$ up to a subsequence. Let $V_x$ be a blow-up at $x.$ By Lemma \ref{lemDimensionReduction} $V_x$ is a minimal cone which is invariant in at least one direction. By Lemma \ref{lemDimensionReduction}, using our assumption this implies that $\mathcal H^{t+1}(\Sigma(V_x))=0$, and thus the singular set of every possible blow-up cone of any minimizer $V$ has zero $\mathcal H^{t+1}$-measure.
Arguing as in Lemma \ref{MinimDim} we obtain $\mathcal H^{t+1}(\Sigma(V))=0.$
\end{proof}
Combining Lemmas \ref{IsolSing}, \ref{MinimDim} and \ref{MeasReduction} we obtain the following corollary. Notice that we will be able to replace $n-1$ by $n-2$ by Theorem \ref{FullRegDim2}.
\begin{corollary}\label{coroZeroHn} Every minimizer satisfies $$\mathcal{H}^{n-1}(\Sigma(u))=0.$$ \end{corollary}
\begin{lemma} \label{l:cover}
Let $u\in {\mathbf H}^\beta (2B)$ be a minimizer of $\mathcal J$ in $2B$ with $\norm{u}_{\dot C^\alpha(2B)}<E_0.$ Then there exists a constant $C$ depending on $n$, $\alpha$ and $E_0$ and a finite collection of balls $\{B(X_i,r_i)\}$ s.t.
\begin{equation}\label{eq:noballs}
\mathcal H^{n-1}\left ( (F(u)\cap B)\setminus \bigcap_{i= 1}^m B(X_i,r_i) \right)\leq C
\end{equation} and
\begin{equation}
\label{eq:balls}
\sum_{i= 1}^m r_i^{n-1} \leq \frac12.
\end{equation} \end{lemma} \begin{proof}
Proof is by contradiction. For $k\in{\mathbb N}$ assume we have $\norm{u_k}_{\dot C^\alpha(2B)}< E_0$ and the left-hand side of \eqref{eq:noballs} is bounded below by $k>0$ for every collection of balls satisfying \eqref{eq:balls}. By Lemma \ref{cacciop} we know the sequence $ u_k$ is bounded in ${\mathbf H}^\beta(B).$
Taking a subsequence we may assume that $u_k$ converges locally uniformly to a minimizer $u$ (see Corollary \ref{c:ContinuityofEnergy}).
By Corollary \ref{coroZeroHn} the set of singularities $\Sigma(u)$ has $\mathcal H^{n-1}$ -measure zero and thus they can be covered with finitely many balls $B_i$ satisfying \eqref{eq:balls}.
Since $F(u)\setminus \Sigma(u)$ is a $C^{1,\gamma}$-surface by Theorem \ref{th:iof}, using the Hausdorff convergence of the free boundaries we apply again Theorem \ref{th:iof} to see that $F(u_k)\cap B_1\setminus \bigcup_{i=1}^M B_i$ are also $C^{1,\gamma}$-surfaces converging to $F(u)\cap B_1\setminus \bigcup_{i=1}^M B_i$ uniformly in the $C^{1}$-norm. This is a contradiction with the assumption that the Hausdorff measure blows up as $k$ goes to $\infty.$ \end{proof}
The fact that the free boundary has finite perimeter follows now from the same iteration argument as \cite[Lemma 5.10]{deSilvaSavinJEMS}.
\begin{lemma}\label{lemFinite}
Let $u$ be as in Lemma \ref{l:cover}.
Then for some constant $C$ depending only on $E_0$,
\begin{equation}
\mathcal H^{n-1}\left( F(u)\cap B\right)\leq C.
\end{equation}
\end{lemma} \begin{proof}
By Lemma \ref{l:cover} we find a finite collection of balls $B_{r_i}$ such that
\begin{equation}
F(u)\cap B \subset \Gamma \cup \bigcup B_{r_i},
\end{equation} with $\mathcal H^{n-1}(\Gamma)\leq C$ and $\sum r_i^{n-1}\leq \frac12.$
Applying Lemma \ref{l:cover} again for each ball $B_{r_i}$ we have \begin{equation}
F(u)\cap B_{r_i} \subset \Gamma_i \cup \bigcup B_{r_{ij}}, \end{equation} with $\mathcal H^{n-1}(\Gamma_i)\leq Cr_i^{n-1}$ and $\sum r_{ij}^{n-1}\leq \frac12r_i^{n-1}.$ Moreover, we have $$ \mathcal H^{n-1}\left((F(u)\cap B_1)\cap \bigcup_{i,j} B_{r_{ij}}\right)\leq\mathcal H^{n-1}(\Gamma)+\sum \mathcal H^{n-1}(\Gamma_i)\leq C \left(1+\sum_{i,j} r_{ij}^{n-1}\right)\leq C\left(1+\frac12\right). $$
Continuing inductively, after $k$ steps we have that \begin{equation}
F(u)\cap B_1 \subset \Gamma' \cup \bigcup_{q=1}^N B_{r_{q}}, \end{equation} with $$ \mathcal H^{n-1}(\Gamma')\leq C \left(\sum_{i=0}^k 2^{-i}\right)\leq 2C, $$ and $\sum r_{q}^{n-1}\leq 2^{-k}.$ This gives the claim. \end{proof}
Finally the fact that $\{u>0\}\cup \Omega$ has locally finite perimeter in $\Omega$ follows from the previous lemma and well-known results of Federer, see for example \cite[Prop. 3.62]{AFP} or \cite[4.5.11]{Federer}.
\subsection{Energy gap} Next we will check that the Allen-Weiss density can also be used to identify singular points. First let us state a useful identity for minimizers (which is also valid in the context of variational solutions in the sense of \cite{WeissWeak}). \begin{lemma}[{See \cite[Proposition 3.4]{Allen}}]\label{LemMagic} Let $u\in {\mathbf H}^\beta_{\loc}(\Omega)$ be a minimizer to \rf{eqLocalizedFunctional} in $\Omega$. For every $B\subset \subset\Omega$ we have \begin{equation}\label{eqMagicFormula}
\int_{B} |y|^{\beta} |\nabla u|^2=\int_{\partial B} |y|^{\beta} u \nabla u \cdot \nu \, d\mathcal{H}^{n}. \end{equation} \end{lemma}
Let $u$ be a minimizer and $x_0\in F(u)$. If we consider a blow-up $u_0$ at $x_0$, then
$$\Psi^{u}_0(x_0)=\Psi^{u_0}_1(0)=\int_{B_1} |y|^{\beta} |\nabla u_0|^2 + m(\{u_0>0\}\cap{\mathbb R}^n\cap B_1) - \alpha \int_{\partial B_1} |y|^{\beta} u_0^2 \, d\mathcal{H}^{n} .$$ By Lemma \ref{LemMagic} we get
$$\Psi^{u_0}_1(0) =\int_{\partial B_1(x_0)} |y|^{\beta} u_0 \nabla u_0 \cdot \nu \, d\mathcal{H}^{n}+ m(\{u_0>0\}\cap{\mathbb R}^n\cap B_1) - \alpha \int_{\partial B_1} |y|^{\beta} u_0^2 \, d\mathcal{H}^{n} .$$
Since $\nabla u_0(x) \cdot \nu (x) = \frac{\alpha }{|x|} u_0(x) $ almost everywhere on the sphere, the first and the third terms cancel out and we obtain $$ \Psi^{u_0}_1(0)= m(\{u_0>0\}\cap B_1'). $$ Thus, the density $\Psi^{u}_0$ at a free boundary point is given by the area of the positive phase of any blow-up at the same point.
We write $\omega_n:=m( B_1')$ for the volume of the $n$-dimensional ball. \begin{proposition}\label{propoAlmostGap} Every homogeneous minimizer $u\in {\mathbf H}^\beta_{\loc}({\mathbb R}^{n+1})$ has density $$\Psi^u_1(0)= m(\{u>0\}\cap B_1') \geq \frac{ \omega_n}2 ,$$ and equality is only attained when $u$ is the trivial minimizer. \end{proposition} \begin{proof}
Let $u$ be a minimizer such that $\Psi^{u}_1(0)\leq \frac{ \omega_n}2$.
Let $x_1\in F_{red}(u) $. Being a regular point, $\Psi^u_0(x_1)=\frac{ \omega_n}2$. On the other hand, by the homogeneity and the continuity in Lemma \ref{lemContinuity}, $$\lim _{r\to\infty}\Psi^u_r(x_1)=\lim _{r\to\infty}\Psi^u_1(x_1/r)=\Psi^u_1(0)\leq \frac{ \omega_n}2.$$ Combining both assertions with the monotonicity of $\Psi$ we get that $\Psi^u_r(x_1)\equiv \frac{ \omega_n}2$. But using the second formula in Theorem \ref{theoMonotonicityMinimum}, one can see that this is true only whenever $\Psi$ is $\alpha$-homogeneous with respect to $x_1$. Thus, $u$ is $1$-symmetric and invariant in the direction of $\langle x_1\rangle$.
Since $\Omega_0$ is a finite perimeter set (see Section \ref{s:finperimeter}), $F_{red}(u)$ has full $\mathcal{H}^{n-1}$ measure on $F(u)$. Thus, we can find $x_1,\dots x_{n-1} \in F_{red}(u) $ linearly independent. By the previous discussion $u$ is invariant on an $(n-1)$-dimensional affine manifold and, thus, it is the trivial solution. \end{proof}
\begin{corollary}[Energy Gap] There exists $\overline{\epsilon}>0$ depending only on $n$ and $\alpha$ such that every minimizer $u\in {\mathbf H}^\beta_{\loc}(\Omega)$ and every singular point $x_0\in \Sigma(u)$ satisfy $$ \Psi^{u}_1(x_0)-\frac{ \omega_n}2 \geq \overline{\epsilon}.$$ \end{corollary} \begin{proof} Assume the conclusion to be false. Then there exist $u_j$ minimizers in $B_1$ with $$ \Psi^{u_j}_1(0)\leq \frac{ \omega_n}2 + 1/j.$$ Passing to a subsequence, $u_j\to u_0$ as in Lemma \ref{lemCompactness}. Using Lemma \ref{lemUSC} we get that $$ \Psi^{u_0}_1 (0)= \lim_j \Psi^{u_j}_1(0) \leq \frac{\omega_n}2 .$$ But then $u_0$ is the trivial cone by Proposition \ref{propoAlmostGap}. Since $F(u_j)\to F(u)$ in the Hausdorff distance, using $\epsilon$-regularity (see Theorem \ref{theoImprovement}) we get that $u_j$ is the trivial cone for $j$ big enough. \end{proof} The value $\overline{\epsilon}$ above depends on the constants and on $\norm{u}_{\dot C^\alpha}$ in a neighborhood of $x_0$. In the next section we will show that $\overline{\epsilon}$ does not depend on $u$ at all.
\section{Full regularity in ${\mathbb R}^{2+1}$}\label{s:r2}
In the case of $n=2$, we prove full regularity of the free boundary for minimizers of our functional. Note that this result does not depend on the previous sections except that we use dimension reduction and blow-ups to deduce regularity of the free boundary.
\begin{theorem} \label{FullRegDim2} Let $n=2$. Then there is no singular minimal cone. In particular, the free boundary $F(u)$ of every minimizer $u$ is $C^{1,\alpha}$ everywhere. \end{theorem} \begin{proof} We follow closely the arguments in \cite[Theorem 5.5]{deSilvaSavinJEMS}, building on \cite{SVcone}. The case $\beta=0$ has been considered in \cite{deSilvaSavinJEMS}. The idea is to construct a competitor by a perturbation argument. We note at this point that the argument is two dimensional in nature and does not generalize to higher dimensions. Recall the functional under consideration: $$
\mathcal J (u,\Omega) = \int_{\Omega} |y|^\beta\,|\nabla u|^2 +m(\left \{ u>0\right \} \cap \mathbb R^n \cap \Omega). $$ Let $V$ be a non trivial minimal cone. Define, as in \cite{deSilvaSavinJEMS}, the Lipschitz continuous function \begin{equation} \psi_R(t)= \left \{ \begin{array}{ccc} 1,\,\,\,\,\,\ 0 \leq t \leq R,\\ 2-\frac{\ln(t)}{\ln(R)},\,\,\,\,\,\ R \leq t \leq R^2,\\ 0,\,\,\,\,\,\ t \geq R^2 \end{array} \right . \end{equation} Define now the bi-Lipschitz change of coordinates $$
Z(x',y)=(x',y)+\psi_R(|(x',y)|)e_1 $$ and set $V^+_R(Z)=V(x',y)$. Clearly, one has $$ D_{(x',y)} Z=\text{Id}+A $$
where $\|A\| \leq |\psi'_R(|(x',y)|) | <<1$. Defining now $V^-_R$ exactly as $V^+_R$ changing $\psi_R$ into $-\psi_R$, the very same computation as in \cite{deSilvaSavinJEMS} gives
$$
\mathcal J(V^+_R, B_{R^2}) +\mathcal J(V^-_R, B_{R^2}) \leq 2 \mathcal J(V, B_{R^2})+\int_{B_{R^2}} |y|^\beta \, |\nabla V|^2 \|A\|^2. $$ Now, we have $$
\int_{B_{R^2}} |y|^\beta \, |\nabla V|^2 \|A\|^2= \int_R^{R^2} \int_{\partial B_r} |y|^\beta \, |\nabla V|^2 \|A\|^2 \, d\mathcal{H}^n\,\,dr. $$
Now since $V$ is homogeneous of degree $\alpha$ by assumption, the function $g(x,y)=|y|^\beta \, |\nabla V|^2$ is homogeneous of degree $\beta+2\alpha-2=-1$. Therefore by a trivial change of variables on the sphere of radius $r$ and using the fact that $n=2$, we get the very same estimate $$
\int_{B_{R^2}} |y|^\beta \, |\nabla U|^2 \|A\|^2 \leq \frac{C}{\ln (R)} \xrightarrow{R\to \infty} 0. $$ The rest of the proof follows verbatim \cite{deSilvaSavinJEMS}, page $1318$ since this is only based on energy considerations and we refer the reader to it. \end{proof}
\section{Uniform bounds around the free boundary}\label{s:UniformBounds} The optimal regularity bound and the non-degeneracy described in Theorem \ref{theoCRS} were obtained in \cite{CaffarelliRoquejoffreSire} with bounds that depend on the seminorm $\norm{u}_{\dot {\mathbf H}^\beta(B_1)}$. As a consequence, this dependence propagates to many of our estimates above. In this chapter we use the semi-norm dependent estimates (e.g. Lemma \ref{lemFinite}) to prove semi-norm \emph{independent} non-degeneracy estimates. Re-running the arguments above yields the semi-norm independent results presented in our main Theorem \ref{t:mainregularity}.
The question of semi-norm independence may seem purely technical; however, independence allows the compactness arguments of the next section to work without additional assumptions on the minimizers involved.
\subsection{Uniform non-degeneracy}\label{s:UniformNondegeneracy} We will begin by showing uniform non-degeneracy from scratch to deduce uniform H\"older character from this fact, reversing the usual arguments in the literature.
The following lemma was shown in \cite[Corollary 4.2]{Allen} in a more general setting. Here we give a more basic approach based on \cite[Lemma 3.4]{AltCaffarelli}. The main difference is that where Alt and Caffarelli could use the energy to directly control the $H^1$ norm of the minimizer, in our case we need to find an alternative because the measure term of the functional is computed on the thin phase (as opposed to the $H^1$ norm which is computed on the whole space). To bypass this difficulty we will use Allen's monotonicity formula.
The drawback of our approach is that we need the ball to be centered on the free boundary, while in the original lemma, Alt and Caffarelli could center the ball in the zero phase, allowing for a slightly better result. \begin{lemma}\label{lemNonDegeneracy} Let $u$ be a minimizer in $B_r$ with $0\in F(u)$. Then $\sup_{\partial B_r} u \geq Cr^\alpha$ with $C$ depending only on $n$ and $\alpha$. \end{lemma} \begin{proof} By rescaling we can assume that $r=1$.
Let $\mathcal{L}u := -\nabla\cdot (|y|^\beta \nabla u)$, consider $\Gamma(x)=\frac{1}{|x|^{n-2\alpha}}$ which is a solution of $\mathcal{L}\Gamma=0$ away from the origin (or $\Gamma(x)=\log |x|$ if $n=1$ and $\alpha=1/2$), and let $$\widetilde{v}(x):=\ell \frac{\max\{1-\Gamma(2x),0\}}{1-\Gamma(2)},$$ where $$\ell:=\sup_{\partial B_1} u.$$
It follows that $u\leq \widetilde{v}$ on $\partial B_1$ and thus $$\mathcal{J}(u, B_1 )\leq \mathcal{J}(\min\{u,\widetilde{v}\},B_1),$$
and observing that $\tilde{v} = 0$ on $B_{1/2}$ and $\tilde{v} > 0$ on the annulus $A:=B_{1}\setminus B_{1/2}$, we get \begin{align*}
\int_{B_{\frac12}}|y|^{\beta} |\nabla u|^2 + m\left(B_{\frac12,+}(u)\right)
\nonumber & \leq \int_{A} |y|^{\beta} (|\nabla (\min\{u,\widetilde{v}\})|^2-|\nabla u|^2) + m(A_{+}'(\min\{u,\widetilde{v}\})) - m(A_{+}'(u)) \\
\nonumber & \leq -2\int_{A} |y|^\beta \nabla\max\{u-\widetilde{v},0\}\cdot \nabla \widetilde{v}.
\end{align*}
By Green's theorem, writing $d\sigma=|y|^\beta d\mathcal{H}^n$ we get \begin{align}\label{eqBoundByEll}
\int_{B_{\frac12}}|y|^{\beta} |\nabla u|^2 + m\left(B_{\frac12,+}(u)\right)
\leq -2\int_{\partial B_{\frac12}} u \partial_\nu \widetilde{v}\, d\sigma
= C_{n, \alpha} \ell \int_{\partial {B_{\frac12}}} u\, d\sigma, \end{align} with $C_{n,\alpha} > 0$.
Using the monotonicity formula and Proposition \ref{propoAlmostGap}, we get that $\psi^u(r) \geq \psi^u(0)\geq \frac{\omega(B_1)}{2}$ and, therefore \begin{equation}\label{eqControlJBelow} \frac{\alpha}{r} \int_{\partial {B_r}} u^2\, d\sigma + \frac{\omega(B_1) r^d}{2}\leq \mathcal{J}_r(u), \end{equation} so using H\"older's inequality and the AM-GM inequality we obtain \begin{equation}\label{eqControlByMoreThanHolder} \int_{\partial {B_{\frac12}}} u\, d\sigma\leq \left(\int_{\partial {B_{\frac12}}} u^2\, d\sigma\right)^\frac12 C_{n,\alpha}^\frac12\leq \frac12 \int_{\partial {B_{\frac12}}} u^2\, d\sigma + \frac12 C_{n,\alpha} \leq C_{n,\alpha} \mathcal{J}_{\frac12}(u). \end{equation}
Combining \rf{eqBoundByEll}, \rf{eqControlJBelow} and \rf{eqControlByMoreThanHolder} we obtain $$ 0 < \mathcal{J}_{\frac12}(u)\leq C_{n, \alpha} \ell \mathcal{J}_{\frac12}(u),$$ and therefore $\ell\geq C_{n,\alpha}^{-1}$.
\end{proof}
To show averaged non-degeneracy we need a mean value principle which is well-known, but we include its proof for the sake of completeness.
\begin{lemma}[Mean value principle]\label{lemMVP}
Let $u\in H^1(\beta,\Omega)$ be a weak solution to $\mathcal{L}u:=\nabla\cdot (|y|^\beta \nabla u)=0$ in $\Omega$, and let $x_0\in {\mathbb R}^n \times\{0\}$ with $B_r(x_0) \subset \Omega$. Then $$u(x_0)=\fint_{B_r} u \, d\omega$$
where the mean is taken with respect to the measure $d\omega:= |y|^\beta \, dx$. \end{lemma}
\begin{proof} Changing variables, we have that \begin{equation*}
A(\rho):=\frac{1}{\rho^{\beta+n+1}}\int_{B_\rho(x_0)} |y|^\beta u(x) dx=\int_{B_1} |y|^\beta u(\rho x+x_0) dx. \end{equation*}
On the other hand, set \begin{align*} \widetilde{A}(\rho)
& := \int_{B_1} |y|^\beta \nabla u(\rho x + x_0)\cdot x \,dx\\
& = \int_{B_\rho(x_0)} \left(\frac{|y|}{\rho}\right)^\beta \frac{\nabla u(x)\cdot (x-x_0)}{\rho} \frac{dx}{\rho^{n+1}}
= \frac{1}{2 \rho^{\beta+n+2}} \int_{B_\rho(x_0)} |y|^\beta \nabla u(x)\cdot \nabla |x-x_0|^2 \, dx. \end{align*}
Since $u$ is a weak solution to $\nabla\cdot (|y|^\beta \nabla u)=0$ in $\Omega$, we can apply Green's formula twice to obtain \begin{align*} \widetilde{A}(\rho)
&= \frac{1}{2 \rho^{\beta+n+2}} \int_{\partial B_\rho(x_0)} |x-x_0|^2 |y|^\beta \nabla u(x)\cdot \nu \, dx = \frac{1}{2 \rho^{\beta+n}} \int_{\partial B_\rho(x_0)} |y|^\beta \nabla u(x) \cdot \nu \, dx=0. \end{align*} Since $u$ is absolutely continuous on lines (see \cite[Theorem 4.21]{EvansGariepy}), for almost every $x$ we have $\int_\rho^r \nabla u(t x + x_0)\cdot x \, dt=u(rx+x_0)-u(\rho x+x_0)$. Applying Fubini's Theorem we get \begin{equation*}
\int_\rho^r \widetilde{A}(t) \, dt = \int_{B_1}|y|^\beta \int_\rho^r \nabla u(t x + x_0)\cdot x \, dt \,dx = \int_{B_1}|y|^\beta (u(rx+x_0)-u(\rho x+x_0))\,dx = A(r)-A(\rho) . \end{equation*} So $A(r)-A(\rho)= 0$ for all $\rho<r$.
On the other hand, taking the mean with respect to the measure $d\omega:= |y|^\beta \, dx$ and using the continuity of $u$ (see \cite[Theorem 2.3.12]{FabesKenigSerapioni}) we obtain
$$\left|u(x_0) -\frac{1}{\omega(B_1)}\lim_{\rho \to 0} A(\rho) \right| = \lim_{\rho \to 0} \frac{1}{\omega(B_\rho(x_0))} \left|\int_{B_\rho(x_0)} (u(x_0) -u(x)) \, d\omega(x)\right| \leq \lim_{\rho \to 0} o_{\rho\to 0} (1)=0.$$ \end{proof}
\begin{corollary}\label{coroMeanBoundedBelow}
Let $u$ be a minimizer in $B_r$ with $0\in F(u)$ and let $d\sigma=|y|^\beta d\mathcal{H}^n$. Then $\fint_{\partial B_r} u \, d\sigma \geq Cr^\alpha$ with $C$ depending only on $n$ and $\alpha$. \end{corollary} \begin{proof} Let $v$ be the $\mathcal{L}-$harmonic replacement of $u$ in $B_r$, that is, the solution to \begin{equation}\label{eqHarmonicReplacement} \begin{cases} \mathcal{L}v=0 & \mbox{ in } B_{r}, \\ v\equiv u &\mbox{ on }\partial B_{r}, \end{cases} \end{equation} see \cite[Theorem 3.17]{HeinonenKilpelainenMartio}. After differentiating with respect to the radius, by the mean value principle we get that $v(0)=\fint_{\partial B_r} u \, d\sigma$. By the comparison principle and the Harnack inequality we get that \begin{equation}\label{eqComparisonAndHarnack} C r^\alpha\leq \sup_{B_{r/2}} u \leq \sup_{B_{r/2}} v \leq C \fint_{\partial B_r} u \, d\sigma. \end{equation} \end{proof}
\subsection{Behavior of the distributional fractional Laplacian}\label{ss:growth}
Next we use an idea of \cite{AltCaffarelli} and investigate the behavior of the distributional $\alpha$-Laplacian of the minimizer introduced in Section \ref{s:Compactness}. As mentioned in the introduction, in \cite{AltCaffarelli} this investigation immediately yields that the positivity set is a set of locally finite perimeter, and more precisely, that it is Ahlfors regular of the correct dimension. However, the nonlocal nature of this problem indicates that the distributional fractional Laplacian may not be supported on the free boundary and thus we cannot expect to immediately gain such strong geometric information.
First we can bound the growth of the fractional Laplacian measure around a free boundary point. Note that this growth is the natural counterpart to the upper Ahlfors regularity in the case of Alt-Caffarelli minimizers. \begin{theorem}\label{theoLambdaBoundAbove} Let $u \in {\mathbf H}^\beta(B_{2r}(x_0))$ be a minimizer of $\mathcal{J}$ in $B_{2r}(x_0)$, and let $x_0\in F(u)$. Then, we have $$\lambda(B_r(x_0)) \leq C r^{n-\alpha}.$$
In particular $\lambda(F(u))=0$. \end{theorem} A glance at \eqref{eqLaplacianTrivial} will convince the reader that these estimates are sharp, for they cannot be improved even in the case of the trivial solution.
\begin{proof}
Without loss of generality we may assume that $x_0=0$. Let $\mathcal{L}u := -\nabla\cdot (|y|^\beta \nabla u)$ and let $v$ be the $\mathcal{L}$-harmonic replacement of $u$ in $B_{2r}$, see \rf{eqHarmonicReplacement}. Write $d\sigma=|y|^\beta d\mathcal{H}^n$ and $M:=\fint_{\partial B_{2r}} u \, d\sigma$. By Harnack's inequality (see \cite{CaffarelliRoquejoffreSire}, for instance) and the mean value principle in Lemma \ref{lemMVP}, $$\inf_{B_r} v\geq C v(0)=CM.$$ We have that $$\lambda(B_r)=\int_{B_r}d\lambda\leq \frac{1}{CM}\int_{B_r}v d\lambda.$$ Since $u\equiv 0$ in the support of $\lambda$ and $u$ is $\mathcal{L}$-subharmonic (see \cite[Lemma 2.2]{AltCaffarelli}) we get $$\int_{B_r}v d\lambda=\int_{B_r}(v-u)d\lambda \leq \int_{B_{2r}}(v-u)d\lambda .$$ By the properties of the measure $\lambda$, we obtain
$$\int_{B_{2r}}(v-u)d\lambda= -\int_{B_{2r}}|y|^\beta \nabla (v-u)\cdot\nabla u = \int_{B_{2r}}|y|^\beta \left(|\nabla u|^2-|\nabla v|^2\right),$$ and using the definition of the functional and the fact that $u$ is a minimizer, we get
$$\int_{B_{2r}}|y|^\beta \left(|\nabla u|^2-|\nabla v|^2\right) = \mathcal{J}(u,B_{2r})-m(B_{2r}^+(u)) -\mathcal{J}(v,B_{2r})+m(B_{2r}') \leq C r^n.$$ All together, we have that $$\lambda(B_r)\leq \frac{1}{CM}Cr^n,$$ and, since uniform non-degeneracy (see Lemma \ref{lemNonDegeneracy}) implies that $M\geq Cr^\alpha$ we can conclude the proof of the first statement.
To show the second one, note that since the free boundary has locally finite $(n-1)$-dimensional Hausdorff measure, given a set $E\subset F(u)$ and $k\in{\mathbb N}$ we can find a collection of balls $I_k=\{B^k_i\}_i$ such that $$E\subset \bigcup_{B\in I_k} B, \quad\quad \sup_{B\in I_k}r(B)\leq 1/k \quad\quad \mbox{and}\quad\quad \sum_{B\in I_k} r(B)^{n-1}\leq 2\mathcal{H}^{n-1}(E).$$ Thus, $$\lambda(E)\leq\sum_{B\in I_k} \lambda(B)\lesssim \sum_{B\in I_k} r(B)^{n-\alpha}\leq \sup_{B\in I_k}r(B)^{1-\alpha} \sum_{B\in I_k} r(B)^{n-1} \xrightarrow{k\to \infty} 0.$$
\end{proof}
Next we study the measure away from the free boundary. We should emphasize here that even though the estimates in Lemma \ref{lemFracLaplacianIn0Phase} and Theorem \ref{theoMeasureAway} depend on $E_0$, they will be used to remove the dependence of our other estimates on $E_0$. More precisely, Theorem \ref{theoMeasureAway} will play a role in establishing the continuity of the Green function in Lemma \ref{lemContinuousGreen}. This qualitative fact is used to prove the quantitative uniform H\"older character in Theorem \ref{theoDimension}.
After proving Theorem \ref{theoDimension}, we may drop the hypothesis $\norm{u}_{{\mathbf H}^\beta(B_2)}\leq E_0$ from both Lemma \ref{lemFracLaplacianIn0Phase} and Theorem \ref{theoMeasureAway}.
\begin{lemma}\label{lemFracLaplacianIn0Phase} If $u \in {\mathbf H}^\beta_{\loc}(B_{2})$ is a minimizer of $\mathcal{J}$ in the ball $B_{2}$ with $\norm{u}_{{\mathbf H}^\beta(B_{2})}\leq E_0 $ and $0\in F(u)$, then for every $x_0=(x',0)\in B_{1,0}(u)$ we get
$$\lim_{y\to 0} |y|^\beta |u_y(x',y)| \approx C {\rm dist}(x_0,F(u))^{-\alpha}.$$ Moreover, for every ball $B$ centered at ${\mathbb R}^n\times\{0\}$ with $B'\subset\subset B_{1,0}(u)$, we have that
$$|y|^\beta |u_y(x',y)| \leq C {\rm dist}(x,F(u))^{-\alpha},$$
for $|y|<C_B{\rm dist}(x,F(u))$, where the constant $C_B$ may depend on $B$. \end{lemma} \begin{proof} Let $u$ be a minimizer, and let $B:=B_r(x_0)$ with $B'\subset B_{1,0}(u)$.
By \cite[Lemma 2.2]{Silvestre}, we can write $u(x',y)=|y|^{1-\beta}g(x') +\mathcal{O}(y^2)$, where $g$ is a $C^{1+\beta}(\frac12B')$ function, with a uniform control on the error term in terms of $\norm{u}_{L^2(B,|y|^\beta)}$. In particular, $\lim_{y\to 0} |y|^{\beta-1} u(x',y)=g(x')$.
Let us define \begin{equation} \widetilde{u}(x',y):=\begin{cases} u(x',y) &\mbox{if }y\geq 0\\ -u(x',-y) &\mbox{if }y< 0. \end{cases} \end{equation}
It is clear that $\mathcal{L}\widetilde{u}\equiv 0$ in $B$. According to \cite[Lemma 3.26, Corollary 3.29]{VitaThesis} $v(x',y)=|y|^\beta y^{-1} \widetilde{u}(x',y)$ is an even $C^\infty(\frac12B)$ function in ${\mathbf H}^{2-\beta}(B)$ (note that $1<2-\beta<3$ is out of the usual range of $\beta$) and satisfying $\nabla\cdot(|y|^{2-\beta}\nabla v)=0$. The mean value principle (see Lemma \ref{lemMVP}) applies also to this case, so
$$g(x_0')=v(x_0)= \frac{1}{\int_{\frac12B}|y|^{2-\beta}}\int_{\frac12B}|y|^{2-\beta} v(x) = C \frac{1}{r^{2-\beta+n+1}} \int_{\frac12B}|y| u(x),$$ and using \emph{P1}-\emph{P3} , if $r={\rm dist}(x_0,F(u))$ we get $$g(x_0')=v(x_0)\approx C r^{\beta-2+1+\alpha}=Cr^{-\alpha}.$$
On the other hand, on the upper half plane we have $u_y=(y^{1-\beta}v)_y=(1-\beta) y^{-\beta} v + y^{1-\beta}v_y$, so $$y^\beta u_y(x',y)=(1-\beta) v(x',y) + y v_y(x',y),$$ and $$\lim_{y\to 0^+} y^\beta u_y(x',y) = (1-\beta) g(x')\approx r^{-\alpha},$$ the limit being uniform on compact subsets of $B$. \end{proof}
\begin{theorem}\label{theoMeasureAway} If $u \in {\mathbf H}^\beta_{\loc}(B_2)$ is a minimizer of $\mathcal{J}$ in the ball $B_2$ with $\norm{u}_{{\mathbf H}^\beta(B_2)}\leq E_0$,
then the measure $\lambda$ is absolutely continuous with respect to the Lebesgue measure, and for $m$-almost every $x\in B_{1}'(u)$ we have that
$$\frac{d\lambda}{dm}(x)=2\lim_{y\to0}|y|^\beta u_y(x',y)\approx \chi_{B_{1,0}(u)}(x) {\rm dist}(x,F(u))^{-\alpha},$$ with constants depending on $n$, $\alpha$ and $E_0$. \end{theorem} \begin{proof}
By Theorem \ref{theoLambdaBoundAbove} we only need to show absolute continuity in $B_{1,0}(u)\cup B_{1,+}'(u)$. For $x=(x',0)\in B_{1,+}'(u)$ by \cite[Lemma 4.2]{CaffarelliSilvestre} we have that
$$\lim_{y\to 0}|y|^\beta u_y(x',y)=0,$$ and, for $x\in B_{1,0}(u)$ we have seen in Lemma \ref{lemFracLaplacianIn0Phase} that
$$\lim_{y\to 0}|y|^\beta u_y(x',y)\approx {\rm dist}(x,F(u))^{-\alpha},$$ showing the second part of the statement.
Consider a ball $B_r(x_0)$ with $x_0\in {\mathbb R}^n\times\{0\}$ and a collection of even smooth functions $\chi_{B_r}\leq \psi_k\leq \chi_{B_{r+\frac1k}}$. Then \begin{equation}\label{eqSandwich}
\lambda(B_r)\leq -\int |y|^\beta \nabla u\cdot \nabla \psi_k\leq \lambda(B_{r+\frac1k}), \end{equation} and for every $\varepsilon>0$ we use the Green's theorem to get
$$-\int |y|^\beta \nabla u\cdot \nabla \psi_k= -\int_{|y|\leq\varepsilon} |y|^\beta \nabla u\cdot \nabla \psi_k - \int_{|y|=\varepsilon } |y|^\beta \psi_k \nabla u \cdot \nu\, dm.$$ Using the symmetry properties and taking limits, \begin{equation}\label{eqControlByUy}
-\int |y|^\beta \nabla u\cdot \nabla \psi_k =2 \lim_{\varepsilon\to 0} \int \varepsilon^\beta \psi_k(x',\varepsilon) u_y (x',\varepsilon)\, dm(x'). \end{equation}
Next we want to apply the dominated convergence theorem.
Let us begin by considering a ball $B_r(x_0)\subset B_1$ centered in the zero phase, with ${\rm dist}(B'_r(x_0),F(u))\geq 2r$. In this case, by Lemma \ref{lemFracLaplacianIn0Phase} we have \begin{equation}\label{eqDominateZero} \varepsilon^\beta u_y(x',\varepsilon)\lesssim r^{-\alpha}, \end{equation} with constants depending perhaps on $u$ and $B_r$ as well.
If instead $B'_r(x_0)\subset\subset B_{1,+}'(u)$, by \cite[Theorem 3.28]{VitaThesis} $u$ is an even $C^\infty$ function on $B'_{r}(x_0)$, so $|y|^\beta u_y=\mathcal{O}(|y|^{1+\beta}).$ Thus \begin{equation}\label{eqDominatePositive} \varepsilon^\beta u_y(x',\varepsilon)\lesssim r^{2-2\alpha} . \end{equation}
In both cases, the dominated convergence theorem applies and $$\lim_{\varepsilon\to 0} \int_{B_{r+\frac1k}\cap \{y=\varepsilon\} } \varepsilon^\beta \psi_k u_y \, dm= \int_{B_{r+\frac1k}'} \psi_k \lim_{\varepsilon\to 0} (\varepsilon^\beta u_y)\, dm,$$ and by \rf{eqSandwich} and \rf{eqControlByUy} we obtain $$\lambda(B_r)\leq 2 \int_{B_{r+\frac1k}'} \psi_k \lim_{\varepsilon\to 0} (\varepsilon^\beta u_y) \, dm\leq \lambda(B_{r+\frac1k}).$$ In particular $\lim_{\varepsilon\to 0} (\varepsilon^\beta u_y)\in L^1_{\loc}(B_{1,0}(u)\cup B_{1,+}'(u))$ and taking limits in $k$ we get $$\lambda(B_r)= 2 \int_{B_{r}'} \lim_{\varepsilon\to 0} (\varepsilon^\beta u_y) \, dm.$$ \end{proof}
A consequence of our control of the behavior of $\lambda$ is that we can establish the existence of exterior corkscrews. We should note that exterior corkscrews can be also obtained by a purely geometric argument given the non-degeneracy and positive density of Theorem \ref{theoCRS} (see, e.g. the proof of Proposition 10.3 in \cite{davidtoroalmostminimizers}). \begin{corollary}\label{coroExteriorCorkscrew} If $u\in $ is a minimizer in $B_2$ with $\norm{u}_{{\mathbf H}^\beta(B_2)}\leq E_0$, then $B_{1,+}'(u)$ satisfies the exterior corkscrew condition, i.e. there exists a constant $C_1$ such that for every $x\in F(u)$ and every $0<r<{\rm dist}(x,\partial B_1)$ one can find $x_0\in B_r(x)$ so that $$B(x_0,C_1r)\cap B_{1,+}' (u)= \emptyset.$$ \end{corollary}
\begin{proof} This is a consequence of Theorems \ref{theoLambdaBoundAbove} and \ref{theoMeasureAway}, and the positive density condition for the zero phase. Indeed, given a ball $B_r\subset {\mathbb R}^{n+1}$, combining both theorems we get $$\begin{aligned}r^{n-\alpha}
& \gtrsim \lambda(B_{1,0}(u)\cap B_r) \geq C_{E_0} \int_{B_{1,0}(u)\cap B_r}{\rm dist}(x,\partial B_1)^{-\alpha}\\
&\geq C \left(\sup_{B_{1,0}(u)\cap B_r} {\rm dist}(x,\partial B_1)\right)^{-\alpha} |B_{1,0}(u)\cap B_r |, \end{aligned}$$ and the positive density condition implies that
$$|B_{1,0}(u)\cap B_r | \geq C_{E_0} r^n .$$ Thus, $$ \sup_{B_{1,0}(u)\cap B_r} {\rm dist}(x,\partial B_1)\geq C_{E_0}r,$$ which is equivalent to the exterior corkscrew condition. \end{proof}
\subsection{Uniform H\"older character}\label{s:UniformFinal}
The uniform non-degeneracy of Section \ref{s:UniformNondegeneracy} lets us conclude uniform control on the H\"older norm of $u$.
\begin{theorem}\label{theoDimension}
Let $u$ be a minimizer of $\mathcal{J}$ in $B_r$ with $0\in F(u)$. Then $|u(x)|\leq C|x|^\alpha$ for every $x\in \partial B_{r/2}$ with $C$ depending only on $n$ and $\alpha$. \end{theorem} \begin{proof} Again we set $v$ to be the $\mathcal{L}$-harmonic replacement of $u$ inside of $B_r$ as in \rf{eqHarmonicReplacement}. Let $\widetilde{u}:=v- u$, so that
$$\mathcal{L}\widetilde{u} = \mathcal{L}v - \mathcal{L}u = -\lambda = - \nabla\cdot (|y|^\beta \nabla u),$$
and $\widetilde{u}\in H^{1,2}_0(B_r; |y|^\beta)$.
Consider the Green function $G: B_r\times B_r\to {\mathbb R}$ such that $\mathcal{L} G(\cdot , z) = \delta_z$, and $G(\cdot , z)\in H^{1,2}_{\loc}(\overline{B_r}\setminus\{z\})$ with null trace on $\partial B_r$
(see \cite[Proposition 2.4]{FabesJerisonKenig}).
By \cite[Proposition 2.1, Lemma 2.7]{FabesJerisonKenig} there exists $p_0>1$ so that $\widetilde{u}$ is the unique function in $ H^{1,p_0}_0(B_r;|y|^\beta)$ such that $\mathcal{L}\widetilde{u}=\lambda$, and moreover \begin{equation}\label{eqGrennRules} \widetilde{u}(z)=\int_{B_r} G(z,x)\, d\lambda(x), \end{equation} for almost every $z\in B_r$.
Below, in Lemma \ref{lemContinuousGreen}, we will see that the equality \rf{eqGrennRules} is in fact valid for every $z\in B_{r/4}$, that is, $\widetilde{u}=\int_{B_r} G(\cdot,x)\, d\lambda(x)$. In particular $$v(0)=\widetilde{u}(0)=\int_{B_r} G(0,x)\, d\lambda(x).$$
Next we use the following estimate (see \cite[Theorem 3.3]{FabesJerisonKenig}): let $z,x\in B_{r/4}$. Then
$$G(z,x)\approx \int_{|x-z|}^r \frac{s \, ds}{w(B(x,s))},$$
where $w$ is the $A_2$ weight $w(x)=|y|^\beta$. Computing, for $x=(x',y)$ we obtain
$$w(B(x,s))\approx s^n \int_{y-s}^{y+s} |t|^\beta\, dt \approx s^n \max\{|y|,s\}^{\beta+1}.$$ First we assume that $n-2\alpha>0$. Thus, if $x\in B_{r/4}'$ then \begin{equation}\label{eqGreenEstimate}
G(z,x) \approx \int_{|x-z|}^r s^{-n-\beta} \, ds \approx |x-z|^{-n-\beta+1} = |x-z|^{2\alpha-n}. \end{equation}
Note that $\lambda(B_r) \leq Cr^{n-\alpha}$ by Theorem \ref{theoLambdaBoundAbove}. Thus, writing $A_{t,s}:=B_s\setminus B_t$, we have that
$$ v(0)= \int_{B_r} G(0,x)\, d\lambda(x) \leq \int_{cr^{2\alpha-n}}^\infty \lambda\left(\{x\in B_{r/4}: G(0,x) > t\}\right) dt + \int_{A_{r/4,r}} G(0,x)\, d\lambda(x).$$ By the strong maximum principle, the Green function in the annulus is bounded by $C r^{n-2\alpha}$. This fact, together with Theorem \ref{theoLambdaBoundAbove}, implies that
$$ v(0) \leq \int_{cr^{2\alpha-n}}^\infty \lambda\left(B_{Ct^{\frac{-1}{(n-2\alpha)}}}\right) dt +Cr^\alpha \leq C \int_{cr^{2\alpha-n}}^\infty t^{-\frac{n-\alpha}{n-2\alpha}} dt +Cr^\alpha = Cr^\alpha.$$
By the mean value theorem we conclude that
$$\fint_{\partial B_r} v \, d\sigma \leq C r^{\alpha},$$ where $d\sigma=|y|^\beta d\mathcal{H}^n$. The theorem follows by observing that, as in \rf{eqComparisonAndHarnack}, the mean of $v$ dominates $u$ by $\sup_{\partial B_{r/2}} u \leq \sup_{\partial B_{r/2}} v \leq C \fint_{\partial B_r} v\, d\sigma$.
In case $n-2\alpha=0$, which could only happen for $n=1$ and $\alpha=1/2$, estimate \rf{eqGreenEstimate} reads as \begin{equation*}
G(z,x) \approx \log\left(\frac{r}{|x-z|}\right), \end{equation*} and the proof follows the same steps.
In case $n-2\alpha<0$, then estimate \rf{eqGreenEstimate} reads as \begin{equation*} G(z,x) \approx r^{n-2\alpha}, \end{equation*} and the estimate is even better compared to the above.
\end{proof}
\begin{lemma}\label{lemContinuousGreen} $\int_{B_r} G(z,x)\, d\lambda(x)$ is continuous in $z\in B_{r/4}$. \end{lemma} \begin{proof}
Let $\varepsilon<r/2$ and let $z_1, z_2\in B_{r/4}$, with $|z_1-z_2|\leq \varepsilon/2$. Then \begin{align}\label{eqBreakGMinusG}
\int_{B_r}|G(z_1,x)-G(z_2,x)|\, d\lambda(x)
\nonumber & \leq \int_{B_r\setminus B_\varepsilon(z_1)}|G(z_1,x)-G(z_2,x)|\, d\lambda(x) \\
& \quad + \int_{B_\varepsilon(z_1)}G(z_1,x)d\lambda(x) + \int_{B_\varepsilon(z_1)}G(z_2,x)d\lambda(x). \end{align}
Next we use \rf{eqGreenEstimate} and Theorems \ref{theoLambdaBoundAbove} and \ref{theoMeasureAway}. By decomposing the domain on dyadic annuli, in case $n-2\alpha>0$ we get \begin{align}\label{eqBoundGreenLambda} \int_{B_\varepsilon(z_1)}G(z_1,x)d\lambda(x) \nonumber & \leq \sum_{j\leq0}\int_{A_{2^{j-1}\varepsilon,2^{j}\varepsilon}(z_1)}G(z_1,x)d\lambda(x)
\lesssim \sum_{j\leq0} \lambda(B_{2^j\varepsilon}(z_1))(2^{j-1}\varepsilon)^{2\alpha-n} \lesssim\varepsilon^\alpha \sum_{j\leq0} 2^{j\alpha}. \end{align} In case $n-2\alpha=0$ we obtain $\varepsilon^\alpha\sum_{j\leq0} 2^{j\alpha}\log\left(\frac{r}{2^j\varepsilon}\right)$ on the right-hand side instead, and in case $n-2\alpha<0$ we obtain $\varepsilon^{n-\alpha}r^{2\alpha-n}\sum_{j\leq0} 2^{j(n-\alpha)}$. In every case, fixing $\varepsilon$ small enough this term can be as small as wanted. The same will happen with the last term on the right-hand side of \rf{eqBreakGMinusG}.
On the other hand, by \cite[Theorem 2.3.12]{FabesKenigSerapioni} Green's function is uniformly continuous on the set $\{(z,x)\in B_r\times B_r: |z-x|>\varepsilon \}$ so $|G(z_1,x_1)-G(z_2,x_2)|\leq \delta_\varepsilon(|z_1-z_2|+|x_1-x_2|)$ with $\delta_\varepsilon(t)\xrightarrow{t\to 0} 0$. Thus,
$$\int_{B_r\setminus B_\varepsilon(z_1)}|G(z_1,x)-G(z_2,x)|\, d\lambda(x) \leq \delta_\varepsilon(|z_1-z_2|)\lambda(B_r)\to 0 .$$
Assuming that $|z_1-z_2|$ is small enough, we obtain that $\int_{B_r}|G(z_1,x)-G(z_2,x)|\, d\lambda(x)$ is as small as wanted and the claim follows. \end{proof}
\begin{remark}\label{remUniformRules}
In light of Theorem \ref{theoDimension}, arguing as in \cite[Theorem 1.1]{CaffarelliRoquejoffreSire} we obtain that every minimizer $u$ in a ball $B_r$ with $0\in F(u)$ has uniform $C^\alpha$ character in $B_{r/2}$. By the Caccioppoli inequality (see Section \ref{secCaccioppoli}) we also obtain the same for the ${\mathbf H}^\beta$ norm. Moreover, using \cite[Theorem 1.2]{CaffarelliRoquejoffreSire} we can find interior corkscrew points with constants not depending on these norms. This allows us to remove the {\it a priori} dependence on $\|u\|_{{\mathbf H}^\beta}$ from all of our results above. \end{remark}
\subsection{Lower estimates for the distributional fractional Laplacian} Next we bound the growth of the measure around a free boundary point from below. None of these results will be used in the present paper, but we include them to give a complete picture of the tools under consideration. \begin{theorem}\label{theoMeasureBall} Let $u \in {\mathbf H}^\beta(B_{2r})$ be a minimizer of $\mathcal{J}$ in $B_{2r}$ such that $0\in F(u)$. Then we have $$\lambda(B_r) \geq C r^{n-\alpha}.$$ \end{theorem}
\begin{proof}
Let $\mathcal{L}u := -\nabla\cdot (|y|^\beta \nabla u)$ and let $v$ be the $\mathcal{L}$-harmonic replacement of $u$ in $B_r$ (see \rf{eqHarmonicReplacement}). Let $\widetilde{u}:=v- u$ and consider the Green function $G: B_r\times B_r\to {\mathbb R}$ as in the proof of Theorem \ref{theoDimension}.
Let $0<\kappa<1$ to be fixed later. By \emph{P1}-\emph{P3} in Theorem \ref{theoCRS} there exists a point $z_0\in B_{\kappa r}$ with \begin{equation}\label{eqU} u(z_0)\approx (\kappa r)^\alpha, \end{equation} with constants depending only on $n$ and $\alpha$ by Remark \ref{remUniformRules}. By \emph{P1} there is a constant $c$ such that for every $z\in B(z_0,c\kappa r)$ we have that $u(z)\approx (\kappa r)^\alpha$. Since $\lambda$ is supported on the zero phase of $u$, the ball $B(z_0,c\kappa r)$ is away from its support, and $$\widetilde{u}(z)=\int_{B_r\setminus B(z_0,c\kappa r) } G(z,x)\, d\lambda(x).$$
Using the strong maximum principle (see \cite[Theorem 6.5]{HeinonenKilpelainenMartio}) and \rf{eqGreenEstimate}, for almost every $z\in B(z_0,c\kappa r/2)$ we get \begin{align*} \widetilde{u}(z)
& \leq \lambda(B_r) \sup_{x\in B_r\setminus B(z_0,c\kappa r) } G(z,x) = \lambda(B_r) \sup_{x\in B_{r/4} \setminus B(z_0,c\kappa r) } G(z,x) \\
& \approx \lambda(B_r)\sup_{x\notin B(z_0,c\kappa r) } |x-z|^{2\alpha-n} =\lambda(B_r) (c \kappa r)^{2\alpha-n}. \end{align*}
That is, \begin{equation}\label{eqTildeU} \widetilde{u}(z)
\lesssim \lambda(B_r) (c\kappa r)^{2\alpha-n}. \end{equation}
On the other hand, note that $u$ is continuous. By the Riesz representation theorem, there exists a probability measure $\omega^z_\mathcal{L}$ such that $$v(z)= \int_{\partial B_r} u(x) d\omega^z_\mathcal{L}(x).$$
We can choose $r$ so that $\partial B_r$ intersects a big part of a corkscrew ball, i.e., assume that there exists a point $\xi_0\in \partial B_r'$ which is the center of a ball $B'(\xi_0, cr)$ where $u$ has positive values. This can be done by the interior corkscrew condition, with all the constants involved depending only on $n$ and $\alpha$. Then, changing the constant if necessary, all points $\xi\in B(\xi_0, cr)$ satisfy that $u(\xi)\geq C r^\alpha$ by the non-degeneracy condition and the optimal regularity. Call $U:=\partial B_r\cap B(\xi_0, cr)$. Then $$v(z)\gtrsim r^\alpha \omega^z_\mathcal{L}(U).$$ But $\omega^z_\mathcal{L}(U)$ is bounded below by a constant by \cite[Lemma 11.21]{HeinonenKilpelainenMartio} and the Harnack inequality (use a convenient Harnack chain). All in all, we have that \begin{equation}\label{eqV}
v(z)\gtrsim r^\alpha. \end{equation}
Combining \rf{eqTildeU}, \rf{eqU} and \rf{eqV} and choosing $\kappa$ small enough, depending in $n$ and $\alpha$, we get $$ \lambda(B_r) \gtrsim \frac{\widetilde{u}(z_0)}{(c \kappa r)^{2\alpha-n}}\geq \frac{ Cr^\alpha - C' (\kappa r)^\alpha }{(c \kappa r)^{2\alpha-n}} \geq C_{n,\alpha} r^{n-\alpha} ,$$ for $\kappa$ small enough.
In case $n-2\alpha=0$, that is for $n=1$ and $\alpha=1/2$, using similar changes as in the proof of Theorem \ref{theoDimension} we get $\widetilde{u}(z)\lesssim \lambda(B_r)\sup_{x\notin B(z_0,c\kappa r)} \log\left(\frac{r}{|x-z|}\right) \approx \lambda(B_r)|\log \kappa|$ instead of \rf{eqTildeU}. In case $n-2\alpha<0$, the proof is even easier than before. \end{proof}
\begin{remark}\label{remDimension} Theorem \ref{theoMeasureBall} implies that the $(n-\alpha)$-Hausdorff measure of the free boundary is locally finite. This does not suffice to show finite perimeter of the positive phase and, therefore, we had to use the approach in Section \ref{s:finperimeter}. \end{remark}
The following theorem summarizes the information that we have gathered so far about the measure $\lambda$.
\begin{theorem}\label{theoMeasureComplete} If $u \in {\mathbf H}^\beta_{\loc}(\Omega)$ is a minimizer of $\mathcal{J}$ in $\Omega$, then the measure $\lambda$ is absolutely continuous with respect to the Lebesgue measure in $\Omega'(u)$. Moreover, given $x_0\in F(u)$ and $r>0$ such that $B_{2r}(x_0)\subset \Omega$, then \begin{equation}\label{eqMeasureBallNice} \lambda(B_r(x_0)) \approx r^{n-\alpha}, \end{equation}
and for almost every $x\in B_{r}'(x_0)$ we have that
$$\frac{d\lambda}{dm}(x)=2\lim_{y\to0}|y|^\beta u_y(x',y)\approx \chi_{\Omega_0(u)}(x) {\rm dist}(x,F(u))^{-\alpha},$$ with constants depending only on $n$ and $\alpha$. \end{theorem}
\section{Rectifiability of the singular set}\label{s:Rect}
In this section we use the Rectifiable-Reifenberg and quantitative stratification framework of Naber-Valtorta \cite{NaberValtortaRectifiable} to prove Hausdorff measure and structure results for the singular set. Recall that $k^*_\alpha$ is the first dimension in which there exists non-trivial $\alpha$-homogeneous global minimizers to \eqref{eqLocalizedFunctional} defined in Section \ref{s:finperimeter}.
\begin{theorem}\label{theoGlobalRectifiability} Let $u\in {\mathbf H}^\beta_{\loc}(\Omega)$ be a minimizer of \rf{eqLocalizedFunctional} in a domain $\Omega$. Then $\Sigma(u)$ is $(n-k^*_\alpha)$-rectifiable and for every $D\subset\subset \Omega$, we have $$\mathcal{H}^{n-k^*_\alpha}(\Sigma(u)\cap D)\leq C_{n,\alpha, {\rm dist}(D,\partial\Omega)}.$$ \end{theorem}
Part of the power of this framework is that it is very general. One needs certain compactness properties on the minimizers and a connection between the drop in the monotonicity formula and the local flatness of the singular set (see Theorem \ref{theoL2Estimate} below). To avoid redundancy and highlight the original contributions of this article, we omit many details here and try to focus on the estimates needed to apply this framework to minimizers of \eqref{eqLocalizedFunctional}. Whenever we omit details we will refer the interested reader to the relevant parts of \cite{EdelenEngelstein}.
The key first step is to introduce the appropriate formulation of quantitative stratification. First introduced by Cheeger and Naber \cite{CheegerNaber} in the context of manifolds with Ricci curvature bounded from below, this is a way to quantify the intuitive fact that $F(u)$ should ``look" $(n-k^*_{\alpha})$-dimensional near a point $x_0 \in F(u)$ at which the blow-ups have $(n-k^*_\alpha)$-linearly independent translational symmetries.
\subsection{Quantitative stratification for minimizers to $\mathcal J$}
We have seen in Section \ref{secReduction} that homogeneous functions have linear spaces of translational symmetry. Here we want to quantify (both in terms of size and stability) how far a function is from having no more than $k$ directions of translational symmetry.
\begin{definition} We write $V^k$ for the collection of linear $k$-dimensional subspaces of ${\mathbb R}^n$. A function $u$ is said to be $k$-symmetric if it is $\alpha$-homogeneous with respect to some point, and there exists a $L\in V^k$ so that $$u(x+v)=u(x), \quad\quad \mbox{for every } v\in L.$$ A function $u$ is said to be $(k,\epsilon)$-symmetric in a ball $B$ if for some $k$-symmetric $\widetilde{u}$ we have
$$r(B)^{-2-n} \int_{B} |y|^{\beta} |u-\widetilde{u}|^2 dy <\epsilon.$$
\end{definition}
Next we define the $k$-stratum $S^k(u)$, the $(k,\epsilon)$-stratum $S^k_\epsilon(u)$ and the $(k,\epsilon,r)$-stratum $S^k_{\epsilon,r}(u)$. A key insight here is to define these strata by the blow-ups having $k$ or fewer symmetries as opposed to exactly $k$ symmetries. \begin{definition} Let $0\leq k\leq n$, $0<\varepsilon<\infty$ and $0<r<{\rm dist}(x,\partial\Omega)$, let $u$ be a continuous function in $\Omega$ and let $x\in F(u)$. We say that: \begin{itemize} \item $x\in S^k(u)$ if $u$ has no $(k+1)$-symmetric blow-ups at $x$. \item $x\in S^k_\epsilon(u)$ if $u$ is not $(k+1,\epsilon)$-symmetric in $B_s(x)$ for every $0<s\leq \min\{1,{\rm dist}(x,\partial\Omega)\}$. \item $x\in S^k_{\epsilon,r}(u)$ if $u$ is not $(k+1,\epsilon)$-symmetric in $B_s(x)$ for every $r\leq s\leq \min\{1,{\rm dist}(x,\partial\Omega)\}$. \end{itemize} If it is clear from the context we will omit $u$ from the notation. \end{definition}
We now detail some standard properties of the strata defined above and how they interact with the free boundary $F(u)$. While the proofs are mostly standard, we give the details as the scaling associated to the problem \eqref{eqLocalizedFunctional} adds some technical difficulties. This proof also provides a blueprint for fleshing out the details in Sections \ref{secToolsBad} and \ref{secToolsGood}.
\begin{lemma}\label{lemProperties} Let $0\leq j \leq k\leq n$, $0< \varepsilon \leq\tau<\infty$, $0<r\leq s<{\rm dist}(x,\partial\Omega)$, and let $u\in {\mathbf H}^\beta_{\loc}(\Omega)$ be a minimizer in $\Omega$. Then: \begin{enumerate} \item $S^0\subset S^1\subset \cdots \subset S^{n-1} = S^n= F(u)$. Moreover, for the reduced boundary, we have that $F_{red}(u) \subset S^{n-1}\setminus S^{n-2}$ and $\Sigma (u) \subset S^{n-k^*_\alpha}$. \item We have $S^j_{\tau}\subset S^k_\epsilon \subset S^k$ and, moreover, $\displaystyle S^k=\bigcup_{\epsilon>0} S^k_\epsilon$. \item Also $S^j_{\tau}\subset S^j_{\tau, r}\subset S^k_{\epsilon, s}$ and, moreover, $\displaystyle S^k_\epsilon= \bigcap_{r>0} S^k_{\epsilon, r}$.
\item The sets $S^k_\epsilon$ are closed, in both $x$ and $u$: if $u_i\xrightarrow{L^2_{\rm loc}(\Omega;|y|^{\beta})} u$ and $x_i\to x$ with $x_i\in S^k_\epsilon(u_i)$, then $x\in S^k_\epsilon(u)$.
\item If $u_i \xrightarrow{L^2_{\rm loc}(\Omega; |y|^{\beta})} u$, $\epsilon_i\to 0$, and $u_i$ are $(k,\epsilon_i)$-symmetric in $B_1$, then $u$ is $k$-symmetric in $B_1$. \end{enumerate} \end{lemma} \begin{proof} \emph{1}. The inclusions $S^k\subset S^{k+1}$ of the first property are trivial. The last equalities are consequences of the non-degeneracy. The fact that $F_{red}(u) \cap S^{n-2} =\emptyset$ can be deduced from the Hausdorff convergence of the free boundaries described in Lemma \ref{lemCompactness} and Theorem \ref{theoImprovement}. Finally, $\Sigma (u) \subset S^{n-k^*_\alpha}$ is a consequence of Lemmas \ref{lemDimensionReduction} and \ref{lemTrivialIsTheChosen}.
\emph{2}. The inclusions $S^j_{\tau}\subset S^k_\epsilon$ of the second property come from the definitions: if $x\notin S^k_\epsilon$ then there exist a ball $B\subset \Omega$ centered at $x$ and a $(k+1)$-symmetric $\widetilde{u}$ so that $r(B)^{-2-n} \int_{B} |y|^{\beta} |u-\widetilde{u}|^2 dy <\epsilon\leq \tau.$ But $\widetilde{u}$ is also $(j+1)$-symmetric. Thus, $x\notin S^j_\tau$.
The fact that $S^k_\epsilon \subset S^k$ is a consequence of the uniform convergence on Lemma \ref{lemCompactness}: if $x\notin S^k$, then $u$ has a $(k+1)$-symmetric blow-up sequence $u_i\to u_0$ at $x$ converging uniformly. Thus, \begin{align*}
\int_{B_{\rho_i}(x)} |y|^{\beta} \left|u(x)- \rho_i^\alpha u_0\left(\frac{x-x_0}{ \rho_i}\right)\right|^2 dx
& = \rho_i^{{\beta}+2\alpha+n+1} \int_{B_1(x)} |y|^{\beta} \left|\frac{u\left(x_0+ \rho_i x\right)}{\rho_i^\alpha}-u_0(x)\right|^2 dx\\
& \leq \rho_i^{n+2} \omega(B_1) \norm{u_i-u_0}_{L^\infty}. \end{align*} That is,
$$ \rho_i^{-n-2} \int_{B_{\rho_i}(x)} |y|^{\beta} \left|u(x)- r_i^\alpha u_0\left(\frac{x-x_0}{ \rho_i}\right)\right|^2 dx \xrightarrow{i\to\infty} 0,$$ and therefore, for every $\varepsilon$ there exists a ball small enough so that $u$ is $(k+1,\varepsilon)$-symmetric in it. In particular $S^k \supset \bigcup_{\epsilon>0} S^k_\epsilon$.
To see the converse, assume that $x\notin \bigcup_\epsilon S^k_\epsilon$. Then for every $i\in{\mathbb N}$ there exist a $(k+1)$-symmetric function $\widetilde u_i$, invariant with respect to $L_i\in V^{k+1}$ and $r_i<\min\{1,{\rm dist}(x,\partial\Omega)\}$ such that
$$\frac{1}{r_i^{n+2}} \int_{B_{r_i}} |y|^{\beta} \left| u(x)-\widetilde{u}_i(x)\right|^2 \, dx < \frac1i.$$ In the case when $r_i$ stays away from zero, since $r_i<1$, we can take a subsequence converging to $r_0\in(0,1)$, and one can see that $u$ is $(k+1)$-symmetric in the ball $B_{r_0}(x_0)$. Otherwise,
consider $u_i:=\frac{u\left(x_0+ r_i x\right)}{r_i^\alpha}$ and $\widetilde{u}_{i,i}=\frac{\widetilde u_i\left(x_0+ r_i x\right)}{r_i^\alpha}$. Taking subsequences, we can assume that $L_i\to L_0$ locally in the Hausdorff distance, and that $u_i \to u_0$ locally uniformly. One can check also using the H\"older character of $u$ that $\{\widetilde{u}_{i,i}\}$ is uniformly bounded in $L^2( B; |y|^{\beta})$, so taking subsequences again, we can assume the existence of $\widetilde{u}_{0}$ so that
$\widetilde{u}_{i,i}\to \widetilde{u}_{0}$ in $L^2( B; |y|^{\beta})$. This function will be $(k+1)$-symmetric, being invariant in the directions of $L_0$. By the triangle inequality we get
$$\int_{B_1} |y|^{\beta} |u_0- \widetilde{u}_0|^2 \, dx \lesssim \int_{B_1} |y|^{\beta} |u_0- u_i|^2 \, dx +\int_{B_1} |y|^{\beta} |u_i- \widetilde{u}_{i,i}|^2 \, dx +\int_{B_1} |y|^{\beta} |\widetilde{u}_{i,i}- \widetilde{u}_0|^2 \, dx .$$ The first and the last integrals converge to zero by our choice of the subsequence. For the middle term just change variables as before:
$$\int_{B_1} |y|^{\beta} |u_i- \widetilde{u}_{i,i}|^2 \, dx = \frac{1}{r_i^{n+2}} \int_{B_{r_i}} |y|^{\beta} \left| u(x)-\widetilde{u}_i(x)\right|^2 \, dx\to 0.$$ Thus we have that $u_0=\widetilde{u_0}$ and, therefore, $x\notin S_k$.
\emph{3}. The inclusions $S^j_{\tau}\subset S^j_{\tau, r}\subset S^k_{\epsilon, s}$ of the third property come from the definitions and thus, $S^k_\epsilon \subset \bigcap_{r>0} S^k_{\epsilon, r}$. The converse implication is also trivial.
\emph{4}. The closedness is obtained by a contradiction argument again. It is straightforward but we write it here for the sake of completeness.
Assume by contradiction that $x\notin S^k_\epsilon(u)$. Then there exist a $(k+1)$-symmetric function $\widetilde{u}$ and a radius $r$ such that
$$\epsilon_0:=\frac{1}{r^{n+2}} \int_{B_{r}(x)} |y|^{\beta} \left| u(x)-\widetilde{u}(x)\right|^2 \, dx <\epsilon.$$ Let $\tau<1$ to be fixed and consider $i_0\in {\mathbb N}$ so that $B_{\tau r}(x_i)\subset B_r(x)$ for every $i\geq i_0$. By the triangle inequality
$$\frac{1}{(\tau r)^{n+2}} \int_{B_{\tau r}(x_i)} |y|^{\beta} \left| u_i(x)-\widetilde{u}(x)\right|^2 \, dx \leq \frac{1}{(\tau r)^{n+2}} \norm{u_i-u} _{L^2(B_{\tau r}(x_i);|y|^{\beta})}^2 + \frac{\epsilon_0}{\tau^{n+2}}.$$
We define $\tau$ so that $\frac{\epsilon_0}{\tau^{n+2}}=\frac{\epsilon+\epsilon_0}{2}$. Choose $i_0$ big enough so that every $i\geq i_0$ satisfies that $\norm{u_i-u} _{L^2(B_{\tau r}(x_i);|y|^{\beta})}^2 < (\tau r)^{n+2} \frac{\epsilon-\epsilon_0}{2}$. Then $x_i\notin S^k_\epsilon(u_i)$, contradicting the hypothesis.
\emph{5}. Assume that $\widetilde u_i$ is invariant with respect to $L_i\in V^{k+1}$ and
$$\int|y|^{\beta} |u_i-\widetilde u_i|^2\leq \epsilon_i.$$ Consider a subsequence $\{u_i\}$ so that the varieties $L_i\to L$ locally in the Hausdorff distance. Using the triangle inequality as in \emph{4} it follows that $u$ is $(k,\delta_i)$-symmetric with $\delta_i\to 0$.
\end{proof}
\begin{proposition}\label{propoSigma} There exists $\epsilon(n,\alpha) > 0$ such that if $u\in {\mathbf H}^\beta_{\loc}(\Omega)$ is a minimizer of $\mathcal{J}$ in a domain $\Omega\subset {\mathbb R}^{n+1}$, then $\Sigma(u)\subset S^{n-k^*_\alpha}_\epsilon(u)$. \end{proposition} \begin{proof} It is enough to show that if $u$ is a minimizer of $\mathcal{J}$ in $B_2(0)$, then $\Sigma(u)\cap B_1(0)\subset S^{n-k^*_\alpha}_\epsilon(u)$.
By contradiction, let us assume that there is a sequence of positive numbers $\epsilon_i\xrightarrow{i\to\infty} 0$, functions $u_i$ minimizing $\mathcal{J}$ in $B_2(0)$ and $x_i\in \Sigma (u_i)\cap B_1(0)$, $r_i \in (0,1]$, with $u_i$ being $(n-k^*_\alpha+1, \epsilon_i)$-symmetric in $B_{r_i}(x_i)$, and let $L_i$ be an $(n-k^*_\alpha+1)$-dimensional subspace that leaves invariant one of the admissible $(n-k^*_\alpha+1)$-symmetric approximants. By rescaling we can assume that $r_i=1$.
Passing to a subsequence we can assume that $L_i \to L_0\in V^{n-k^*_\alpha+1}$ locally in the Hausdorff distance and $x_i\to x_0$. By the compactness results in Lemma \ref{lemCompactness} we have a uniform limit $u_0$ which is a minimizer as well, and it is $(n-k^*_\alpha+1)$-symmetric with invariant manifold $L_0$. By Lemma \ref{lemConeBlowup} any blow-up $u_{0,0}$ at $x_0$ will be $(n-k^*_\alpha+1)$-symmetric as well. Applying Lemma \ref{lemDimensionReduction} $(n-k^*_\alpha+1)$ times we find that the restriction of $u_{0,0}$ to the orthogonal manifold $L_0^{\perp}$ is a $(k^*_\alpha-1)$-dimensional minimal cone which, by Lemma \ref{lemTrivialIsTheChosen} is the trivial solution, and so is $u_{0,0}$. Thus, $x_0$ is a regular point for $u_0$.
On the other hand, the Hausdorff convergence of Lemma \ref{lemCompactness} together with the improvement of flatness of Theorem \ref{theoImprovement} imply that for $i$ big enough $x_i\in F_{\rm red}(u_i)$, reaching a contradiction. \end{proof}
\subsection{The Refined Covering Theorem}
Our estimates on the size and structure of the singular set $\Sigma(u)$ come from similar results concerning the $S^k_{\epsilon}(u)$. In particular, we prove the following covering result:
\begin{theorem}\label{theoMain} Let $u\in {\mathbf H}^\beta(B_5)$ be a minimizer to \rf{eqLocalizedFunctional} in $B_5$ with $0\in F(u)$. For given real numbers $\epsilon>0$, $0<r\leq 1$ and every natural number $1\leq k\leq n-1$, we can find a collection of balls $\{B_r(x_i)\}_{i=1}^N$ with $N\leq C_{n,\alpha,\epsilon} r^{-k}$ such that $$S^k_{\epsilon,r}(u)\cap B_1\subset \bigcup_i B_r(x_i).$$
In particular, $|B'_r(S^k_{\epsilon,r}\cap B_1)|\leq C_{n,\alpha,\epsilon} r^{n-k}$ for every $0<r\leq 1$ and $$\mathcal{H}^k(S^k_{\epsilon}(u) \cap B_1)\leq C_{n,\alpha,\epsilon}.$$ \end{theorem}
From Proposition \ref{propoSigma} and Theorem \ref{theoMain} we can conclude the following corollary which comprises the second part of Theorem \ref{theoGlobalRectifiability} above.
\begin{corollary}\label{coroSingular} If $u\in {\mathbf H}^\beta(B_5)$ is a minimizer to \rf{eqLocalizedFunctional} in $B_5$ with $0\in F(u)$, then $\Sigma(u)$ is $(n-k^*_\alpha)$-rectifiable and for every $0<r\leq 1$ we have
$$|B_r (\Sigma(u)\cap B_1)|\leq C_{n,\alpha} r^{k^*_\alpha}.$$ In particular, $$\mathcal{H}^{n-k^*_\alpha}(\Sigma(u)\cap B_1)\leq C_{n,\alpha}.$$ \end{corollary}
Rectifiability is encoded in the following result. We omit the details of proof here but it is a consequence of the packing result above, the Rectifiable-Reifenberg theorem of \cite{NaberValtortaRectifiable} and Theorem \ref{theoL2Estimate} below. For more details see Sections 2 and 8 of \cite{EdelenEngelstein} (particularly Theorem 2.2 in the former and the proof of Theorem 1.12 in the latter).
\begin{theorem}\label{theoMain2} Let $u$ be a non-negative, even minimizer to \rf{eqLocalizedFunctional} in a domain $\Omega$. Then $S^k_\epsilon(u)$ is $k$-rectifiable for every $\epsilon$ and, hence, each stratum $S^k(u)$ is $k$-rectifiable as well. \end{theorem}
The proof of Theorem \ref{theoMain} follows from inductively applying the following, slightly more technical, packing result (for details see Section 4 of \cite{EdelenEngelstein}).
\begin{theorem}\label{theoTreeConstruction} Let $\epsilon>0$. There exists $\eta (n,\alpha, \epsilon)$ such that, for every minimizer $u\in {\mathbf H}^\beta(B_5)$ of $\mathcal{J}$ in $B_5$ with $0\in F(u)$ and $0<R<1/10$, there is a finite collection $\mathcal{U}$ of balls $B$ with center $x_B \in S^k_{\epsilon, \eta R}$ and radius $R \leq r_B \leq 1/10$ which satisfy the following properties: \begin{enumerate}[A)] \item Covering control: $$ S^k_{\epsilon, \eta R} \cap B_1 \subset \bigcup_{B\in \mathcal{U}} B.$$ \item Energy drop: For every $B\in \mathcal{U}$, $$\mbox{either}\quad r_B=R, \quad\quad \mbox{or}\quad \sup_{2B} \Psi^u_{2r_B}\leq \sup_{B_2}\Psi^u_2 -\eta.$$ \item Packing: $$\sum_{B\in \mathcal{U} } r_B^k \leq c(n, \alpha, \epsilon).$$ \end{enumerate} \end{theorem}
We construct the balls of Theorem \ref{theoTreeConstruction} using a ``stopping time" or ``good ball/bad ball" argument. Much of this argument uses harmonic analysis and geometric measure theory and is completely independent of the original problem \eqref{eqLocalizedFunctional}. However, there are a few places in which we need to connect the behavior of minimizers to the geometric structure of the singular set. Here we will sketch the ``good ball/bad ball" argument, taking for granted the estimates needed to apply this argument to our functional. In the next few subsection we will provide these estimates. For more details on the construction itself we refer the reader to Section 7 in \cite{EdelenEngelstein}.
\noindent {\bf Outline of the Construction in Theorem \ref{theoTreeConstruction}} To find this covering we define good and bad balls as follows: imagine our ball, $B$, has radius 1. We say that $B$ is a \emph{good ball}, if at every point in $x\in S^k_\varepsilon(u)\cap B$ the monotone quantity centered at that point at some small scale, $\rho$, is not much smaller than the monotone quantity on ball $B$ (we say these points have ``small density drop"). A ball $B$ is a \emph{bad ball} if all the points in $S^k_\varepsilon(u)\cap B$ with small density drop are contained in a small neighborhood of a $(k-1)$-plane. This good/bad is a dichotomy follows from Theorem \ref{theoKeyDichotomy} in Section \ref{secToolsBad}.
In a good ball of radius $r$ we cover $S^k_\varepsilon(u)$ with balls of radius $\rho r$ iterating the construction until we find a bad ball or until the radius of the ball becomes very small. In a bad ball, we cover $S^k_\varepsilon(u)$ away from the $(k-1)$-plane without much care. Close to the $(k-1)$-plane we cover $S^k_\varepsilon(u)$ with balls of radius $\rho r$ iterating the construction until we reach a good ball or until the radius of the ball becomes very small.
Inside long strings of good balls, the packing estimates follow from powerful tools in geometric measure theory (see Theorem \ref{theoDiscreteReifenberg} below) and the connection between the drop in monotonicity and the local flatness of the singular strata (see Theorem \ref{theoL2Estimate} below). We give more details in Section \ref{secToolsGood}.
Inside long strings of bad balls each of which is near the $(k-1)$-plane of the previous bad ball, we have even better packing estimates than expected (as we are effectively well approximated by planes which are lower dimensional). This leaves only points which are in many bad balls and in most of those balls they are far away from the $(k-1)$-plane. However, at these points the monotone quantity drops a definite amount many times, which contradicts either finiteness or monotonicity. This implies that the points and scales inside the bad balls which are not close to the $(k-1)$-plane form a negligible set (the technical term is a Carleson set). We give more information about the bad balls in Section \ref{secToolsBad}.
\subsection{Tools for bad balls: key dichotomy}\label{secToolsBad}
\begin{theorem}[Key dichotomy]\label{theoKeyDichotomy} Let $\epsilon, \rho, \gamma, \eta'>0$ be fixed numbers with $\rho \gamma<2$. There exists an $\eta_0 (n, \alpha, \epsilon, \rho, \gamma, \eta')<\rho/100$ such that for every $\eta \leq \eta_0$, every $r>0$, every $E>0$ and every minimizer $u\in {\mathbf H}^\beta(B_{4r})$ of $\mathcal{J}$ in $B_{4r}$ with $0\in F(u)$ and $\sup_{B_r}\Psi^u_{2r}\leq E$, then either \begin{itemize} \item $\Psi_{\gamma \rho r}^u\geq E-\eta'$ on $S^k_{\epsilon,\eta r}\cap B_r$, or \item there exists $\ell \in L^{k-1}$ so that $\{x\in B_r: \Psi^u_{2\eta r}(x)\geq E-\eta\}\subset B_{\rho r}(\ell)$. \end{itemize} \end{theorem}
The key dichotomy is a direct consequence of the Lemma \ref{lemPreKey} below. The core idea is to make effective the following assertion: if $u$ is $k$-symmetric, then along the invariant manifold the Allen-Weiss density is constant, and every point away from the manifold will have $(k+1)$-symmetric blow-ups by Lemma \ref{lemConeBlowup}.
\begin{lemma}\label{lemPreKey} Let $\epsilon, \rho, \gamma, \eta'>0$ be fixed numbers with $\gamma\rho <2$. There exist $\eta_0, \theta>0$ such that for every $\eta<\eta_0$, every $E>0$ and every minimizer $u$ of $\mathcal{J}$ in $B_4$ with $0\in F(u)$ and $\sup_{B_1}\Psi^u_2\leq E$, if there exist $w_0,\dots, w_k\in B_1$ and affine manifolds $L^i:=\langle w_0,\dots, w_i\rangle\in V^{i}$ with $$w_i\notin B_\rho(L^{i-1}),\quad\quad \mbox{ and } \quad\quad \Psi^u_{2\eta}(w_i) \geq E-\eta \quad\quad\mbox{for every }i\in\{0,\cdots,k\},$$ then, \begin{equation}\label{eqSmallJump} \Psi^u_{\gamma\rho}(x)\geq E-\eta' \quad\quad \mbox{on }B_\theta(L^k)\cap B_1 \end{equation} and \begin{equation}\label{eqCloseStrata} S^k_{\epsilon,\eta}\cap B_1\subset B_\theta(L^k) \end{equation} \end{lemma}
The proof follows (with only minor modifications) the proof in \cite[Lemma 3.3]{EdelenEngelstein}. We end this subsection by formally defining the good/bad balls alluded to above:
\begin{definition}\label{defGoodBad} Let $x\in B_2$, $0< R<r<2$ and $u$ be a minimizer to $\mathcal J$ in $B_5$. We say that the ball $B_r(x)$ is \emph{good} if $$\Psi^u_{\gamma\rho r}\geq E-\eta' \quad\quad \mbox{on } S^k_{\epsilon, \eta R}\cap B_r(x), $$ and otherwise we say that $B_r(x)$ is \emph{bad}. \end{definition} By Theorem \ref{theoKeyDichotomy} in any bad ball $B$ there exists an affine $(k-1)$-manifold $\ell_B$ with \begin{equation}\label{eqBadEstimate} \{w\in B: \Psi^u_{2\eta r}(w)\geq E-\eta\}\subset B_{\rho r}(\ell_B^{k-1}). \end{equation}
\subsection{Tools for good balls: packing estimates and GMT}\label{secToolsGood}
In this section we control the local flatness of the singular strata by the drop in monotonicity. To do this we introduce a key tool from geometric measure theory which estimates the flatness of a set. Given a Borel measure $\mu$, a point $x$ and a radius $r$, the beta coefficient is defined as follows: \begin{equation}\label{eqDefBeta} \beta^k_{\mu,2}(B_r(x))^2:=\beta^k_{\mu,2}(x,r)^2 =\inf_{L \in V^k_a} \frac{1}{r^{k}} \int_{B_r(x)} \frac{{\rm dist}(z,L)^2}{r^2}\, d\mu (z) \end{equation} where $V^k_a$ stands for the collection of $k$-dimensional affine sets of ${\mathbb R}^n$. The beta coefficients are meant to measure in a scale invariant way how far is a measure from being flat, in this case in the $L^2$ distance, although other $L^p$ versions have been used in the literature for $1\leq p\leq\infty$ quite often, dating back to \cite{JonesTraveling} (for the $L^\infty$ version) and David-Semmes \cite{DavidSemmes} (for the $L^p$ version).
If we control the size of the $\beta^k$'s we can conclude size and structure estimates on the measure $\mu$. The following theorem says exactly this and represents a major technical achievement. It differs (importantly) from prior work in this area by the lack of {\it a priori} assumptions on the upper or lower densities of the measure involved.
\begin{theorem}[Discrete-Reifenberg Theorem, see {\cite[Theorem 3.4]{NaberValtortaRectifiable}}]\label{theoDiscreteReifenberg} Let $\{B_{r_q}(q)\}_q$ be a collection of disjoint balls, with $q\in B_1(0)$ and $0<r_q\leq1$, and let $\mu$ be the packing measure $\mu:=\sum_q r_q^k\delta_q$, where $\delta_q$ stands for the Dirac delta at $q$. There exist constants $\tau_{DR}, C_{DR}>0$ depending only on the dimension such that if $$\int_0^{2r}\int_{B_r(x)} \beta^k_{\mu,2}(z,s)^2 \, d\mu (z) \frac{ds}{s} \leq \tau_{DR} r^k \quad\quad \mbox{for every } x\in B_1(0), \, 0<r\leq 1,$$ then $$\mu(B_1(0))=\sum_q r_q^k\leq C_{DR}.$$ \end{theorem}
To obtain the packing estimates required for the Discrete-Reifenberg Theorem, we need to control the beta coefficients. The key estimate of this entire framework lies in the following theorem, which shows the drop in monotonicity at a given point and a given scale controls the beta coefficient at a comparable scale. \begin{theorem}\label{theoL2Estimate} Let $\epsilon>0$ be given. There exist $\delta(n,\alpha, \epsilon)$ and $c(n, \alpha, \epsilon)$ such that for every $u\in {\mathbf H}^\beta(B_{5r})$ minimizing $\mathcal{J}$ in $B_{5r}(x)$ with $x\in F(u)$ and \begin{equation}\label{eq0DeltaNonK1Epsilon} \begin{cases} u \mbox{ is }(0,\delta)\mbox{-symmetric in } B_{4r}(x)\\ u \mbox{ is not }(k+1,\epsilon)\mbox{-symmetric in } B_{4r}(x), \end{cases} \end{equation} and every Borel measure $\mu$, we have that \begin{equation}\label{eqL2Estimate} \beta^k_{\mu,2}(B_r(x))^2
\leq \frac{ c(n,\alpha, \epsilon)}{r^k} \int_{B_r(x)} \left( \Psi_{4r}^u (w)-\Psi_{r}^u (w)\right) \, d\mu (w). \end{equation}
\end{theorem}
We follow the proof of \cite[Theorem 5.1]{EdelenEngelstein} closely. First the authors give an explicit formula for the beta coefficients.
\begin{lemma}\label{lemExplicitBetas} Let $X$ be the center of mass of a Borel measure $\mu$ on $B=B_r(x)$. Let $\{\lambda_i\}_{i=1}^n$ be the decreasing sequence of eigenvalues of the non-negative bilinear form $$Q(v,w):= \fint_{B}(v\cdot (z-X)) (w\cdot (z-X))\, d\mu (z),$$ and let $\{v_i\}_{i=1}^n$ be a corresponding orthonormal sequence of eigenvectors, that is $v_i\cdot v_j=\delta^{ij}$ and $Q(v_i , v)=\lambda_i v_i\cdot v$. Then $$\beta_{\mu,2}^k(B)^2= \frac{1}{r^{k}} \int_{B} \frac{{\rm dist}(z,L^k)^2}{r^2}\, d\mu (z) = \frac{\mu(B)}{r^{k}}\frac{(\lambda_{k+1} + \dots + \lambda_n)}{r^2},$$ where $L^k := X+ \mathrm{span}\langle v_1,\dots, v_k \rangle$. \end{lemma}
Next we find a relation between the eigenvalues of $Q$ and Allen-Weiss' energy.
\begin{lemma} \label{lemLambda2} Under the hypothesis of Lemma \ref{lemExplicitBetas}, for every $u\in {\mathbf H}^\beta(B_{5r})$ minimizing $\mathcal{J}$ in $B_{5r}(x)$ and every $i\leq n$, we have that \begin{equation}\label{eqEigenAndMonotonicity}
\lambda_i \frac{2}{r^{n+2}}\int_{A_{2r,3r}(x)} |y|^{\beta} (v_i\cdot Du(z))^2\, dz\leq C \fint_{B_r(x)} \left( \Psi_{4r}^u (w)-\Psi_{r}^u (w)\right) \, d\mu (w) . \end{equation} \end{lemma} \begin{proof} The argument follows as in \cite[(18) and below]{EdelenEngelstein}. In formula \emph{(18)} one needs to change $u(z)$ by $\alpha u(z)$, which can be done with exactly the same argument. \end{proof}
Finally, using compactness, we bound the left-hand side of \rf{lemLambda2} from below. \begin{lemma}\label{lemViBelow} Let $\epsilon>0$ be given. There exists a $\delta(n,\alpha, \epsilon)$ and $c(n, \alpha, \epsilon)$ such that, for every orthonormal basis $\{v_i\}_{i=1}^n$ and every $u\in {\mathbf H}^\beta(B_{5r})$ minimizing $\mathcal{J}$ in $B_{5 r}(x)$ with $x\in F(u)$ and satisfying \rf{eq0DeltaNonK1Epsilon}, we have that \begin{equation}\label{eqViBelow}
\frac{1}{c(n, \alpha, \epsilon)} \leq r^{-n} \int_{A_{2r,3r}(x)} |y|^{\beta} \sum_{i=1}^{k+1} (v_i\cdot Du(z))^2\, dz.\end{equation} \end{lemma} \begin{proof} The proof follows that of \cite[(19)]{EdelenEngelstein} and we omit it.
\end{proof}
\begin{proof}[Proof of Theorem \ref{theoL2Estimate}] By Lemmas \ref{lemExplicitBetas}, \ref{lemViBelow} and \ref{lemLambda2} we get that \begin{align*} \beta^k_{\mu,2}(B)^2
&\leq \frac{\mu(B)}{r^{k+2}}(n-k) \lambda_{k+1} \\
& \leq \frac{\mu(B)}{r^k} (n-k) c(n,\alpha, \epsilon) \sum_{i=1}^{k+1}\frac{\lambda_i}{r^{n+2}}\int_{A_{2r, 3r}(x)} |y|^{\beta} (v_i\cdot Du(z))^2 \, dz\\
&\leq \frac{ c(n,\alpha, \epsilon)}{r^k} \int_{B_r(x)} \left( \Psi_{4r}^u (w)-\Psi_{r}^u (w)\right) \, d\mu (w). \end{align*}
\end{proof}
\renewcommand{Acknowledgements}{Acknowledgements} \begin{abstract} M.E. was partially supported by an NSF postdoctoral fellowship, NSF DMS 1703306 and by David Jerison's grant DMS 1500771. A.K. acknowledges Financial support from the Spanish Ministry of Economy and Competitiveness, through the Mar\'ia de Maeztu Programme for Units of Excellence in R\&D (MDM-2014-0445). M.P. was funded by the European Research Council under the grant agreement 307179-GFTIPFD. G.S. has received funding from the European Union's Horizon 2020 research and innovation programme under Marie Sk{\l}odowska-Curie grant agreement No 665919. A.K., M.P. and G.S. were also partially funded by 2017-SGR-0395 (Catalonia) and MTM-2016-77635-P (MINECO, Spain). Y.S. is partially supported by the Simons foundation.
M.E. would also like to thank Nick Edelen for many fruitful conversations regarding the quantitative stratification and rectifiable Reifenberg framework.
A.K., M.P., G.S. would like to thank Xavier Cabr\'e, Tom\'as Sanz, Matteo Cozzi, Albert Mas, Maria del Mar Gonz\'alez, Luis Silvestre and Stefano Vita for some conversations around \cite{VitaThesis}. They would also like to thank Mihalis Mourgoglou for some conversations regarding the degenerate elliptic measure. \end{abstract}
\appendix \section*{Appendix} \section{Relation with the nonlocal Bernoulli problem}
As in \cite[Lemma 2.1]{DipierroValdinoci}, we see that the study of minimizers of $\mathcal{J}$ includes the study of minimizers of $J$.
\begin{proposition}\label{propoExtension} If $f$ is a minimizer of $J$ in the unit ball of ${\mathbb R}^n$ then $f*P_y$ is a minimizer of $\mathcal{J}$ in every ball $B$ such that $B'\subset \subset B_1'$.
If $u=f*P_y$ is a minimizer of $\mathcal{J}$, then $f$ is a minimizer for $J$. In particular, if $u$ is a minimizer of $\mathcal{J}$ in every ball, positive outside the hyperplane $\{y=0\}$, and $u(x,y)=\mathcal{O}(|(x,y)|^\alpha)$, then $u|_{{\mathbb R}^n\times\{0\}}$ is a minimizer for $J$ in every ball. \end{proposition}
We follow \cite[Lemma 2.1]{DipierroValdinoci}, that is, we use the following result from \cite[Section 7]{CaffarelliRoquejoffreSavin}. \begin{lemma}[see {\cite[Section 7]{CaffarelliRoquejoffreSavin}}]\label{lemCRS} Let $f,g$ satisfy that $J_0(f,B_1), J_0(g,B_1) < \infty$, and suppose that $f-g$ is compactly supported in $B_1\subset{\mathbb R}^n$. Then we have that \begin{equation*} J_0(g,B_1) - J_0(f,B_1)
= c_{n,\alpha} \inf \int_{\Omega} |y|^\beta (|\nabla v(x,u)|^2 - |\nabla (f*P_y)(x)|^2), \end{equation*} where the infimum is taken among all the symmetric bounded Lipschitz domains $\Omega$ with the property that $\Omega\cap ({\mathbb R}^n\times\{0\})\subset B_1$ and among all symmetric functions $v$ with trace $g$ satisfying that $v-f*P_y$ is compactly supported on $\Omega$. \end{lemma}
\begin{proof}[Proof of Proposition \ref{propoExtension}] Let $f$ be a minimizer of $J$ in the unit ball of ${\mathbb R}^n$ and let $B_r$ be a ball such that $B_r'\subset \subset B_1'$. We want to show that $u:=f*P_y$ is a minimizer of $\mathcal{J}$ in $B_r$.
Let $v:{\mathbb R}^{n+1}\to {\mathbb R}$ so that $v\equiv u$ in ${\mathbb R}^{n+1}\setminus B_r$ and $v \in H^1(\beta, B_r)$. Let $g$ be the trace of $v$ in ${{\mathbb R}^n\times\{0\}}$. By Lemma \ref{lemCRS} we have that \begin{equation}\label{eqQuality}
J_0(g,B_1) - J_0(f,B_1)\leq c_{n,\alpha} \int_{B_{r+\varepsilon}} |y|^\beta (|\nabla v|^2 - |\nabla u|^2) \end{equation} for every $\varepsilon>0$.
In particular, since $g|_{(B')^c }\equiv 0$, $g$ is an admissible competitor for $f$ and $J(f,B_1)\leq J(g,B_1)$, i.e., \begin{align}\label{eqFinestQuality} J_0(g,B_1) - J_0(f,B_1) & \geq -m(\{g>0\}\cap B_1) + m(\{f>0\}\cap B_1)\\ \nonumber & = m(\{u>0\}\cap B_r')-m(\{v>0\}\cap B_r'). \end{align} The proposition follows combining \rf{eqQuality} and \rf{eqFinestQuality} and letting $\varepsilon\to 0$.
The converse follows the same sketch: every global minimizer can be expressed as the Poisson extension of its restriction to the hyperplane by Proposition \ref{propoUniquenessDirichlet} and it is left to the reader. \end{proof}
As a consequence of the previous proposition, all the results that we have proven for minimizers of $\mathcal{J}$ also apply to minimizers of $J$: \begin{corollary} If $u:{\mathbb R}^n\to {\mathbb R}$ is a minimizer to $J$ in $B_2\subset {\mathbb R}^n$ and $0\in F(u)$, then $\norm{u}_{C^\alpha(B_1)}\leq C$, it satisfies the nondegeneracy condition $u(x)\geq C {\rm dist}(x,F(u))^\alpha$ for $x\in B_1$, the positive phase satisfies the corkscrew condition, every blow-up limit is $\alpha$-homogeneous, and the boundary condition in \rf{eqEulerLagrangeFrac} is satisfied at $F_{{\rm red}}(u)$.
Moreover, the positive phase $\{u>0\}\cap B_1$ is a set of finite perimeter, the singular set is an $(n-3)$-rectifiable set, it is discrete whenever $n=3$ and it is empty if $n\leq 2$.
All the constants depend only on $n$ and $\alpha$. \end{corollary}
\section{Uniqueness of extensions}
In Proposition \ref{propoExtension} we have used the following result, included in \cite[Proposition 3.1]{CaffarelliRoquejoffreSire}. Here we provide a proof which is different than the one appearing in \cite{CaffarelliRoquejoffreSire}.
\begin{proposition}\label{propoUniquenessDirichlet}
Let $\alpha\in(0,1)$, $\beta=1-2\alpha$, and set $\mathcal{L}u=-\dive(|y|^{\beta}\nabla u)$ in $\mathbb R^{n+1}$. Suppose that $v:\overline{\mathbb R^{n+1}_+}\to\mathbb R$ is nonnegative outside $\mathbb R^n$, it is a solution to $\mathcal{L}v=0$ in $\mathbb R^{n+1}_+$ with $v(x',0)=0$ for all $x'\in\mathbb R^n$ and $|v(x)|\leq C|x|^{\alpha}$. Then $v\equiv 0$. \end{proposition} \begin{proof}
First, since $|y|^{\beta}$ is $C^{\infty}$ away from the hyperplane $\mathbb R^n$, $v\in C^{\infty}_{\loc}(\mathbb R^{n+1}_+)$. Let now $i\in\{1,\dots n\}$, and set \[ f_m(x)=\frac{v\left(x+\frac{1}{m}e_i\right)-v(x)}{1/m}. \] Let $B_r=B_r(x',0)$ be a ball centered at $(x',0)\in\mathbb R^n \times\{0\}$ with radius $r$, and let $B_{2r}$ be its double ball. Set also $w(x)=w(x',y)=y^{\beta}$ for $y>0$. Since $f_m$ is a solution of $\mathcal{L}f_m=0$ in $B_r^+=B\cap\mathbb R^{n+1}_+$, \cite[Theorem 2.4.3]{FabesKenigSerapioni} shows that \[
\max_{B_r^+}|f_m(x)|\leq C\left(\frac{1}{w(B_{2r}^+)}\int_{B_{2r}^+}|f_m|^2w\right)^{1/2}. \] From convergence of difference quotients (similarly to \cite[Theorem 3, page 277]{Evans}), if $v\in H^1(\beta,B_{2r}^+)$, the last estimate will imply that $f_m$ is uniformly bounded in $B_r^+$ by a constant $C_r$. Therefore, from the boundary Caccioppoli estimate (\cite[(2.4.2)]{FabesKenigSerapioni}) we have that \[
\int_{ B_{r/2}^+}|\nabla f_m|^2w\leq\frac{C}{r^2}\int_{ B_r^+}|f_m|^2w\leq \frac{C}{r^2}\int_{ B_r^+}C_r^2w\leq C_{n,r,w}<\infty, \] hence $\{f_m\}$ is bounded in $H^1(\beta,B_{r/2}^+)$. From weak compactness, a subsequence of $\{f_m\}$ converges to a solution of $\mathcal{L}u=0$ in $B_{r/2}^+$, and since $f_m\to\partial_i v$ pointwise, we obtain that $\partial_iv$ is an $H^1(\beta,B_{r/2}^+)$ solution in $B_{r/2}^+$. Hence $\partial_iv$ is a solution to $\mathcal{L}u=0$ in $\mathbb R^{n+1}_+$.
Now, for $x=(x',y)\in\mathbb R^{n+1}_+$, let $R=|x|$. We distinguish between two cases: $y>R/16$, and $y<R/16$.
In the first case, set $B_R$ to be the ball of radius $R$, centered at $x$. Note then that $B_{R/16}\subseteq\mathbb R^{n+1}_+$. Then, from \cite[Theorem 2.3.1]{FabesKenigSerapioni}, Caccioppoli's estimate and the assumption $|v(x)|\leq C|x|^{\alpha}$, \begin{align*}
|\partial_iv(x)|{^2}&\leq \frac{C}{w(B_{R/32})}\int_{B_{R/32}}|\partial_i v|^2w\leq \frac{C}{w(B_{R/32})}\frac{C}{R^2}\int_{B_{R/16}}|v|^2w\leq \frac{C}{R^2}\frac{w(B_{R/16})}{w(B_{R/32})}\sup_{B_{R/16}}|v|\leq CR^{2\alpha-2}. \end{align*} In the second case, let $B_R$ be the ball centered at $(x',0)$ with radius $R$, and denote $B_R^+=B_R\cap\mathbb R^{N+1}_+$. Then $x\in B_{R/8}^+$, therefore from \cite[Theorem 2.4.3]{FabesKenigSerapioni} and the boundary Caccioppoli estimate, \begin{align*}
|\partial_iv(x)|{^2}&\leq {\frac{C}{w(B_{R/8}^+)}\int_{B_{R/8}^+}|\partial_i v|^2w\leq\frac{C}{w(B_{R/8}^+)}\frac{C}{R^2}\int_{B_{R/4}^+}|v|^2w\leq \frac{C}{R^2}\frac{w(B_{R/4}^+)}{w(B_{R/8}^+)}\sup_{B_{R/4}^+}|v|\leq CR^{2\alpha-2}.} \end{align*}
So, in all cases, $|\partial_i v(x)|\leq C|x|^{\alpha-1}$. Letting $R\to\infty$ and using the maximum principle, we find that $\partial_iv=0$ for any $i=1,\dots n$. Therefore $v$ does not depend on the first $n$ variables, { so} $v(x',y)=v(y)$. {Hence,} in $\mathbb R^{n+1}_+$, \[ 0=-\dive(y^{\beta} {\nabla} v(y))=-{ \partial_y}(y^{\beta}v'(y))\,\,\Rightarrow\,\, y^{\beta}v'(y)=\tilde{c}, \] for some constant $\tilde{c}$. From \cite[Theorem 2.4.6]{FabesKenigSerapioni}, $v$ is H{\"o}lder continuous up to the boundary, therefore for any $y>0$, \[ v(y)=v(y)-v(0)=\int_0^yv'=\int_0^y\tilde{c}s^{-\beta}\,ds=\frac{\tilde{c}}{1-\beta}y^{1-\beta}, \] which implies that \[
|\tilde{c}|=(1-\beta)y^{\beta-1}|v(y)|=(1-\beta)y^{\beta-1}|v(0,y)|\leq (1-\beta)y^{\beta-1}y^{\alpha}=(1-\beta)y^{-\alpha}, \] for any $y>0$. Letting $y\to\infty$ we obtain that $\tilde{c}=0$, hence $v'(y)=0$ as well, which implies that $v$ is a constant. Since $v$ vanishes on $\mathbb R^n$, this implies that $v\equiv 0$. \end{proof}
\end{document} | arXiv |
\begin{document}
\begin{center} {\large \bf Berry phase in the simple harmonic oscillator } \end{center} \begin{center} JeongHyeong Park$^\dag$\footnote[4]{E-mail address:
[email protected]} and Dae-Yup Song$^\ddag$\footnote[6]{E-mail address:
[email protected]} \end{center} $~~~~~~~~^\dag$Department of Mathematics, Honam University, Kwangju 506-714, Korea\\ $~~~~~~~~~^\ddag$Department of Physics, Sunchon National University, Sunchon 540-742, Korea
\begin{center} Short title: Berry's phase \\ Classification numbers: 03.65.Bz 03.65.Ca 03.65.Fd \end{center}
\begin{abstract} Berry phase of simple harmonic oscillator is considered in a general representation. It is shown that, Berry phase which depends on the choice of representation can be defined under evolution of the half of period of the classical motions, as well as under evolution of the period. The Berry phases do {\em not} depend on the mass or angular frequency of the oscillator. The driven harmonic oscillator is also considered, and the Berry phase is given in terms of Fourier coefficients of the external force and parameters which determine the representation. \end{abstract}
\section{Introduction} The (time-dependent) harmonic oscillator gives a system whose quantum states are described by solutions of the classical equation of motion (classical solutions). This fact has been recognized by Lewis \cite{Lewis} who found, in an application of asymptotic theory of Kruskal \cite{Kruskal}, that there exists a quantum mechanically invariant operator. This invariant operator determined by the classical solutions has then been used to construct wave function of (generalized) harmonic oscillator systems \cite{Yeon,Ji,Lee}. An alternative and simple (at least in conceptually) way to find the wave functions in terms of classical solutions is to use the Feynman path integral method \cite{Song1}. As observed by Feynman and Hibbs \cite{FH}, the kernel (propagator) for a general quadratic system is almost determined by classical action. The classical action can be given in terms of two linearly independent classical solutions and, for a general quadratic system, the exact kernel has been found by requiring the kernel to satisfy the initial condition and Schr\"{o}dinger equation \cite{Song1}. Further, it has been shown that \cite{Song2}, the wave functions of a general quadratic system can be obtained from those of unit mass harmonic oscillator system through unitary transformations.
A perceptive and interesting observation made by Berry \cite{Berry} is that, if an eigenstate of Hamiltonian is adiabatically carried around by cyclic Hamiltonian, the change of the phase of wave function separates into the obvious dynamical part and an additional geometric part. That the geometric change (Berry Phase) still remains some naturalness even if the cycled wave function is not an eigenstate of the Hamiltonian and even if the carrying is not adiabatic was pointed out by Aharonov and Anandan \cite{AA}. In a recent paper by one of the authors \cite{Song3}, it was shown that the Berry phase for harmonic oscillator of $\tau$-periodic Hamiltonian can be defined only if the two linearly independent classical solutions are finite all over the time. Moreover, in the cases of the two linearly independent solutions finite all over the time, it was shown that there exists at least a representation where the Berry phase can be defined under the $\tau$- or $2\tau$-evolution.
The Berry phase is known to be closely related to the classical Hannay angle \cite{BH}. By adopting the fact that a Gaussian wave packet could be a wave function of a harmonic oscillator system, the approach of Aharonov and Anandan \cite{AA} has been used to calculate Berry's phase, and the relation between the phase and Hannay angle is given for the wave packet \cite{GC}.
For the simple harmonic oscillator (SHO) of the equation of motion \begin{equation} M(\ddot{x} +w^2 x)=0 \end{equation} with constant mass $M$ and constant angular frequency $w$, the wave functions become stationary if we choose $\cos wt$ and $\sin wt$ as two solutions, while in general they describe the states of pulsating probability distribution \cite{Song2}.
In this paper, we will calculate Berry phase of the SHO system in general representations, and will show that Berry phase of the SHO system indeed depends on the choice of representation. We will consider a general representation made from the classical solutions $\cos wt$ and $C\sin(wt+\beta)$ of (1), where $C$ is a nonzero constant and the constant $\beta$ is not one of $(2n+1)\pi/2$ ($n=0, \pm 1, \pm 2,\cdots$). The period of the classical solutions is $\tau_0$ (=$2\pi/w$). It will be shown that, due to a quasiperiodicity of all classical solution of SHO, Berry phase can be defined under the $\tau_0/2$-evolution as well as $\tau_0$-evolution, which may not be possible for a general oscillator system considered in \cite{Song3}. The Berry phase will be evaluated in terms of the parameters $C$ and $\beta$; it turns out that the Berry phase does not depend on the mass $M$ or angular frequency $w$ of the oscillator. For the presence of external force, as already noted in \cite{Moore,Lee}, the phase can not be defined if a particular solution of the equation of motion diverges as the time goes to infinity. For the cases where the particular solution is periodic, the Berry phase whose leading order is order of $1/\hbar$ will be evaluated in terms of the Fourier coefficients of the external force. In view of view analyses on general harmonic oscillator of \cite{Song3}, the SHO is rather special in the sense that any number can be the period of the Hamiltonian. As will be mentioned in detail, we thus need care in applying the formulas to find Berry phase of the SHO system.
\section{The wave functions and (quasi)periodicity} As is well-known \cite{Song2,KL,Yeon,Ji,Lee}, the wave functions of SHO can be written as \begin{eqnarray} \psi_n(x,t) &=&
{1\over \sqrt{2^n n!}}({\Omega \over \pi\hbar})^{1\over 4}
{1\over \sqrt{\rho(t)}}[{u(t)-iv(t) \over \rho(t)}]^{n+{1\over 2}}
\exp[{x^2\over 2\hbar}(-{\Omega \over \rho^2(t)}
+i M(t){\dot{\rho}(t) \over \rho(t)})] \cr & &~~~~~~~~
\times H_n(\sqrt{\Omega \over \hbar} {x \over \rho(t)}), \end{eqnarray} where the $u(t),v(t)$ are two linearly independent solutions of (1) and $\rho(t)$ is defined as \begin{equation} \rho(t)= \sqrt{u^2(t)+v^2(t)}. \end{equation} Without losing generality, the two linearly independent solutions $u(t),v(t)$ can be written as \begin{equation} u(t)= \cos wt ~~~~~~v(t)=C\sin (wt+ \beta). \end{equation}
If we choose $C=1$ and $\beta=0$ , it gives the "stationary representation" where the wave functions are of stationary probability distribution \[ \tilde\psi_n(x,t)= {1\over \sqrt{2^n n!}}({Mw \over \pi\hbar})^{1/4} \exp[-i(n+{1\over 2})wt -{Mw\over \hbar} x^2]H_n(\sqrt{Mw\over \hbar} x). \] The $\tilde\psi_n(x,t)$ is the eigenstate of the Hamiltonian of SHO \begin{equation} H={p^2\over 2M}+{Mw^2 \over 2}x^2, \end{equation} with eigenvalue $E_n=(n+{1\over 2})\hbar w$. The Hamiltonian does not depend on time, so the Hamiltonian is periodic with any period. The analyses in \cite{Song3} might thus suggest that, in the stationary representation where the Berry phase can be defined for an evolution of any time, the phase is zero. Indeed, phase change of $\tilde\psi_n$ over $t$ evolution is simply equals to the dynamical phase change $-E_n t/\hbar$, so that Berry phase is zero.
\begin{figure}
\caption{\small The representation space of SHO: the horizontal and vertical axes denote $u$ and $v$, respectively. Some of the curves which would depict classical motions of the SHO are given. Though the classical motion for the dashed line may be allowed, representation corresponding to the line does not exist.}
\end{figure}
In the figure 1, some trajectories of $(u(t),v(t))$ which would depict classical motions of SHO are given. They make closed curves, since both of $u(t)$ and $v(t)$ are periodic. Different closed curves in the figure 1 give different sets of wave functions, while it is not possible to construct a representation corresponding to the dashed line in {\bf b} of the figure. Classical motion of SHO may be depicted as the mass $M$ circulating along a closed curve with uniform angular velocity $w$, so that it needs a period $\tau_0$ for a complete circulation. As noted in \cite{Ji,Lee,Song3}, if $\rho(t)$ is periodic with some period, the wave functions are (quasi)periodic with the period of $\rho(t)$. Since $\rho(t)$ is the distance from the origin to a point of a curve, as in the figure 1, it is clear that in general $\rho(t)$ is periodic with the half of the period of classical motion. The circle of radius 1 in {\bf a} gives rise to the stationary representation. $\rho(t)$ is constant along the circle and thus any number can be a period of $\rho$, which is compatible with that Berry phase in the stationary representation must be 0.
From the continuity, one may find the relation \begin{equation} u(\tau_0/2+t)-iv(\tau_0/2+t)= \exp(-i\pi)[u(t)-iv(t)], \end{equation} which gives the quasiperiodicity relation of the wave functions \begin{equation} \psi_n(x, \tau_0/2+t)= \exp[i\chi_n(\tau_0/2)]\psi_n(x,t) \end{equation} with the over all phase change \begin{equation} \chi_n(\tau_0/2)=-(1/2+n)\pi. \end{equation} Since a phase is defined only up to an additive constant $2\pi$, (8) can be written in different ways. For example, a relation $\chi_n(\tau_0/2)=(n-1/2)\pi$ is equivalent to (8).
\section{Berry Phase} The Berry phase $\gamma_n$ is given from the overall phase change by subtracting the dynamical phase change \begin{equation} \gamma_n(\tau_0/2)=\chi_n(\tau_0/2) - \delta_n(\tau_0/2), \end{equation} and the dynamical phase change $\delta_n$ is given as \cite{Song3} \begin{eqnarray} \delta_n(\tau')&=&-{i\over \hbar}\int_0^{\tau'}
\int_{-\infty}^{\infty}\psi_n^*(t) H \psi_n(t) dx dt \cr &=&-{1 \over 2}(n+{1\over 2})\int_0^{\tau'}
[ {\Omega \over M\rho^2}
+ {M\dot{\rho}^2 \over \Omega}
+{\rho^2(t)Mw^2 \over \Omega}] dt. \end{eqnarray} For the classical solutions of (4), $\Omega$ defined as $M(u\dot{v}-v\dot{u})$ is \begin{equation} \Omega= MCw\cos\beta, \end{equation} and thus the $\delta_n(\tau_0/2)$ is written as \begin{equation} \delta_n(\tau_0/2)= -(n+{1\over 2})\pi {1+C^2 \over 2C\cos \beta}. \end{equation} The Berry phase of the wave function $\psi_n$ for the $ \tau_0/2$-evolution, therefore, can be given as \begin{equation} \gamma_n(\tau_0/2)=(n+{1\over 2})\pi[-1+{1+C^2 \over 2C\cos\beta}]. \end{equation} The Berry phase under $\tau_0$-evolution can be obtained from $\gamma_n(\tau_0/2)$ \begin{equation} \gamma_n(\tau_0)=2\gamma_n(\tau_0/2)=
(n+{1\over 2})\pi[{1-2C\cos\beta +C^2 \over C\cos\beta}]. \end{equation} The $\gamma_n$ depends on $C$ and $\beta$, but does {\em not} depend on the parameters of the Hamiltonian $M$ and $w$.
The fact that the phase $\gamma_n$ is defined up to an additive constant $\pm2\pi$ says that there are infinitely many representation which give the same Berry phase. For example, if $\cos \beta = 1/2$, the values of $C$ which gives $\gamma_n(\tau_0/2)=0$ can be written as \[ (\cdots, 9/2\pm \sqrt{(9/2)^2 -1},5/2\pm \sqrt{(5/2)^2 -1},
-3/2\pm \sqrt{(3/2)^2 -1},-7/2\pm \sqrt{(7/2)^2 -1}, \cdots). \]
In order to compare with the the result of Ge and Child \cite{GC}, by defining \[ \alpha(t)={1\over 2\hbar}({\Omega \over \rho^2} -i M {\dot{\rho} \over \rho}), \] one may find the relation \begin{equation} -{i\over 2}\int_0^{\tau_0} {\dot{\alpha} \over \alpha+\alpha^*} dt
= {1-2C\cos\beta +C^2 \over 2 C\cos\beta}\pi \end{equation} which gives the Berry phase $\gamma_0(\tau_0)$ in agreement with the result in (14).
In order to compare the results here with the formulae in \cite{Song3}, we need to consider several cases differently. If we take $\tau$ as one of $(2m-1)\tau_0/2$ $(m=1,2,\cdots)$, the Floquet theorem \cite{Magnus} is satisfied by that classical solutions are 2$\tau$-periodic. If we take $\tau$ as one of $m\tau_0$ $(m=1,2,\cdots)$, the Floquet theorem is satisfied by that classical solutions are $\tau$-periodic. In the above cases, the corresponding formula for Berry phases \cite{Song3} can be used to obtain the Berry phase in (14) or its integral multiples. In a general oscillator, Berry phase may not be defined under $\tau/2$-evolution, and thus the equation (13) has no corresponding formula in \cite{Song3}. If we take $\tau$ as being not one of $m\tau_0/2$ $(m=1,2,\cdots)$, the fact that Berry phase can be defined for any evolution in stationary representation is in agreement with the analyses in \cite{Song3}. Again, a formula corresponding to this case can be used to show that Berry phase is 0 in the stationary representation.
\section{Driven harmonic oscillator} For the SHO with a driving force $F(t)$, the wave function is given as \cite{Song1,Song2,Song3,Lee} \begin{eqnarray} \psi_n^F &=&
{1\over \sqrt{2^n n!}}({MwC\cos\beta \over \pi\hbar})^{1\over 4}
{1\over \sqrt{\rho(t)}}[{\cos wt-iC\sin wt \over \rho(t)}]^{n+{1\over 2}}
\exp[{i\over \hbar}(M\dot{x}_px+\delta(t))] \cr & &~~~~~~~~
\times \exp[{(x-x_p)^2\over 2\hbar}(-{\Omega \over \rho^2(t)}
+i M(t){\dot{\rho}(t) \over \rho(t)})]
H_n(\sqrt{\Omega \over \hbar} {x -x_p \over \rho(t)}), \end{eqnarray} where \begin{equation} \delta(t)={M\over 2}\int_{t_0}^t[w^2 x_p^2(z) -\dot{x}_p^2(z) ] dz \end{equation} with arbitrary constant $t_0$. The $x_p$ defined by the relation \begin{equation} \ddot{x}_p + w^2 x_p = {F(t)\over M} \end{equation} may be the classical coordinate of $x$. $F$ denotes the driving force, and we only consider the periodic $F$ satisfying $F(t+\tau_f)=F(t)$.
If $x_p$ is finite all over the time and there exist two positive integers $N,p$ of no common divisor except 1 such that $\tau_0/\tau_f=p/N$, the Berry phase can be defined under $N\tau_0$-evolution in a general representation. The periodic $F(t)$ can be written as \begin{equation} F(t)=\sum_{n=-\infty}^\infty f_n e^{in w_f t}, \end{equation} where \begin{equation} f_n={1\over \tau_f}\int_0^{\tau_f} F(t) e^{-inw_f t} dt \end{equation} with $w_f= 2\pi / \tau_f.$ The $x_p(t)$ finite all over the time can be written as \begin{equation} x_p=\sum_{n=-\infty}^\infty {f_n \over M(-n^2w_f^2 + w^2)}e^{in w_f t}
+D e^{i wt} + D^*e^{-i wt}. \end{equation} If $p=1$, $f_N$ must be zero for the finiteness of $x_p$, which will be assumed from now on. In (21), complex number $D$ is a free parameter and different choice of $D$ gives different wave functions of the driven system. The minimum period of $x_p$ is $N\tau_0$ ($=p\tau_f$) in general so that \begin{equation} x_p(t+N\tau_0) = x_p(t). \end{equation} It has been known that \cite{Song3} the Berry phase of a driven system separates into the contribution from undriven system and that from the presence of driving force. The contribution from the driving force is written as \[ {1\over \hbar}\int_0^{\tau'} M\dot{x}_p^2 dt, \] where $\tau'$ is the period needed for the Berry phase. The Berry phase $\gamma_n^F$ of driven SHO system under $N\tau_0$ evolution is thus given as \begin{equation} \gamma_n^F(N\tau_0)=
N\gamma_n(\tau_0)+ {1\over \hbar}\int_0^{N\tau_0} M\dot{x}_p^2 dt. \end{equation} After some algebra, from (13) and (20) one may find the relation \begin{eqnarray} \gamma_n^F(N\tau_0) &=&\pi(n+{1\over 2})N[{1-2C\cos\beta +C^2 \over C\cos\beta}]\cr
&& +2\pi{N^3p^2 \over \hbar Mw^3}
\sum_{n=-\infty}^\infty {n^2|f_n|^2 \over (p^2n^2 -N^2)^2}
+ 2\pi{ MNw \over \hbar} |D|^2. \end{eqnarray}
There exist the representations where Berry phase can {\em not} be defined even with periodic $x_p$. For example, in the representation of $D\neq 1$, if $\tau_f/\tau_0$ is an irrational number Berry phase can not defined under any evolution of finite time. However, for the SHO with driving force, there is a special representation of $C=1, \beta=0$ and $D=0$ where Berry phase can be defined for any periodic $x_p$. In this representation, Berry phase under $\tau_f$-evolution is written as \begin{equation} \gamma_n^F(\tau_f)= {2\pi w_f \over \hbar M}\sum_{n=-\infty}^\infty
{n^2|f_n|^2 \over (n^2w_f^2 -w^2)^2}. \end{equation} The presence of this special representation is due to the fact that, Berry phase can be defined for any evolution in the stationary representation of the (undriven) SHO system.
\section{Conclusion} We have calculated Berry phase of the SHO system in terms of classical solutions, and have explicitly shown that the phase depends on the choice of representations. The Berry phase of undriven system does not depend on the mass or angular frequency of the SHO, while it depends on two parameters which comes from choosing classical solutions.
For a driven system of periodic $x_p$, the Berry phase, if it exists, separates into the contribution from the undriven system and that from the presence of driving force. For the SHO system with driving force, the contribution from the presence of of driving force which is the order of $1/\hbar$ depends on mass, angular frequency, and two more real parameters coming from the way of choosing classical solutions. There is a special representation where Berry phase can be defined for any periodic $x_p$, while in some representations Berry phase can not be defined. For a general harmonic oscillator of time-dependent mass and frequency with driving force, it may not be the case that there exists such a special representation since stationary representation for the system without driving force may not exist in general.
While classical solutions of the SHO system is $\tau_0$-periodic, the solutions have an additional property that absolute value of any classical solution is $\tau_0/2$-periodic. This fact enables one to define the Berry phase under $\tau_0/2$-evolution as well as $\tau_0$-evolution. For a general harmonic oscillator of time-dependent mass and frequency, if there exists such an additional periodicity in the absolute value of all classical solution, making use of the periodicity, it will be possible to define the Berry phase for the evolution of shorter period than that of the Hamiltonian.
\noindent {\bf Acknowledgments} \newline JHP wishes to acknowledge the partial financial support of the Korea Research Foundation made in the program, year of 1998. DYS acknowledges the partial financial support from Sunchon National University (Non-Directed Research Fund).
\end{document} | arXiv |
The Journal of Mathematical Neuroscience
Investigating the Correlation–Firing Rate Relationship in Heterogeneous Recurrent Networks
Andrea K. Barreiro1 &
Cheng Ly2
The Journal of Mathematical Neuroscience volume 8, Article number: 8 (2018) Cite this article
The structure of spiking activity in cortical networks has important implications for how the brain ultimately codes sensory signals. However, our understanding of how network and intrinsic cellular mechanisms affect spiking is still incomplete. In particular, whether cell pairs in a neural network show a positive (or no) relationship between pairwise spike count correlation and average firing rate is generally unknown. This relationship is important because it has been observed experimentally in some sensory systems, and it can enhance information in a common population code. Here we extend our prior work in developing mathematical tools to succinctly characterize the correlation and firing rate relationship in heterogeneous coupled networks. We find that very modest changes in how heterogeneous networks occupy parameter space can dramatically alter the correlation–firing rate relationship.
One prominent goal of theoretical neuroscience is to understand how spiking statistics of cortical networks are modulated by network attributes [9, 28, 42]. This understanding is essential to the larger question of how sensory information is encoded and transmitted, because the statistics of neural activity impact population coding [7, 15–17, 37]. One family of statistics that is implicated in nearly all population coding studies is trial-to-trial variability (and co-variability) in spike counts; there is now a rich history of studying how these statistics arise, and how they effect coding of stimuli [1, 10, 18, 25, 33]. Recent work has demonstrated that the information content of spiking neural activity depends on spike count correlations and its relationship (if any) with stimulus tuning [1, 6, 18, 25, 44].
An important relationship observed in many experimental studies is that pairwise correlations on average increase with firing rates. This has been observed in vitro [8] and in several visual areas: area MT [2], V4 [5] (when measured between cells in the same attentional state), V1 [19, 36], and notably, in ON–OFF directionally sensitive retinal ganglion cells [11, 44]. The retinal studies involved cells with a clearly identified function, and therefore allowed investigation of the coding consequences of the observed correlation–firing rate relationship. These studies found that the stimulus-dependent correlation structure observed compared favorably to a structure in which stimulus-independent correlations were matched to their (stimulus-)averaged levels. This finding reflects a general principle articulated in other studies [18, 25], that stimulus-dependent correlations are beneficial when they serve to spread the neural response in a direction orthogonal to the signal space.
These findings thus provide strong motivation for understanding what network mechanisms can produce this positive (and perhaps beneficial) correlation–firing rate relationship. This correlation–firing rate trend has been explained theoretically in feedforward networks driven by common input [8, 26, 38]; however, many cortical networks are known to be dominated by strong recurrent activity [24, 34, 39]. On the other hand, theoretical studies of the mechanisms for correlations in recurrent networks have largely analyzed homogeneous networks (i.e., identical cells, aside from excitatory/inhibitory cell identity) [9, 13, 27, 28, 40, 41], and have not considered how correlations vary with firing rates with realistic intrinsic heterogeneity. Thus, how spike count correlations can vary across a population of heterogeneously-tuned, recurrently connected neurons is not yet well understood despite its possible implications for coding.
In a previous paper, we addressed this gap by developing a mathematical method to compactly describe how second-order spike count statistics vary with both intrinsic and network attributes [4]. Specifically, we adapted network linear response theory [14, 27, 43] to account for heterogeneous and recurrent networks, which in turn allows us to identify important network connections that contribute to correlations via a single-cell susceptibility function [8]. Here, we will use this method to survey a broad family of recurrent networks to understand how three factors influence the relationship between correlations and firing rates; how the neurons occupy parameter space, the strength of recurrent excitation, and the strength of background noise. This work thus provides a more complete picture of how even modest changes in important circuit parameters can alter the correlation–firing rate relationship.
First, we summarize how a network linear response theory allows us to use the single-neuron firing rate function to approximate correlations. We then apply this theory to examine three factors that can modulate the correlation–firing rate relationship: direction in effective parameter space, strength of recurrent excitation, and strength of background noise.
Using the Single-Neuron Firing Rate Function to Characterize Correlation Susceptibility
The response of a neuron is generally a nonlinear function of model parameters. However, background noise linearizes this response, so that we can use a linear theory to describe the change in response to small changes in parameters. We assume we have some way to approximate the change in firing rate which occurs as a result of a change in parameter X:
$$\begin{aligned} \nu_{i} (t) = & \nu_{i,0} + (A_{X,i} * X_{i}) (t); \end{aligned}$$
\(\nu_{i,0}\) is the baseline rate (when \(X = 0\)) and \(A_{X,i}(t)\) is a susceptibility function that characterizes the firing rate response [8, 20, 41]. The parameter which is modulated has often been chosen to be a current bias μ [8, 41]; however, it could also be the mean or variance of a conductance, a time constant, or a spike generation parameter [29, 30].
In a coupled network, the parameter change \(X_{i}\) arises from inter-neuron coupling; substituting \(X_{i}(t) \rightarrow \sum_{j} (\mathbf {J}_{X,ij} * \nu_{j} )\) and moving to the spectral domain, we find
$$\begin{aligned} \bigl\langle \tilde{y}(\omega) \tilde{y}^{\ast}(\omega) \bigr\rangle = & \bigl( \mathbf {I}- \tilde{\mathbf {K}}(\omega) \bigr)^{-1} \bigl\langle \tilde {y}^{0}(\omega) \tilde{y}^{0 \ast}(\omega) \bigr\rangle \bigl( \mathbf {I}- \tilde{\mathbf {K}}^{\ast}(\omega) \bigr)^{-1}, \end{aligned}$$
where \(\tilde{y}_{i} = \mathcal{F} [ y_{i} - \nu_{i} ]\) is the Fourier transform of the mean-shifted process (\(\nu_{i}\) is the average firing rate of cell i) and \(\tilde{f} = \mathcal{F} [ f ]\) for all other quantities; \(\tilde{\mathbf {K}}_{ij} (\omega) = \tilde {A}_{X, i}(\omega) \tilde{\mathbf {J}}_{X, ij}(\omega)\) is the interaction matrix in the frequency domain (which may depend on the parameter being varied, i.e. X); \(\langle\tilde{y}^{0}(\omega) \tilde{y}^{0 \ast}(\omega) \rangle\) is the power spectrum of the uncoupled neural response. Using the usual series expansion for \(( \mathbf {I}- \tilde{\mathbf {K}}(\omega) )^{-1}\), we see that the covariance \(\tilde{\mathbf {C}} (\omega) \equiv\langle\tilde{y}(\omega) \tilde{y}^{\ast}(\omega ) \rangle\) naturally decomposes into contributions from different graph motifs:
$$\begin{aligned} \tilde{\mathbf {C}} (\omega) =& \bigl( \mathbf {I}- \tilde{\mathbf {K}}(\omega) \bigr)^{-1} \tilde{\mathbf {C}}^{0} (\omega) \bigl( \mathbf {I}- \tilde{ \mathbf {K}}^{\ast}(\omega) \bigr)^{-1} \\ = & \tilde{\mathbf {C}}^{0} (\omega) + \tilde{\mathbf {K}}(\omega) \tilde { \mathbf {C}}^{0} (\omega) + \tilde{\mathbf {C}}^{0} (\omega) \tilde{\mathbf {K}}^{*}(\omega) + \tilde{\mathbf {K}}(\omega) \tilde{\mathbf {C}}^{0} (\omega) \tilde{ \mathbf {K}}^{*}(\omega) + \cdots . \end{aligned}$$
Each instance of \(\tilde{\mathbf {K}}\) includes the susceptibility function in the spectral domain, \(A_{X}(\omega)\), which modulates the effect of each directed connection by the responsiveness of the target neuron to the parameter of interest. As noted by previous authors [27], the validity of the expansion in Eq. (3) relies on the spectral radius of K̃, \(\rho(\tilde{K})\), remaining less than one. We checked that this remains true for all networks we examined in this paper, with a maximum of \(\rho(\tilde{K}) = 0.9564\).
We next justify why—at least for long-time correlations—we can estimate each of these terms using the single-neuron firing rate function. First, in the small frequency limit, \(A_{X,i}(\omega)\) will coincide with the derivative of the firing rate with respect to the parameter:
$$\lim_{\omega\rightarrow0} \tilde{A}_{X,i}(\omega) = \frac{d\nu _{i}}{d X}. $$
For the common input motif, \(\tilde{\mathbf {K}}(\omega) \tilde{\mathbf {C}}^{0} (\omega) \tilde{\mathbf {K}}^{*}(\omega)\), we can write
$$\begin{aligned} \bigl( \tilde{\mathbf {K}} \tilde{\mathbf {C}}^{0} \tilde{\mathbf {K}}^{*} \bigr)_{ij} = &\sum_{k \rightarrow i, k\rightarrow j} \tilde{\mathbf {K}}_{ik} \tilde{\mathbf {C}}^{0}_{kk} \tilde{ \mathbf {K}}_{jk} \end{aligned}$$
$$\begin{aligned} = &\sum_{k \rightarrow(i,j), k \in E} (\tilde{A}_{g_{E},i} \tilde{ \mathbf {J}}_{ik} ) \tilde{\mathbf {C}}^{0}_{kk} (\tilde {A}_{g_{E},j} \tilde{\mathbf {J}}_{jk} ) \\ &{} + \sum _{k \rightarrow(i,j), k \in I} (\tilde{A}_{g_{I},i} \tilde{\mathbf {J}}_{ik} ) \tilde{\mathbf {C}}^{0}_{kk} (\tilde{A}_{g_{I},j} \tilde{ \mathbf {J}}_{jk} ) \end{aligned}$$
$$\begin{aligned} = & \vert \tilde{\mathbf {J}}_{E} \vert ^{2} \sum _{k \rightarrow(i,j), k \in E} \tilde{A}_{g_{E},i} \tilde{A}_{g_{E},j} \tilde{\mathbf {C}}^{0}_{kk} \\ &{} + \vert \tilde{\mathbf {J}}_{I} \vert ^{2} \sum_{k \rightarrow(i,j), k \in I} \tilde{A}_{g_{I},i} \tilde {A}_{g_{I},j} \tilde{\mathbf {C}}^{0}_{kk} \end{aligned}$$
$$\begin{aligned} = & \vert \tilde{\mathbf {J}}_{E} \vert ^{2} \tilde{A}_{g_{E},i} \tilde{A}_{g_{E},j} \sum _{k \rightarrow(i,j), k \in E} \tilde {\mathbf {C}}^{0}_{kk} \\ &{} + \vert \tilde{\mathbf {J}}_{I} \vert ^{2} \tilde {A}_{g_{I},i} \tilde{A}_{g_{I},j} \sum_{k \rightarrow(i,j), k \in I} \tilde{ \mathbf {C}}^{0}_{kk}, \end{aligned}$$
where we have separated excitatory (E) and inhibitory (I) contributions, using \(g_{E}\) and \(g_{I}\) to denote the mean conductance of each type, and assumed that the synaptic coupling kernels, \(\tilde {\mathbf {J}}_{jk}\), depend only on E/I cell identity (and have thus dropped the first subscript, which adds no additional information).
This provides a formula for the long-time covariance, but we are typically interested in the correlation; fortunately, the small frequency limit also allows us to obtain a normalized correlation measure from the cross-spectrum, because
$$\begin{aligned} \lim_{T \rightarrow\infty} \rho_{T,ij} = & \lim_{T \rightarrow \infty} \frac{\operatorname {Cov}_{T}(n_{i},n_{j})}{\sqrt{ \operatorname {Var}_{T}(n_{i}) \operatorname {Var}_{T}(n_{j})}} = \frac{\tilde{\mathbf {C}}_{ij}(0)}{\sqrt{\tilde{\mathbf {C}}_{ii}(0) \tilde {\mathbf {C}}_{jj}(0)}}, \end{aligned}$$
where \(\operatorname {Cov}_{T}(n_{i},n_{j})\) and \(\operatorname {Var}_{T}(n_{i})\) denote covariance and variance of spike counts in a time window T; that is, \(\rho_{T,ij}\) is the Pearson correlation coefficient (which varies between −1 and 1).
Finally, letting \(\omega\rightarrow0\) and normalizing with the assumption that spiking is close to a Poisson process, as expected for an asynchronously firing network, so that \(\tilde{\mathbf {C}}_{kk} \approx \nu_{k}\):
$$\begin{aligned} \frac{ ( \tilde{\mathbf {K}} \tilde{\mathbf {C}}^{0} \tilde{\mathbf {K}}^{*} )_{ij}}{\sqrt{\tilde{\mathbf {C}}_{ii} \tilde{\mathbf {C}}_{jj} }} = & \underbrace{ \frac{1}{\sqrt{\nu_{i} \nu_{j}}} \frac{d\nu_{i}}{d g_{E}} \frac{d\nu_{j}}{d g_{E}} }_{\substack{\text{modulation by}\\ \text{firing rate function}}} \vert \tilde{\mathbf {J}}_{E} \vert ^{2} \underbrace{ \biggl[ \sum_{\substack{k \rightarrow(i,j), \\ k \in E}} \tilde{\mathbf {C}}^{0}_{kk} \biggr]}_{\text{total E common input}} \\ &{} + \underbrace{ \frac{1}{\sqrt{\nu_{i} \nu_{j}}} \frac{d\nu_{i}}{d g_{I}} \frac{d\nu_{j}}{d g_{I}} }_{\substack{\text{modulation by}\\ \text{firing rate function}}} \vert \tilde{\mathbf {J}}_{I} \vert ^{2} \underbrace{ \biggl[ \sum _{ \substack{k \rightarrow(i,j),\\ k \in I} } \tilde{\mathbf {C}}^{0}_{kk} \biggr]}_{\text{total I common input}} . \end{aligned}$$
The above expression summarizes how the impact of excitatory and inhibitory common input are modulated by the single-neuron firing rate function, ν, and its derivatives.
Thus far, we have presented results previously obtained by others [27, 40, 41]. We now depart from these authors by focusing specifically on the behavior of the term in Eq. (9); and for simplicity, the behavior of this modulating factor for two identical neurons; e.g.
$$\begin{aligned} \frac{1}{\sqrt{\nu_{i} \nu_{j}}} \frac{d\nu_{i}}{d g_{I}} \frac{d\nu _{j}}{d g_{I}} \rightarrow& \frac{1}{\nu} \biggl( \frac{d\nu}{d g_{I}} \biggr)^{2}. \end{aligned}$$
In principle, analogous expressions govern larger motifs, such as chains, that involve additional evaluations of ν and its derivatives; for example, excitatory length-3 chains arising from \(\tilde{\mathbf {K}}^{3} \tilde{\mathbf {C}}^{0}\) would be:
$$\begin{aligned} \frac{ ( \tilde{\mathbf {K}}^{3} \tilde{\mathbf {C}}^{0} )_{ij}}{\sqrt {\tilde{\mathbf {C}}_{ii} \tilde{\mathbf {C}}_{jj} }} = & \underbrace{ \biggl[ \frac{1}{\sqrt{\nu_{i} \nu_{j}}} \frac{d\nu_{i}}{d g_{E}} \frac{d\nu_{l}}{d g_{E}} \frac{d\nu_{k}}{d g_{E}} \biggr]}_{\text{modulation by firing rate function}} \times \vert \tilde{\mathbf {J}}_{E} \vert ^{3} \times \underbrace{ \biggl[ \sum_{l \rightarrow i} \sum _{\substack{k \rightarrow l,\\ l \in E}} \sum_{\substack{j \rightarrow k,\\ k \in E}} \tilde{ \mathbf {C}}^{0}_{jj} \biggr]}_{\text{all $E \rightarrow E \rightarrow E \rightarrow E$ paths}} . \end{aligned}$$
However, we found that, for a wide range of networks, direct common input—and inhibitory common input in particular—was the dominant contributor to pairwise correlations (Fig. 6(A)).
We now examine how different network mechanisms modulate the correlation–firing rate relationship, focusing on three factors: direction in effective parameter space, strength of recurrent excitation, and strength of background noise.
Direction in Parameter Space Modulates the Correlation–Firing Rate Relationship
Suppose we have a firing rate function of one or more intrinsic parameters (for exposition purposes, assume a function of two parameters \((x_{1}, x_{2})\)), i.e.
$$\nu= F(x_{1}, x_{2}). $$
The parameters \(x_{j}\) might include input bias, membrane time constant, ionic/synaptic reversal potentials, or a spiking threshold. Based on our arguments from the previous section, we will approximate correlation susceptibility by the quantity
$$\hat{S} = \frac{1}{F} \biggl( \frac{\partial F}{\partial x_{1}} \biggr)^{2}, $$
where \(x_{1}\) is an appropriately chosen parameter. Specifically, we will find, empirically, that the inhibitory common input is the dominant term, and therefore will use \(x_{1} = g_{I}\), the mean inhibitory conductance. Thus, the units of Ŝ in all figures are Hz/[\(g_{I}^{2}\)], where \(g_{I}\) is dimensionless (see Eq. (17)).
Heterogeneous firing rates can arise when each neuron occupies a different location in parameter space (i.e. a different \((x_{1},x_{2})\) point), thus causing both firing rate F and susceptibility Ŝ to vary from neuron to neuron. We now ask: how does Ŝ change with firing rate?
Note that this question, as stated, is ill-posed because F and Ŝ are both functions of two parameters (\(x_{1}\) and \(x_{2}\)): there is not necessarily a one-to-one or even a functional relationship between these quantities. Suppose that, locally, a population of neurons can be described as lying along a parameterized path in the \((x_{1}, x_{2})\) plane: i.e., \((x_{1}(s), x_{2}(s))\), for \(s_{\mathrm{min}}< s< s_{\mathrm{max}}\). Then we can compute the directional derivative:
$$ \frac{d\hat{S}}{dF} = \frac{d\hat{S}/ds}{ dF/ds} = \frac{\nabla\hat{S} \cdot d\mathbf {x}}{ \nabla F \cdot d\mathbf {x}}, $$
where we have expressed the directional derivatives in terms of the local direction of the path: i.e. \(d \mathbf {x}= (\frac{dx_{1}}{ds}, \frac {dx_{2}}{ds})\) and the gradients of F and Ŝ. However, this depends not only on the functions F and Ŝ, but also on the direction dx.
To gain some intuition for why (and when) direction in \((x_{1},x_{2})\) space matters, we consider the networks studied in [4]. Previously, we simplified the firing rate function by setting all but two parameters (inhibitory conductance, \(\langle g_{I,i} \rangle\), and threshold, \(\theta_{i}\)) to their population average; i.e.
$$\begin{aligned} F \bigl( \langle g_{I,i} \rangle, \theta_{i} \bigr) \equiv& f \bigl( \langle g_{I,i} \rangle, \langle\sigma_{g_{I},i} \rangle_{p}, \bigl\langle \langle g_{E,i} \rangle \bigr\rangle _{p}, \langle\sigma_{g_{E},i} \rangle _{p}, \langle\sigma_{i} \rangle_{p}, \theta_{i} \bigr) , \end{aligned}$$
and \(\langle \cdot \rangle_{p}\) denotes the population average. In Figs. 1(A) and (B), we show F and \(\hat{S} \equiv ( \frac{\partial F}{\partial x_{1}} )^{2} /F\) thus computed, for the asynchronous network studied in that paper. We then surveyed a sequence of diagonal paths through the center (i.e., midpoint of the ranges of threshold and inhibitory conductance), with each path plotted in a different color. In Fig. 1(C) we plot firing rate (solid lines) and susceptibility (dashed-dotted lines) along each curve. Finally, in Fig. 1(D) we plot the susceptibility versus the firing rate, along each path.
Firing rate and susceptibility (Ŝ), computed for the asynchronous (Asyn) network studied in [4], with all other parameters besides threshold θ and mean inhibitory conductance \(\langle g_{I}\rangle\) set to their average values (thereby leaving a two-dimensional parameter space). Here, we traverse the plane on straight-line paths defined by their angle through the origin. Although the units of \(g_{I}\) are dimensionless, they are shown as the units for Ŝ for completeness. The units of θ (i.e., voltage) are also scaled to be dimensionless
This last panel shows that there is striking variability in the susceptibility-firing rate relationship, depending on the direction the path takes through the center of the \((\theta, \langle g_{I} \rangle )\) plane. Any given firing rate (below \(\sim15\mbox{ Hz}\)) is consistent with either increase or decrease of susceptibility. All curves go through a single point in the \((\theta, \langle g_{I} \rangle)\) plane, and therefore a single point in the \((F, \hat{S})\) plane; here the direction of the Ŝ–F relationship (i.e., whether Ŝ increases or decreases with F) can change rapidly with angle, as for the red, orange, and yellow curves.
We then extended these observations by traversing the phase space with two additional families of straight-line paths (Fig. 2); the radial paths are repeated in Figs. 2(A) and (B). For horizontal (Figs. 2(C) and (D)) and vertical (Figs. 2(E) and (F)) families, paths no longer intersect at a single point; nevertheless, a given firing rate is consistent with both increases and decreases in susceptibility. This is pronounced in Fig. 2(F), where at 15 Hz susceptibility decreases with firing rate in the orange, yellow and light green paths, but increases for the light blue, medium blue, royal blue, and indigo paths.
Susceptibility (Ŝ) vs. firing rate, computed for the asynchronous network studied in [4], with all other parameters besides threshold θ and mean inhibitory conductance \(\langle g_{I}\rangle\) set to their average values (thereby leaving a two-dimensional parameter space: the other (averaged) parameters are \(\langle\sigma_{g_{I},i} \rangle_{p} = 0.6602\) (see Eq. (30)), \(\langle\sigma_{g_{E},i} \rangle_{p} = 0.0026\) (see Eq. (29)), \(\langle\sigma_{i} \rangle_{p} = 6.3246\), \(\langle\langle g_{E},i \rangle \rangle_{p} = 0.0053\) (see Eq. (17)). Here, we traverse the plane on three different families of straight-line paths. The dashed lines show paths visualized in [4]: \(\theta= 1\) (vertical, aqua blue) and \(\langle g_{I}\rangle= 1.83\) (horizontal, orange/yellow)
We performed the same computations on the strong asynchronous network studied in [4] that has larger excitatory coupling strength: results are shown in Fig. 3. A given firing rate could be consistent with either increase or decrease of susceptibility; we see this in the radial paths (Figs. 3(A) and (B)) and horizontal paths (Figs. 3(C) and (D)) for rates between 5–10 Hz. However, vertical paths (Figs. 3(E) and (F)) always have susceptibility increasing with firing rate (except perhaps at the highest firing rates). As in the asynchronous network, direction of susceptibility (increase vs. decrease) can change rapidly with angle, for paths that intersect a single point; see Figs. 3(A)–(B), red to orange to yellow.
Susceptibility (Ŝ) vs. firing rate, computed for the strong asynchronous (Strong Asyn) network studied in [4], with all other parameters besides threshold θ and mean inhibitory conductance \(\langle g_{I}\rangle\) set to their average values (thereby leaving a two-dimensional parameter space: the other (averaged) parameters are \(\langle\sigma_{g_{I},i} \rangle_{p} = 0.5884\) (see Eq. (30)), \(\langle\sigma_{g_{E},i} \rangle_{p} = 0.0378\) (see Eq. (29)), \(\langle\sigma_{i} \rangle_{p} = 4.7434\), \(\langle \langle g_{E},i \rangle \rangle_{p} = 0.0611\), see Eq. (17)). Here, we traverse the plane on three different families of straight-line paths. Dashed lines show paths visualized in [4]: \(\theta= 1\) (vertical, aqua blue) and \(\langle g_{I}\rangle= 1.46\) (horizontal, yellow/green)
Quantifying the Likelihood of a Positive Correlation–Firing Rate Relationship
In the previous section, we saw that the path a network occupies in effective parameter space can have a dramatic effect on the correlation–firing rate relationship: we next seek to formalize these observations. Let dx be the local direction in which we want to consider the behavior of F and Ŝ. If \(\nabla\hat{S} \cdot d \mathbf {x}\) and \(\nabla F \cdot d \mathbf {x}\) are of the same sign, then Ŝ increases with F; if \(\nabla\hat {S} \cdot d \mathbf {x}\) and \(\nabla F \cdot d \mathbf {x}\) have opposite signs, then Ŝ decreases with F. The more aligned ∇Ŝ and ∇F, the more paths lead to \(\frac{d\hat{S}}{dF} > 0\); the more anti-aligned ∇Ŝ and ∇F, the more paths lead to \(\frac{d\hat{S}}{dF} < 0\). For example, consider the limiting cases where: (i) if ∇Ŝ and ∇F point exactly in the same direction, then \(\nabla\hat {S} \cdot d \mathbf {x}\) and \(\nabla F \cdot d \mathbf {x}\) are always same-signed for any dx; (ii) if ∇Ŝ and ∇F point in opposite directions, then \(\nabla\hat{S} \cdot d \mathbf {x}\) and \(\nabla F \cdot d \mathbf {x}\) are never same-signed. Figure 4 illustrates how the alignment of these two quantities alters the region where correlation increases with firing rate.
Where ∇Ŝ and ∇F are similarly aligned, Ŝ and F will both increase along most paths through that point. In each panel, gray shows the part of the x-plane where \(\frac{d\hat{S}}{dF} = \frac{\nabla\hat{S} \cdot d\mathbf {x}}{\nabla F \cdot d\mathbf {x}} > 0\), black where \(\frac{d\hat{S}}{dF}<0\). From left to right: ∇Ŝ and ∇F positively aligned; ∇Ŝ and ∇F orthogonal; ∇Ŝ and ∇F negatively aligned
To examine the utility of this idea, we reconsider the radial paths along which we previously computed susceptibility. The paths all go through a single point, so we can check the \(\operatorname{sign} ( (\nabla\hat{S} \cdot \mathbf {x}) (\nabla F \cdot \mathbf {x}) )\) for all directions through this point. In Figs. 5(A) and (C), white indicates positive and black negative. Figures 5(B) and (D) repeat the susceptibility-firing rate curves from Fig. 2(B) and Fig. 3(B). For the asynchronous network (Fig. 5(A)), the red, indigo, and royal blue paths are predicted to have negative \(d\hat {S}/dF\), as we can confirm in Fig. 5(B). Yellow, green, and light blue curves are predicted to have positive \(d\hat{S}/dF\). The orange curve is close to \(dF = 0\); the true blue curve is close to \(dS = 0\). For the strong asynchronous network, only the red curve is in the negative region of Fig. 5(C); this is also the only path with \(d\hat {S}/dF < 0\) in Fig. 5(D).
Using a single number to predict \(d\hat{S}/dF\). (A) Paths through parameter space for the asynchronous network: white shows the part of the x-plane where \(\frac{d\hat{S}}{dF} = \frac{\nabla \hat{S} \cdot d\mathbf {x}}{\nabla F \cdot d\mathbf {x}} > 0\), black where \(\frac{d\hat{S}}{dF}<0\), where ∇Ŝ and ∇F are computed at the center of the diagram. (B) Correlation susceptibility vs. firing rate, for each path illustrated in (A). (C)–(D) As in (A)–(B), but for the strong asynchronous network
Of course, this prediction only applies to portions of the path near the point at which we computed the gradients; away from this point, gradients will change along with the direction of the Ŝ vs. F curve. For example, the royal blue curve in Fig. 5(B) increases with firing rate for small firing rates, and the light blue, true blue, and royal blue curves in Fig. 5(D) decrease with firing rate, (for large firing rates).
This motivates a direction-independent measure to quantify the fraction of paths that lead to an increase of correlation with firing rate. This is given exactly in terms of the angle between ∇Ŝ and ∇F:
$$ \cos\theta= \frac{\nabla\hat{S} \cdot\nabla F}{\Vert \nabla \hat{S} \Vert \Vert \nabla F \Vert } $$
and in particular the fraction of x directions in which Ŝ increases with F is given by
$$ \frac{1}{\pi} \biggl( \pi-\cos^{-1} \biggl( \frac{\nabla\hat{S} \cdot\nabla F}{\Vert \nabla\hat{S} \Vert \Vert \nabla F \Vert } \biggr) \biggr) . $$
Because cos−1 has range \([0, \pi]\), this varies between 0 and 1. The more aligned ∇Ŝ and ∇F, the more paths lead to \(\frac{d\hat{S}}{dF} > 0\); the more anti-aligned ∇Ŝ and ∇F, the more paths lead to \(\frac{d\hat{S}}{dF} < 0\).
Strength of Recurrent Excitation Modulates the Correlation–Firing Rate Relationship
Our use of inhibitory susceptibility (i.e. Eq. (10)) to characterize the relationship between correlations and firing rates relied on intermediate assumptions, specifically:
Second-order motifs dominate pairwise correlations.
Inhibitory common input is the dominant second-order motif.
Asynchronous spiking assumption: \(\operatorname {Var}_{T}(n_{i}) = T \nu_{i} \Rightarrow\tilde{\mathbf {C}}_{ii} = \nu_{i}\).
Here, we check that these conditions are still satisfied for a range of neural network models.
In [4], we examined two spiking regimes achieved by varying the strength of excitation: both recurrent excitation \(W_{EE}\) and excitatory input into the inhibitory population \(W_{IE}\). We next applied our theory to a dense grid of parameters (different networks), each identified by its location on the \((W_{EE}, W_{IE})\) plane. Both excitatory strengths were varied from a minimum of their values for the asynchronous network (\(W_{EE} = 0.5\) and \(W_{IE} = 5\)) to a maximum of 1.4 times their value in the strong asynchronous network (to \(W_{EE} = 12.6\) and \(W_{IE} = 11.2\)). The parameter \(W_{XY}\) is a dimensionless scale factor (see Eqs. (17)–(20)); for example in an all-to-all homogeneous network, \(W_{XY}=1\) is when the presynaptic input is exactly the average population firing rate (filtered by the synapse).
To quantify the importance of paths of different length through the network, we can define the contributions at any specific order k by using the interaction network K:
$$\begin{aligned} \tilde{\mathbf {R}}_{ij}^{k} = & \frac{ ( \sum_{l=0}^{k} \tilde{\mathbf {K}}^{l} \tilde{\mathbf {C}}^{0} (\tilde{\mathbf {K}}^{*})^{k-l} )_{ij}}{\sqrt {\tilde{\mathbf {C}}_{ii} \tilde{\mathbf {C}}_{jj} }} . \end{aligned}$$
Then we regressed the total correlation (\(\tilde{\mathbf {C}}_{ij}/\sqrt {\tilde{\mathbf {C}}_{ii}\tilde{\mathbf {C}}_{jj}}\)) against the contributions at each specific order (\(\tilde{\mathbf {R}}^{k}_{ij}\)); the corresponding fraction of variance explained (\(R^{2}\) value) gives a quantitative measure of how well the total correlation can be predicted from each term.
Similarly, we quantity the importance of specific second-order motif types, by regressing the total contribution from second-order motifs (\(\tilde{\mathbf {R}}^{2}_{ij}\)) against the contribution from specific motifs. We performed this computation for each network (a total of 225 networks), and summarize the results in Fig. 6; to present the data compactly, we collapse \(R^{2}\) values (all values are \(\in[0,1]\)) for a fixed \(W_{EE}\) and varying \(W_{IE}\) by showing mean and standard deviation only (standard deviation as error bars). Second-order contributions dominate for small to moderate \(W_{EE}\) (Fig. 6(A)); other orders only compete when \(W_{EE}\) has already exceeded the level of the strong asynchronous network (where the network is close to the edge of instability, and at the limit of validity for mean-field, asynchronous assumptions).
Second-order motifs dominate pairwise correlations in a wide range of networks; inhibitory common input is the dominant second-order motif. (A) Fraction of variance explained (\(R^{2}\)) from linear regressions of total correlation (\(\tilde{\mathbf {C}}_{ij}/\sqrt{\tilde {\mathbf {C}}_{ii}\tilde{\mathbf {C}}_{jj}}\)) against contributions from first order (blue), second order (red), third-order (green), and fourth-order (magenta) motifs. (B) Contributions up to fourth order (\(\tilde{\mathbf {R}}^{k}_{ij}\), for \(k=1,\ldots,4\)) vs. total correlation (\(\tilde{\mathbf {C}}_{ij}/\sqrt{\tilde {\mathbf {C}}_{ii}\tilde{\mathbf {C}}_{jj}}\)) for all E-E cell pairs in a network, for two individual networks included in panel A. (C) Fraction of variance explained (\(R^{2}\)) from linear regressions of contributions to pairwise correlations from second-order motifs (\(\tilde{R}^{2}_{ij}\)) against contributions from the distinct types of second-order motifs: inhibitory common input (magenta), excitatory common input (red), decorrelating chains (green), and correlating chains (blue). Both (A,C): Each data point represents the mean value from 15 networks with \(W_{IE}\) between 5 and 11.2; error bars show standard deviation across these values. \(W_{XY}=1\) is when the presynaptic input is exactly the average population firing rate (filtered by the synapse) in an all-to-all homogeneous network
To illustrate the meaning of this statistic, in Fig. 6(B) we show contributions up to fourth order (\(\tilde{\mathbf {R}}^{k}_{ij}\), for \(k=1,\ldots,4\)) vs. total correlation (\(\tilde{\mathbf {C}}_{ij}/\sqrt{\tilde{\mathbf {C}}_{ii}\tilde{\mathbf {C}} _{jj}}\)) for all E–E cell pairs in a network, for two individual networks included in the summary panel. Note that the second-order contributions cluster near the unity line in both cases, indicating that second-order contributions are the best predictor of total correlations, consistent with the \(R^{2}\) values stated.
Of the second-order motifs, inhibitory common input is the dominant contribution at any value of \(W_{EE}\), except perhaps the last, at which point the excitatory population has unrealistically high firing rates (Fig. 6(C)). To summarize, we have confirmed that far from being limited to a few examples, the conditions we identified in [4] as allowing us to focus on susceptibility to inhibition to explain pairwise correlations, appear to hold up for a variety of networks.
We note that our findings about the relative magnitudes of terms of different orders are purely empirical; that is, they are based on concrete numerical observations, rather than a priori estimates. Thus, it should be reassessed if anything about the parameters or network connectivity is changed. In particular, a likely reason for the relative weakness of first-order terms is that in these networks excitation is almost always weaker than inhibition; while first-order terms are always excitatory, second-order terms can involve excitation and/or (comparatively strong) inhibition.
Having confirmed the validity of our approach, we computed the susceptibility with respect to inhibition, for each of the networks examined in the previous section (because of instability, we restricted these computations to excitatory strengths ×1.2 the values used in [4]). Because background noise values varied slightly, we actually examined two network families; one in which we chose σ values as in the asynchronous network, another in which we chose σ values as in the strong asynchronous network. Also as in [4], we estimated susceptibility using network-averaged values of \(g_{E}\), \(g_{I}\), \(\sigma_{g_{E}}\), and \(\sigma_{g_{I}}\).
Surprisingly, the difference in background noise dwarfed the recurrent excitation strengths, at least in accessing the relationship between Ŝ and firing rate. In Fig. 7, we show Ŝ vs. F curves, for a set of representative networks, on a single plot. Color indicates \(W_{EE}\) (shade of blue) and \(W_{IE}\) (shade of red); values of \(W_{EE}\) are 0.50 (as in the asynchronous network from [4]), 6.45, 9 (as in the strong asynchronous network from [4]), and 10.7, values of \(W_{IE}\) are 5 (as in the asynchronous network from [4]), 7.1, 8 (as in the strong asynchronous network from [4]), and 8.6. For reference, \(W_{XY}=1\) is when the presynaptic input is exactly the average population firing rate (filtered by the synapse) in an all-to-tall homogeneous network, so these coupling strengths vary significantly. We see that, for the full range of recurrent excitation values, Ŝ vs. F curves in Fig. 7(A) are mostly decreasing; Ŝ vs. F curves in Fig. 7(B) are mostly increasing for low F, and saturating for high F.
Firing rate vs. susceptibility (Ŝ), computed for a family of networks generated by modulating the strength of excitation (\(W_{EE}\) and \(W_{IE}\)). (A) Background noise values \(\sigma_{E}\), \(\sigma_{I}\) set as in the asynchronous network from [4]. (B) Background noise values \(\sigma_{E}\), \(\sigma _{I}\) set as in the strong asynchronous network from [4]. Sixteen curves are chosen, for a survey of the range of networks achievable by varying strength of recurrent excitation. Values of \(W_{EE}\) are 0.50 (as in the asynchronous network from [4]), 6.45, 9 (as in the strong asynchronous network from [4]), and 10.7. Values of \(W_{IE}\) are 5 (as in the asynchronous network from [4]), 7.1, 8 (as in the strong asynchronous network from [4]), and 8.6. Again, \(W_{XY}=1\) is when the presynaptic input is exactly the average population firing rate (filtered by the synapse) in an all-to-all homogeneous network, so the coupling strengths vary significantly
Background Noise Modulates the Correlation–Firing Rate Relationship
To further explore the role of background noise, we repeated the susceptibility calculation on additional families of networks, now allowing background noise strengths \(\sigma_{E}\) and \(\sigma_{I}\) (i.e. to the excitatory and inhibitory populations) to vary separately. Input to excitatory cells was varied between \(\sigma_{E} = 1.5\) and 2.5; input to inhibitory cells was varied between \(\sigma_{I} = 1.5\) and 3. These noise values are relatively large; see Eq. (17) and note that voltage is of order 1. In Fig. 8(A) we display susceptibility vs. firing rate curves for 12 \((\sigma_{E}, \sigma_{I})\) pairs; asterisks indicate \(\sigma_{E}\) and \(\sigma_{I}\) by color (green intensity for \(\sigma_{E}\) and blue intensity for \(\sigma_{I}\)). Within each panel curves are colored as in Fig. 7: red intensity for \(W_{IE}\) and blue intensity for \(W_{EE}\).
The strength of background noise modulates the correlation–firing rate relationship. (A) Each panel shows firing rate vs. susceptibility (Ŝ), computed for a family of networks generated by modulating the strength of excitation (\(W_{EE}\) and \(W_{IE}\)) with various background noise levels (see Eq. (17) for \(\sigma_{E}\) and \(\sigma_{I}\) definitions). (B) Population-averaged effective parameters \(\langle g_{I} \rangle\) and \(\mathcal{E}_{\mathrm{rev}}\), for each network displayed in (A); see Eq. (28) for \(\mathcal{E}_{\mathrm{rev}}\)
Surprisingly, most network families (i.e. \((\sigma_{E}, \sigma_{I})\)) were associated with a decrease in correlation with firing rate. The exceptions are \((0.15,0.25)\) (as in the strong asynchronous network in [4]) and \((0.15,0.3)\). This latter was most robustly associated with a positive correlation–firing rate relationship. Furthermore, the shape of susceptibility-firing curves did not appear to vary much with the strength of recurrent excitation (i.e., curves within each panel are similar).
We next sought to investigate possible mechanisms for a positive correlation–firing rate relationship, by examining the effective parameters that govern the neural response: in essence, the network's "operating point" (see Eq. (26)). Possible choices include \(g_{I}\), \(g_{E}\), \(\sigma_{g_{E}}\), \(\sigma_{g_{I}}\), and the effective reversal potential \(\mathcal{E}_{\mathrm{rev}}\); we found \(\sigma _{g_{E}}\) and \(\sigma_{g_{I}}\) to be largely functions of \(g_{E}\) and \(g_{I}\), while \(\mathcal{E}_{\mathrm{rev}}\) has a (nonlinear) functional relationship with \(g_{E}\) and \(g_{I}\). Thus two parameters suffice, and here we chose to use \(g_{I}\) and \(\mathcal{E}_{\mathrm{rev}}\). In Fig. 8(B), we plot average parameter values for each curve, color-coded by \((\sigma_{E}, \sigma_{I})\). Any given color is consistent with a relatively tight range of \(g_{I}\) and (comparatively) broad range of \(E_{\mathrm{rev}}\). As \(\sigma_{I}\) increases (increasing blue intensity), inhibitory conductance \(g_{I}\) increases and reversal potential \(\mathcal{E}_{\mathrm{rev}}\) decreases. However, it was not apparent that any particular region in \((g_{I}, \mathcal{E}_{\mathrm{rev}})\) parameter space was associated with a positive correlation–firing rate relationship, in that the values of \(g_{I}\) and \(\mathcal{E}_{\mathrm{rev}}\) that supported a positive relationship supported negative relationships as well.
In this paper, we showed that using a single-cell firing rate function to examine the relationship between correlations and firing rates is feasible for a wide range of heterogeneous, recurrent networks. We focused on three factors that can modulate the correlation–firing rate relationship: how the network occupies effective parameter space, strength of recurrent excitation, and strength of background noise. Although there are many sets of parameters known to vary within a heterogeneous network of neurons, we have already demonstrated vastly different correlation–firing rate relationships with our methods, with a theory that can be readily applied to other model networks.
One possible application of this work is in designing neural networks for computational experimentation; just as modelers now modify cortical networks to obey experimental constraints on firing rates [3, 35], we could also include a constraint on the desired correlation–firing rate relationship. Here we showed that we can quickly assess a wide range of possible network configurations for a positive correlation–firing rate relationship: in Sect. 2.5, for example, we performed calculations on \(15 \times15 \times12 = 2700\) heterogeneous networks, with a nominal amount of computing time.
One surprising finding in our computations was the relative insensitivity of the slope of the correlation–firing rate relationship to recurrent excitation (\(W_{EE}\), \(W_{IE}\)), as demonstrated in Figs. 7 and 8. This is striking in contrast to the strong sensitivity on display in Figs. 2(B) and 3(B). This difference is explained as follows: in every case where we computed the susceptibility for a self-consistent network (i.e. a solution of Eqs. (26)–(30) and (32)–(33)), the source of heterogeneous firing rates was neural excitability, set via a spiking threshold θ. The resulting effective parameters, such as inhibitory conductance \(\langle g_{I} \rangle\), did not deviate strongly from their population mean values. In essence, all of these networks took a horizontal path through \((\theta, \langle g_{I} \rangle)\) parameter space, as in Figs. 2(C), (D) and Figs. 3(C), (D). If we were to generate networks where heterogeneity arises from another source—such as the strength or frequency of inhibitory connections [23]—we might see different results. We look forward to exploring this in future work.
A priori, there is no reason to expect that the correlation–firing rate relationship in these recurrent networks can be simplified to a feedforward motif based on shared inhibitory input; this was purely an empirical observation (see Fig. 6(B)). We remark that others have shown that the effective input correlation can be canceled to have near zero input (and thus output) correlation on average in balanced networks [28, 40], in contrast to some of the models considered here (i.e., strong asynchronous regime with more net excitation). The conditions for correlation cancellation in these model networks is beyond the scope of this study, but note that others have shown correlation cancellation does not always hold ([21, 22] via altering connection probabilities). The studies that demonstrate correlation cancellation often have faster (or equal) inhibitory synaptic time scales than excitatory: \(\tau_{I} \leq\tau_{E}\) [21, 28, 32] ([40] used current-based instantaneous synapses or \(\tau_{I}=\tau_{E}=10\mbox{ ms}\)) while in our networks the inhibitory synapses have longer time scales (Table 1). Note that Fig. S4 of [28] shows that having effectively zero input correlation does not hold as the inhibitory time scales increase beyond the excitatory time scales. Finally, system size is another key parameter that could certainly affect the magnitude of the recurrent feedback. In contrast to [28, 40], we did not account for how system size would affect correlation cancellation in these heterogeneous networks.
Table 1 Intrinsic parameters that are fixed throughout
Although affirmative answers to whether correlations increase with firing rate in experiments were cited in the Introduction [2, 5, 8, 11, 19, 36, 44] we also note that many experiments have shown that the average correlation across a population can decrease with firing rate when a global state changes or a stimulus is presented. A recent review [9] shows that stimulus-induced decorrelation (with increased firing rate) occurs in a variety of brain regions and animals. This is slightly different from the situation we examine here, where we consider the relationship between correlations and firing rates within a stimulus condition. Regardless, the fact that the relationship between correlation and firing rate is not obvious points to the continued need for theoretical studies into the mechanisms of spike statistics modulation.
Finally, our finding that correlations often decrease, rather than increase, with firing rate stands in apparent contradiction to earlier work on feedforward networks [8, 38]. On closer inspection, we can identify several reasons why our results differ; with conductance inputs (rather than currents) we have a quantitatively different relationship between input parameters and firing rates; furthermore with more adjustable single-neuron parameters, the same sets of firing rates may be observed with single-neuron parameters set in different ways. While the current clamp experiments described in [8] found a consistent increase of correlations with firing rates, we can hypothesize that the parallel dynamic clamp experiments in which pairwise correlations arise from common inhibitory input, would in fact show a decrease with firing rate. However, we also predict that whether an increase or decrease with firing rate is observed would depend on whether firing rates are modulated by varying the level of inhibitory input, or by otherwise varying the excitability of the cells (perhaps pharmacologically).
Neuron Model and Network Setup
We considered randomly connected networks of excitatory and inhibitory neurons. Each cell is a linear integrate-and-fire model with second-order alpha-conductances, i.e. membrane voltage \(v_{i}\) was modeled with a stochastic differential equation, as long as the voltage is below threshold \(\theta_{i}\):
$$\begin{aligned} \tau_{m} \frac{dv_{i}}{dt} =& -v_{i}- g_{E,i}(t) (v_{i}-\mathcal{E}_{E}) - g_{I,i}(t) (v_{i}-\mathcal{E}_{I}) + \sigma_{i} \sqrt{\tau_{m}} \xi_{i}(t). \end{aligned}$$
When \(v_{i}\) reaches \(\theta_{i}\), a spike is recorded and voltage is reset to 0 following a refractory period:
$$\begin{aligned} v_{i}(t) \geq \theta_{i} \quad\Rightarrow\quad v_{i}(t + \tau_{\mathrm{ref}}) = & 0, \end{aligned}$$
Each neuron receives Gaussian white background noise with magnitude \(\sigma_{i}\) depending only on the cell type; that is, \(\langle\xi_{i}(t) \rangle= 0\) and \(\langle\xi_{i}(t) \xi_{i} (t+s) \rangle= \delta(s)\). The membrane time constant, \(\tau_{m}\), and excitatory and inhibitory synaptic reversal potentials, \(\mathcal{E}_{E}\) and \(\mathcal{E}_{I}\), are the same for every cell in the network (see Table 1). The thresholds \(\theta_{i}\) are a significant source of heterogeneity, and they are selected from a log–normal distribution with mean 1 and variance \(e^{(0.2)^{2}}-1\); since the system size is moderate, the \(\theta_{i}\)'s were set to have C.D.F. (cumulative distribution function) values equally spaced from 0.05 to 0.95 for both E and I cells.
Each cell responds to synaptic input through conductance terms, \(g_{E,i}\) and \(g_{I,i}\), which are each governed by a pair of differential equations:
$$\begin{aligned} \tau_{d,X} \frac{dg_{X,i}}{dt} = & -g_{X,i} + g^{(1)}_{X,i} , \end{aligned}$$
$$\begin{aligned} \tau_{r,X} \frac{dg^{(1)}_{X,i}}{dt} = & -g^{(1)}_{X,i} + \tau _{r,X} \alpha_{X} \biggl( \frac{W_{YX}}{N_{YX}} \biggr)\sum _{j\in X,j \rightarrow i} \sum_{k}\delta(t - t_{j,k}) , \end{aligned}$$
where \(Y = \{E,I\}\) denotes the type of cell i and \(X = \{E,I\}\) denotes the type of the source neuron j. Each spike is modeled as a delta-function that impacts the auxiliary variable \(g^{(1)}_{X,i}\); here \(t_{j,k}\) is the kth spike of cell j. The rise and decay time constants \(\tau_{r,X}\) and \(\tau_{d,X}\) and pulse amplitude \(\alpha_{X}\) depend only on the type of the source neuron, that is they are otherwise the same across the population. The parameter \(W_{YX}\) denotes the strength of \(X \rightarrow Y\) synaptic connections, which are (once given the type of source and target neurons) identical across the population. The "raw" synaptic weight (listed in Table 2) is divided by \(N_{YX}\), the total number of \(X \rightarrow Y\) connections received by each Y-type cell.
Table 2 Excitatory connection strengths mediate different firing regimes
Table 2 show connectivity parameters for the two example networks we discuss in Sect. 2.2. For Figs. 1–3, five parameters are set as stated in this table. In Sect. 2.4 and Figs. 6–7, \(W_{EE}\) was varied between 0.5 and 12.6 and \(W_{IE}\) between 5 and 11.2. In Sect. 2.5 and Fig. 8, \(W_{EE}\) was varied between 0.5 and 10.8 and \(W_{IE}\) between 5 and 9.6; \(\sigma_{E}\) was varied between 1.5 and 2.5 and \(\sigma_{I}\) between 1.5 and 3.
Linear Response Theory
In general, computing the response of even a single neuron to an input requires solving a complicated, nonlinear stochastic process. However, it often happens that the presence of background noise linearizes the response of the neuron, so that we can describe this response as a perturbation from a background state. This response is furthermore linear in the perturbing input and thus referred to as linear response theory [31]. The approach can be generalized to yield the dominant terms in the coupled network response as well. We will use the theory to predict the covariance matrix of spiking activity. The derivation is presented in full in [20, 29, 30]; here, we present only the main points.
We assume we have some way to approximate the change in firing rate which occurs as a result of a change in parameter:
$$\begin{aligned} \nu_{i} (t) = & \nu_{i,0} + (A_{X,i} * \epsilon X_{i}) (t); \end{aligned}$$
\(\nu_{i,0}\) is the baseline rate (when \(X = 0\)) and \(A_{X,i}(t)\) is a susceptibility function that characterizes this firing rate response up to order ϵ [8, 20, 41].
In order to consider joint statistics, we need the trial-by-trial response of the cell. First, we propose to approximate the response of each neuron by
$$\begin{aligned} y_{i}(t) \approx& y_{i}^{0}(t) + \biggl( A_{X, i} * \sum_{j} (\mathbf {J}_{X,ij} * y_{j} ) \biggr) (t) ; \end{aligned}$$
that is, each input \(X_{i}\) has been replaced by a filtered version of the presynaptic firing rates \(y_{j}\).
In the frequency domain this becomes
$$\begin{aligned} \tilde{y}_{i}(\omega) = & \tilde{y}_{i}^{0} + \tilde{A}_{X, i}(\omega) \biggl( \sum_{j} \tilde{\mathbf {J}}_{X, ij}(\omega) \tilde{y}_{j} (\omega) \biggr), \end{aligned}$$
where \(\tilde{y}_{i} = \mathcal{F} [ y_{i} - \nu_{i} ]\) is the Fourier transform of the mean-shifted process (\(\nu_{i}\) is the average firing rate of cell i) and \(\tilde{f} = \mathcal{F} [ f ]\) for all other quantities. In matrix form, this yields a self-consistent equation for ỹ in terms of \(\tilde{y}^{0}\):
$$\begin{aligned} \bigl( \mathbf {I}- \tilde{\mathbf {K}}(\omega) \bigr) \tilde{y} = \tilde {y}^{0} \quad \Rightarrow\quad \tilde{y} = \bigl( \mathbf {I}- \tilde{\mathbf {K}}(\omega) \bigr)^{-1} \tilde{y}^{0}, \end{aligned}$$
where \(\tilde{\mathbf {K}}_{ij} (\omega) = \tilde{A}_{X, i}(\omega) \tilde{\mathbf {J}}_{X, ij}(\omega)\) is the interaction matrix in the frequency domain. The cross-spectrum is then computed via
$$\begin{aligned} \bigl\langle \tilde{y}(\omega) \tilde{y}^{\ast}(\omega) \bigr\rangle = & \bigl( \mathbf {I}- \tilde{\mathbf {K}}(\omega) \bigr)^{-1} \bigl\langle \tilde {y}^{0}(\omega) \tilde{y}^{0 \ast}(\omega) \bigr\rangle \bigl( \mathbf {I}- \tilde{\mathbf {K}}^{\ast}(\omega) \bigr)^{-1} . \end{aligned}$$
To compute the interaction matrix for a network of conductance-based neurons, we use the effective time constant approximation (as in the supplemental for [41]). We first separate each conductance into mean and fluctuating parts, e.g., \(g_{E,i} \rightarrow\langle g_{E,i} \rangle+ ( g_{E,i} - \langle g_{E,i} \rangle )\) (see the discussion in [12]). Next we identify an effective conductance \(g_{0,i}\) and potential \(\mathcal{E}_{\mathrm{rev},i}\), and treat the fluctuating part of the conductances as noise, i.e. \(g_{E,i} - \langle g_{E,i} \rangle \rightarrow\sigma_{g_{E},i} \xi_{E,i}(t)\), so that Eq. (17) becomes
$$\begin{aligned} \tau_{m} \frac{dv_{i}}{dt} = & - g_{0,i} (v_{i} - \mathcal{E}_{\mathrm{rev},i}) + \sigma_{g_{E},i} \xi_{E,i}(t) (v_{i} - \mathcal{E}_{E}) \\ &{}+ \sigma_{g_{I},i} \xi_{I,i}(t) (v_{i} - \mathcal{E}_{I}) + \sqrt{\sigma_{i}^{2} \tau_{m}} \xi _{i}(t), \end{aligned}$$
$$\begin{aligned} g_{0,i} = & 1 + \langle g_{E,i}\rangle+ \langle g_{I,i} \rangle , \end{aligned}$$
$$\begin{aligned} \mathcal{E}_{\mathrm{rev},i} = & \frac{ \langle g_{E,i} \rangle\mathcal {E}_{E} + \langle g_{I,i} \rangle\mathcal{E}_{I}}{g_{0,i}} , \end{aligned}$$
$$\begin{aligned} \sigma_{g_{E},i}^{2} = & \operatorname {Var}\bigl[ g_{E,i}(t) \bigr] = \mathrm {E}\bigl[ \bigl( g_{E,i}(t) - \langle g_{E,i} \rangle \bigr) ^{2} \bigr] , \end{aligned}$$
$$\begin{aligned} \sigma_{g_{I},i}^{2} = & \operatorname {Var}\bigl[ g_{I,i}(t) \bigr] = \mathrm {E}\bigl[ \bigl( g_{I,i}(t) - \langle g_{I,i} \rangle \bigr) ^{2} \bigr] . \end{aligned}$$
The parameters which govern the firing rate response will now be the conductance mean and variance, e.g. \(\langle g_{E,i} \rangle\) and \(\sigma_{g_{E},i}^{2}\); we next compute how these depend on incoming firing rates for second-order α-function synapses (Eqs. (19) and (20)). We first simplify the equation for the auxiliary variable (Eq. (20)):
$$\begin{aligned} \tau_{r,X} \frac{dg^{(1)}_{X,i}}{dt} = & -g^{(1)}_{X,i} + \tau _{r,X} \hat{\alpha}_{X,i} \sum _{k} \delta(t-t_{k}) \end{aligned}$$
so that \(\hat{\alpha}_{X,i}\) includes all factors that contribute to the pulse size in Eq. (20), including synapse strength and pulse amplitude. The time constants \(\tau_{r,X}\), \(\tau_{d,X}\) and synapse jump sizes \(\hat{\alpha}_{X,i}\) generally depend on cell type. Then assuming that each spike train is a Poisson process with a constant mean firing rate: i.e., each spike train is modeled as a stochastic process \(S(t)\) with
$$\bigl\langle S(t) \bigr\rangle = \nu; \qquad \bigl\langle S(t)S(t+\tau) \bigr\rangle - \nu^{2} = \nu\delta(\tau), $$
a straightforward but lengthy calculation shows that
$$\begin{aligned} \bigl\langle g_{X,i}(t) \bigr\rangle = & \hat{\alpha}_{X,i} \nu_{X,i} \tau _{r,X} , \end{aligned}$$
$$\begin{aligned} \operatorname {Var}\bigl[ g_{X,i}(t) \bigr] = & \biggl( \frac{1}{2} \hat { \alpha}_{X,i}^{2} \nu_{X,i} \tau_{r,X} \biggr) \biggl( \frac{\tau _{r,X}}{\tau_{r,X} + \tau_{d,X}} \biggr) \\ = &\bigl\langle g_{X,i}(t) \bigr\rangle \times\frac{\hat{\alpha}}{2} \times \biggl( \frac{\tau _{r,X}}{\tau_{r,X} + \tau_{d,X}} \biggr) , \end{aligned}$$
where \(\nu_{X,i}\) is the total rate of type-X spikes incoming to cell i. Notice that modulating the rate of an incoming spike train will impact both the mean and variance of the input to the effective equation, Eq. (26) (via \(\mathcal{E}_{\mathrm{rev},i}\) and \(\sigma _{g_{X},i}\)). Furthermore, this impact may differ for excitatory and inhibitory neurons, giving us a total of four parameters that can be varied in the effective equation.
Therefore, we have four susceptibility functions to compute, \(\tilde {A}_{\langle g_{E} \rangle, i}(\omega)\), \(\tilde{A}_{\langle g_{I} \rangle, i}(\omega)\), \(\tilde{A}_{\sigma_{g_{E}}^{2}, i}(\omega)\), and \(\tilde{A}_{\sigma_{g_{I}}^{2}, i}(\omega)\). The first two capture the change in firing rate as a result of a change in mean conductance—\(\langle g_{E,i} \rangle\rightarrow\langle g_{E,i} \rangle_{0} + \langle g_{E,i} \rangle_{1} \exp(\imath\omega t)\) or \(\langle g_{I,i} \rangle\rightarrow\langle g_{I,i} \rangle_{0} + \langle g_{I,i} \rangle_{1} \exp(\imath\omega t)\)—while the final two address a change in variance—\(\sigma_{g_{E},i}^{2} \rightarrow ( \sigma_{g_{E},i}^{2} )_{0} + ( \sigma_{g_{E},i}^{2} )_{1} \exp(\imath\omega t)\) or \(\sigma_{g_{I},i}^{2} \rightarrow ( \sigma_{g_{I},i}^{2} )_{0} + [4] ( \sigma_{g_{I},i}^{2} )_{1} \exp(\imath\omega t)\). Since the corresponding Fokker–Planck equation required to obtained these entities is linear, we can compute both susceptibilities separately and combine them to get the net effect. With these pieces, we now have the interaction matrix:
$$\begin{aligned} \tilde{\mathbf {K}}_{ij} (\omega) = & \textstyle\begin{cases} \tilde{A}_{\langle g_{E} \rangle, i}(\omega) \tilde{\mathbf {J}}_{ij}(\omega) + \tilde{A}_{\sigma_{g_{E}}^{2}, i}(\omega) \tilde {\mathbf {L}}_{ij}(\omega), & j \text{ excitatory,} \\ \tilde{A}_{\langle g_{I} \rangle, i}(\omega) \tilde{\mathbf {J}}_{ij}(\omega) + \tilde{A}_{\sigma_{g_{I}}^{2}, i}(\omega) \tilde {\mathbf {L}}_{ij}(\omega), & j \text{ inhibitory,} \end{cases}\displaystyle \end{aligned}$$
where \(\tilde{\mathbf {L}}(\omega)\) plays a similar role as \(\tilde{\mathbf {J}}\), but for the effect of incoming spikes on the variance of conductance. Its relationship to \(\tilde{\mathbf {J}}\) (either in the frequency or time domain) is given by the same simple scaling shown in Eq. (33): i.e., for j excitatory,
$$\begin{aligned} \tilde{\mathbf {L}}_{ij} (\omega) = & \tilde{\mathbf {J}}_{ij}(\omega) \times \biggl( \frac{\hat{\alpha}_{E,i}}{2} \biggr) \times \biggl( \frac{\tau_{r,E}}{\tau_{r,E} + \tau_{d,E}} \biggr), \end{aligned}$$
where the first factor comes from the effective spike amplitude \(\hat {\alpha}_{E,i}\) (and is the scale factor proposed in [29], Eq. (64)), and the second arises from using second-order (vs. first-order) alpha-functions.
To implement this calculation, we first solve for a self-consistent set of firing rates: that is, \(\nu_{i}\) is the average firing rate of Eq. (26), along with Eqs. (27)–(30) and (32)–(33). We then apply Richardson's threshold integration method [29, 30] directly to Eq. (26) to compute the unperturbed power spectrum (\(\langle\tilde{y}^{0}(\omega ) \tilde{y}^{0 \ast}(\omega) \rangle\)) and susceptibility functions. The software we used to implement this calculation is described more fully in [4] and can be found at https://github.com/andreakbarreiro/LR_CondBased.
Computing Statistics from Linear Response Theory
Linear response theory yields the cross-spectrum of the spike train, \(\langle\tilde{y}_{i}(\omega) \tilde{y}_{j}^{\ast}(\omega) \rangle\), for each distinct pair of neurons i and j (see Eq. (25)). The cross-correlation function, \(\mathbf {C}_{ij}(\tau)\), measures the similarity between two processes at time lag τ, while the cross-spectrum measures the similarity between two processes at frequency ω:
$$\begin{aligned} \mathbf {C}_{ij} (\tau) \equiv& \bigl\langle \bigl(y_{i}(t)- \nu_{i} \bigr) \bigl(y_{j}(t + \tau )-\nu_{j} \bigr) \bigr\rangle , \end{aligned}$$
$$\begin{aligned} \tilde{\mathbf {C}}_{ij}(\omega) \equiv& \bigl\langle \tilde{y}_{i} (\omega) \tilde{y}_{j} (\omega) \bigr\rangle . \end{aligned}$$
The Weiner–Khinchin theorem [31] implies that \(\{ \mathbf {C}_{ij}, \tilde{\mathbf {C}}_{ij} \} \) are a Fourier transform pair: that is,
$$\begin{aligned} \tilde{\mathbf {C}}_{ij} (\omega) = & \int_{-\infty}^{\infty} \mathbf {C}_{ij} (t) e^{-2 \pi\imath\omega t}\, dt . \end{aligned}$$
In principle, the cross-correlation \(\mathbf {C}(t)\) and cross-spectrum \(\tilde{\mathbf {C}}(\omega)\) matrices are functions on the real line, reflecting the fact that correlation can be measured at different time scales. In particular, for a stationary point process the covariance of spike counts over a window of length T, \(n_{i}\) and \(n_{j}\), can be related to the cross-correlation function \(\mathbf {C}_{ij}\) by the following formula [17]:
$$\begin{aligned} \operatorname {Cov}_{T}(n_{i}, n_{j}) = & \int_{-T}^{T} \mathbf {C}_{ij}(\tau) (T - \vert \tau\vert )\, d\tau . \end{aligned}$$
The variance of spike counts over a time window of length T, \(n_{i}\), is likewise given by integrating the autocorrelation function \(\mathbf {C}_{ii}\):
$$\begin{aligned} \operatorname {Var}_{T}(n_{i}) = & \int_{-T}^{T} \mathbf {C}_{ii}(\tau) (T - \vert \tau\vert )\, d\tau . \end{aligned}$$
By normalizing by the time window and taking the limit as \(T\rightarrow \infty\),
$$\begin{aligned} \lim_{T \rightarrow\infty} \frac{\operatorname {Cov}_{T}(n_{i}, n_{j})}{T} = & \lim_{T \rightarrow\infty} \int_{-T}^{T} \mathbf {C}_{ij}(\tau) \biggl(1 - \frac{\vert \tau\vert }{T} \biggr)\, d\tau \\ = & \int_{-\infty}^{\infty} \mathbf {C}_{ij}(\tau)\, d\tau = \tilde{\mathbf {C}}_{ij}(0) , \end{aligned}$$
we can see that, for an integrable cross-correlation function, we can use \(\tilde{\mathbf {C}}_{ij}(0)\) as a measure of long-time covariance.
Similarly, the long-time limit of the Pearson correlation coefficient of the spike counts,
$$\begin{aligned} \lim_{T \rightarrow\infty} \rho_{T,ij} = & \lim_{T \rightarrow \infty} \frac{\operatorname {Cov}_{T}(n_{i},n_{j})}{\sqrt{ \operatorname {Var}_{T}(n_{i}) \operatorname {Var}_{T}(n_{j})}} = \frac{\tilde{\mathbf {C}}_{ij}(0)}{\sqrt{\tilde{\mathbf {C}}_{ii}(0) \tilde {\mathbf {C}}_{jj}(0)}} , \end{aligned}$$
gives us a normalized measure of long-time correlation.
excitatory
inhibitory
Asyn:
Strong Asyn:
strong asynchronous
Averbeck BB, Latham PE, Pouget A. Neural correlations, population coding and computation. Nat Rev Neurosci. 2006;7:358–66.
Bair W, Zohary E, Newsome WT. Correlated firing in macaque visual area mt: time scales and relationship to behavior. J Neurosci. 2001;21(5):1676–97.
Barreiro AK, Gautam SH, Shew W, Ly C. A theoretical framework for analyzing coupled neuronal networks: application to the olfactory system. PLoS Comput Biol. 2017;13(10):e1005780.
Barreiro AK, Ly C. When do correlations increase with firing rate in recurrent networks? PLoS Comput Biol. 2017;13(4):e1005506.
Cohen MR, Maunsell JHR. Attention improves performance primarily by reducing interneuronal correlations. Nat Neurosci. 2009;12:1594–600.
da Silveira RA, Berry MJ. High-fidelity coding with correlated neurons. PLoS Comput Biol. 2014;10(11):e1003970.
Dayan P, Abbott LF. Theoretical neuroscience: computational and mathematical modeling of neural systems. London: Taylor & Francis; 2001.
de la Rocha J, Doiron B, Shea-Brown E, Josić K, Reyes A. Correlation between neural spike trains increases with firing rate. Nature. 2007;448:802–6.
Doiron B, Litwin-Kumar A, Rosenbaum R, Ocker GK, Josić K. The mechanics of state-dependent neural correlations. Nat Neurosci. 2016;19(3):383–93.
Ecker AS, Berens P, Tolias AS, Bethge M. The effect of noise correlations in populations of diversely tuned neurons. J Neurosci. 2011;31(40):14272–83.
Franke F, Fiscella M, Sevelev M, Roska B, Hierlemann A, Azeredo da Silveira R. Structures of neural correlation and how they favor coding. Neuron. 2016;89:409–22.
Gerstner W, Kistler WM, Naud R, Paninski L. Neuronal dynamics: from single neurons to networks and models of cognition. Cambridge: Cambridge University Press; 2014.
Helias M, Tetzlaff T, Diesmann M. The correlation structure of local neuronal networks intrinsically results from recurrent dynamics. PLoS Comput Biol. 2014;10(1):e1003428.
Hu Y, Trousdale J, Josic K, Shea-Brown E. Motif statistics and spike correlations in neuronal networks. J Stat Mech Theory Exp. 2013;2013(3):03012.
Hu Y, Zylberberg J, Shea-Brown E. The sign rule and beyond: boundary effects, flexibility, and noise correlations in neural population codes. PLoS Comput Biol. 2014;10(2):e1003469.
Josić K, Shea-Brown E, Doiron B, de la Rocha J. Stimulus-dependent correlations and population codes. Neural Comput. 2009;21:2774–804.
Kay SM. Fundamentals of statistical signal processing, volume 1: estimation theory. New York: Prentice Hall; 1993.
Kohn A, Coen-Cagli R, Kanitscheider I, Pouget A. Correlations and neuronal population information. Annu Rev Neurosci. 2016;39:237–56.
Lin IC, Okun M, Carandini M, Harris KD. The nature of shared cortical variability. Neuron. 2015;87:644–56.
Lindner B, Doiron B, Longtin A. Theory of oscillatory firing induced by spatially correlated noise and delayed inhibitory feedback. Phys Rev E. 2005;72(6):061919.
Litwin-Kumar A, Doiron B. Slow dynamics and high variability in balanced cortical networks with clustered connections. Nat Neurosci. 2012;15(11):1498–505.
Litwin-Kumar A, Doiron B. Formation and maintenance of neuronal assemblies through synaptic plasticity. Nat Commun. 2014;5:5319.
Ly C. Firing rate dynamics in recurrent spiking neural networks with intrinsic and network heterogeneity. J Comput Neurosci. 2015;39:311–27.
Martin KAC. Microcircuits in visual cortex. Curr Opin Neurobiol. 2002;12(4):418–25.
Moreno-Bote R, Beck J, Kanitscheider I, Pitkow X, Latham P, Pouget A. Information-limiting correlations. Nat Neurosci. 2014;17:1410–7.
Ostojic S, Brunel N, Hakim V. How connectivity, background activity, and synaptic properties shape the cross-correlation between spike trains. J Neurosci. 2009;29:10234–53.
Pernice V, Staude B, Cardanobile S, Rotter S. How structure determines correlations in neuronal networks. PLoS Comput Biol. 2011;7(5):e1002059.
Renart A, de la Rocha J, Bartho P, Hollender L, Parga N, Reyes A, Harris KD. The asynchronous state in cortical circuits. Science. 2010;327:587–90.
Richardson MJE. Firing-rate response of linear and nonlinear integrate-and-fire neurons to modulated current-based and conductance-based synaptic drive. Phys Rev E. 2007;76:021919.
Richardson MJE. Spike-train spectra and network response functions for non-linear integrate-and-fire neurons. Biol Cybern. 2008;99:381–92.
Risken H. The Fokker–Planck equation: methods of solutions and applications. New York: Springer; 1989.
Rosenbaum R, Smith MA, Kohn A, Rubin JE, Doiron B. The spatial structure of correlated neuronal variability. Nat Neurosci. 2017;20:107–14.
Ruff DA, Cohen MR. Attention can either increase or decrease spike count correlations in visual cortex. Nat Neurosci. 2014;17(11):1591–7.
Sanchez-Vives MV, McCormick DA. Cellular and network mechanisms of rhythmic recurrent activity in neocortex. Nat Neurosci. 2000;3(10):1027–34.
Schuecker J, Schmidt M, van Albada SJ, Diesmann M, Helias M. Fundamental activity constraints lead to specific interpretations of the connectome. PLoS Comput Biol. 2017;13(2):e1005179.
Schulz DPA, Sahani M, Carandini M. Five key factors determining pairwise correlations in visual cortex. J Neurophysiol. 2015;114:1022–33.
Shamir M, Sompolinsky H. Implications of neuronal diversity on population coding. Neural Comput. 2006;18:1951–86.
Shea-Brown E, Josic K, de la Rocha J, Doiron B. Correlation and synchrony transfer in integrate-and-fire neurons: basic properties and consequences for coding. Phys Rev Lett. 2008;100:108102.
Shu Y, Hasenstaub A, McCormick DA. Turning on and off recurrent balanced cortical activity. Nature. 2003;423(6937):288–93.
Tetzlaff T, Helias M, Einevoll GT, Diesmann M. Decorrelation of neural-network activity by inhibitory feedback. PLoS Comput Biol. 2012;8:e1002596.
Trousdale J, Hu Y, Shea-Brown E, Josic K. Impact of network structure and cellular response on spike time correlations. PLoS Comput Biol. 2012;8(3):e1002408.
van Vreeswijk C, Sompolinsky H. Chaos in neuronal networks with balanced excitatory and inhibitory activity. Science. 1996;274:1724–6.
Zhao L, Beverlin B II, Netoff T, Nykamp DQ. Synchronization from second order network connectivity statistics. Front Comput Neurosci. 2011;5:1–16.
Zylberberg J, Cafaro J, Turner MH, Shea-Brown E, Rieke F. Direction-selective circuits shape noise to ensure a precise population code. Neuron. 2016;89(2):369–83.
This work was motivated in part by useful feedback we received at the Third Annual International Conference on Mathematical Neurosciences, held in Boulder CO in May–June 2017. We would like to thank the organizers for an enjoyable and stimulating conference.
Software used to generate the computational results shown here can be found at: https://github.com/andreakbarreiro/LR_CondBased.
Department of Mathematics, Southern Methodist University, Dallas, USA
Andrea K. Barreiro
Department of Statistical Science and Operations Research, Virginia Commonwealth University, Richmond, USA
Cheng Ly
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AKB and CL designed the project. AKB and CL wrote software. AKB performed simulations. AKB and CL designed figures. AKB and CL wrote the paper. All authors read and approved the final manuscript.
Correspondence to Andrea K. Barreiro.
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.
Barreiro, A.K., Ly, C. Investigating the Correlation–Firing Rate Relationship in Heterogeneous Recurrent Networks. J. Math. Neurosc. 8, 8 (2018). https://doi.org/10.1186/s13408-018-0063-y
DOI: https://doi.org/10.1186/s13408-018-0063-y
Recurrent network
Spike count correlation
Linear response
Third Annual International Conference on Mathematical Neuroscience | CommonCrawl |
I love watches, and I had an idea for a weird kind of watch movement (all of the stuff that moves the hands). It is made up of a a central wheel, with one of the hands connected to it (in this case, it will be the hour hand). This hand goes through a pivot, and then displays the time. I attached a video of a 3d mock up here, because it is kinda hard to explain. My question is, is there any functions that would be able to graph the movement of the end of the hand? I don't want to make the real prototype just yet.
I will take the origin to be the place the hand slides through, $y$ vertical positive up, $x$ horizontal positive right. Let the hand have length $L$ and the circle radius $R$. It appears $L$ is a little greater than $2R$, so it sticks out of the pivot even when the left end is at the farthest left point.
Denote with $l$ the length of the hand and with $R$ the radius of the circle.
As an exercise you can eliminate angle $\alpha$ and obtain an implicit relation between coordinates of point $M$, but there is not much that you can do with it. It is better to work with parametric equations. Select $l,R$ and calculate coordinates for a range of $\alpha$ angles.
Not the answer you're looking for? Browse other questions tagged geometry functions trigonometry or ask your own question.
Function for this game movement graph?
How can I relate the lengths in an analog watch and the angles in sexagesimal system? | CommonCrawl |
\begin{document}
\title{An improved approximation algorithm for the minimum cost subset
$k$-connected subgraph problem} \author{Bundit Laekhanukit\thanks{ School of Computer Science, McGill University. \hbox{Email: [email protected]}.}
\thanks{ This research was supported by Natural Sciences and Engineering Research Council of Canada (NSERC) grant no.~288334, 429598 and by Harold H Helm fellowship.}\thanks{
Most of the work was done while the author was in Department of Combinatorics and Optimization, University of Waterloo, Canada.} }
\maketitle
\begin{abstract}
The minimum cost subset $k$-connected subgraph problem is a
cornerstone problem in the area of network design with vertex
connectivity requirements. In this problem, we are given a graph
$G=(V,E)$ with costs on edges and a set of terminals $T$. The goal
is to find a minimum cost subgraph such that every pair of terminals
are connected by $k$ openly (vertex) disjoint paths.
In this paper, we present an approximation algorithm for the subset
$k$-connected subgraph problem which improves on the previous best
approximation guarantee of $O(k^2\log{k})$ by Nutov (FOCS 2009).
Our approximation guarantee, $\alpha(|T|)$, depends upon the
number of terminals: \[ \alpha(|T|) \ \ =\ \ \begin{cases}
O(k \log^2 k) & \mbox{if } 2k\le |T| < k^2\\ O(k \log k) & \mbox{if } |T| \ge k^2 \end{cases} \]
So, when the number of terminals is {\em large enough}, the
approximation guarantee improves significantly.
Moreover, we show that, given an approximation algorithm for
$|T|=k$, we can obtain almost the same approximation guarantee
for any instances with $|T|> k$.
This suggests that the hardest instances of the problem are when
$|T|\approx k$.
\end{abstract}
\section{Introduction} \label{sec:intro}
We present an improved approximation algorithm for the {\em minimum
cost subset $k$-connected subgraph problem}. In this problem (subset $k$-connectivity, for short), we are given a graph $G=(V,E)$ with edge costs and a set of terminals $T\subseteq V$. The goal is to find a minimum cost subgraph such that each pair of terminals is connected by $k$ openly (vertex) disjoint paths. This is a fundamental problem in network design which includes as special cases the minimum cost Steiner tree problem (the case $k=1$) and the minimum cost $k$ vertex-connected spanning subgraph problem (the case $T=V$). However, the subset $k$-connectivity problem is significantly harder than these two special cases. Specifically, an important result of Kortsarz, Krauthgamer and Lee~\cite{KKL04} shows that the problem does not admit an approximation guarantee better than $2^{\log^{1-\epsilon}{n}}$ for any $\epsilon>0$ unless $\mathrm{NP}\subseteq \mathrm{DTIME}(n^{O(\mathrm{polylog}(n))})$. In contrast, polylogarithmic approximation guarantees are known for the minimum cost $k$-vertex connected spanning subgraph problem. The first such result was obtained by Fakcharoenphol and Laekhanukit \cite{FL08} using
the {\em Halo-set decomposition}, introduced by Kortsarz and Nutov~\cite{KN05}.
Subsequently, Nutov~\cite{Nutov09a} improved the approximation guarantee to $O\left(\log k \cdot \log\frac{n}{n-k}\right)$.
Since the hardness result of Kortsarz et~at.~\cite{KKL04} no non-trivial approximation algorithm was known for the general case of the subset-$k$-connectivity problem until the work of Chakraborty, Chuzhoy and Khanna~\cite{CCK08}. They presented an $O(k^{O(k^2)}\cdot\log^4|T|)$-approximation algorithm for the rooted version of our problem, namely the {\em rooted subset $k$-connectivity problem}. There, given a root vertex $r$ and a set of terminals $T$, the goal is to find a minimum cost subgraph that has $k$ openly disjoint paths from the root vertex $r$ to every terminal in $T$. Chakraborty et~al. showed how to solve the subset $k$-connectivity problem by applying the rooted subset $k$-connectivity algorithm $k$ times, thus obtaining an $O(k^{O(k^2)}\cdot\log^4|T|)$-approximation algorithm.
Recently, in a series of developments
\cite{CCK08,CK08,CK09,Nutov12,Nutov09c}, the approximation guarantees for the rooted subset $k$-connectivity problem has been steadily improved. This has culminated in an $O(k\log{k})$ guarantee due to Nutov~\cite{Nutov12}, thus implying an approximation guarantee of $O(k^2\log{k})$ for the subset $k$-connectivity problem.
There is a trivial way to obtain an approximation bound of $O(|T|^2)$.
So, with the current progress on the rooted subset $k$-connectivity problem, the application of the rooted subroutine is only useful when the number of terminals is large enough, say $|T|\ge 2k$.
The main contribution of this paper is to show that, in this case, only a polylogarithmic number of applications of the rooted subset $k$-connectivity algorithm are required to solve the subset $k$-connectivity problem.
Given an approximation algorithm for the rooted subset $k$-connectivity problem, we show that only $O(\log^2{k})$ applications of the algorithm are required, and we can save a factor of $O(\log{k})$ since some of these applications are applied to instances with lower costs.
Moreover, as the number of terminal increases above $k^2$, we are able to save an additional $O(\log k)$ factor. Thus, given an approximation algorithm for the rooted subset $k$-connectivity problem in~\cite{Nutov12} due to Nutov (and with careful analysis), we achieve an $\alpha(|T|)$-approximation guarantee where \[ \alpha(|T|) \ \ =\ \ \begin{cases}
O(k \log^2 k) &\ \ \ \mbox{if } 2k\le |T| < k^2\\ O(k \log k) &\ \ \ \mbox{if } |T| \ge k^2 \end{cases} \]
As we may combine our algorithm with the trivial $O(|T|^2)$-approximation algorithm for the case $|T|<2k$, we obtain an approximation guarantee of $O(k^2)$, which improves upon the previous best approximation guarantee of $O(k^2\log{k})$ for all cases.
Moreover, for $|T| \geq 2k$, we obtain a significant improvement of a factor of $k$.
Observe, however, that for the case $|T|\approx k$ the guarantee is still quadratic. At first, this may seem paradoxical since we may hope that the problem is easier when the number of terminals is small. Our results suggest that this is not the case. Indeed, it appears that the hardest instances of subset $k$-connectivity may have at most $k$ terminals.
Precisely, we show that, given an $\alpha(k)$-approximation algorithm for the subset $k$-connectivity problem with $|T|=k$, there is an $(\alpha(k)+f(k))$-approximation algorithm for any instance with $|T|>k$, where $f(k)$ is the best known approximation guarantee for the rooted subset $k$-connectivity problem.
Furthermore, we give an approximation preserving reduction from the rooted subset $k$-connectivity problem to the subset $k$-connectivity problem, showing a strong connection between the two problems.
\ \\ {\bf Related Work.} Some very special cases of the subset $k$-connectivity problem are known to have constant factor approximation algorithms.
For $k=1$, the minimum cost Steiner tree problem, the best known approximation guarantee is $1.39$ due to Byrka, Grandoni, Rothvo{\ss} and Sanit{\`a}~\cite{BGRS10}.
For $k=2$, a factor two approximation algorithm was given by Fleischer, Jain and Williamson~\cite{FJW06}.
The subset $k$-connectivity problem also has an $O(1)$-approximation algorithm when edge costs satisfy the triangle inequality; see Cheriyan and Vetta~\cite{CV07}.
The most general problem in this area is the vertex-connectivity survivable network design problem (VC-SNDP). In VC-SNDP, the connectivity requirement for each pair of vertices can be arbitrary.
Recently, Chuzhoy and Khanna~\cite{CK09} showed that there is an $O(k^3\log n)$-approximation algorithm for VC-SNDP.
The problems where requirements are edge and element connectivity (EC-SNDP and Element-SNDP) are also very well studied.
Both problems admit $2$-approximation algorithms via iterative rounding.
For EC-SNDP, a $2$-approximation algorithm was given by Jain~\cite{Jain01}.
For Element-SNDP a $2$-approximation algorithm was given by Fleischer, Jain and Williamson~\cite{FJW06}.
The vertex-cost versions of these problems have also been studied in literature. Nutov~\cite{Nutov10-nodecost} gave an approximation guarantee of $O(k\log|T|)$ for vertex-cost EC-SNDP using a technique, called spider decomposition. Later on, in~\cite{Nutov12}, Nutov applied the spider decomposition technique to other vertex-cost problems, giving approximation guarantees of $O(k\log|T|)$ for Element-SNDP, $O(k^2\log|T|)$ for the rooted subset $k$-connectivity problem, $O(k^3\log|T|)$ for the subset $k$-connectivity problem and $O(k^4\log^2|T|)$ for VC-SNDP.
\section{Preliminaries and Results} \label{sec:prelim}
We begin with some formal definitions. Let $G=(V,E)$ denote the graph for an instance of the problem. For a set of edges $F$, the graph $G'=(V,E\cup F)$ is denoted by $G+F$; for a vertex $v$, the graph obtained from $G$ by removing $v$ is denoted by $G-v$. For any set of vertices $U\subseteq V$, let $\Nbr{U}$ denote the set of {\em neighbors} of $U$; that is, $\Nbr{U}=\{v\in V-U: \exists (u,v)\in E, u\in U\}$. Define a set $\inv{U}$ to be $V-(U\cup \Nbr{U})$, which is the {\em vertex-complement} of $U$.
For any pair of vertices $s,t\in V$, two $s,t$-paths are {\em openly disjoint} if they have no vertices except $s$ and $t$ in common.
Let $T\subseteq V$ be a set of vertices called {\em terminals}. Without loss of generality, assume that no two terminals of $T$ are adjacent in $G$. This assumption can be easily justified by subdividing every edge joining two terminals; that is, if there is an edge $(s,t)$ joining two terminals, then we replace $(s,t)$ by two new edges $(s,u)$ and $(u,t)$ and set costs of the new edges so that $c(s,t)=c(s,u)+c(u,t)$, where $c(.)$ is a cost function.
The set of terminals $T$ is {\em $k$-connected in $G$} if the graph $G$ has $k$ openly disjoint $s,t$-paths between every pair of terminals $s,t\in T$.
Thus, by Menger Theorem, the removal of any set of vertices with size at most $k-1$ leaves all the remaining terminals in the same component of the remaining graph.
By the {\em subset connectivity} of $G$ on $T$, we mean the maximum integer $\ell$ such that $T$ is $\ell$-connected in $G$. A {\em deficient set} is a subset of vertices $U\subseteq V$ such that both $U$ and $\inv{U}$ contain terminals of $T$ and
$|\Nbr{U}|<k$. Observe that the vertex-complement $\inv{U}$ is also a deficient set. Similarly, given a designated {\em root} vertex $r$, the graph is {\em $k$-connected from $r$ to $T$} if $G$ has $k$ openly disjoint $r,t$-paths for every terminal $t\in T$ ($r$ may or may not be in $T$).
By the {\em rooted connectivity} of $G$ from $r$ to $T$, we mean the maximum integer $\ell$ such that $G$ is $\ell$-connected from $r$ to $T$.
In the {\em subset $k$-connectivity problem}, we are given a graph $G=(V,E)$ with a cost $c(e)$ on each edge $e\in E$, a set of terminals $T\subseteq V$, and an integer $k\ge 0$. The goal is to find a set of edges $\widehat{E}\subseteq E$ of minimum cost such that $T$ is $k$-connected in the subgraph $\widehat{G}=(V,\widehat{E})$.
In the {\em rooted subset $k$-connectivity problem}, our goal is to find a set of edges $\widehat{E}\subseteq E$ of minimum cost such that the subgraph $\widehat{G}=(V,\widehat{E})$ is $k$-connected from $r$ to $T$, for a given root $r$.
Nutov~\cite{Nutov12} recently gave an $O(k\log{k})$-approximation algorithm for the rooted subset $k$-connectivity problem. The approximation guarantee improves by a logarithmic factor for the problem of increasing the rooted connectivity of a graph by one.
\begin{theorem}[Nutov 2009~\cite{Nutov12}]
\label{thm:rooted-kconn}
There is an $O(k\log{k})$-approximation algorithm for the rooted
subset $k$-connectivity problem. Moreover, consider the restricted
version of the problem where the goal is to increase the rooted
connectivity from $\ell$ to $\ell+1$.
Then the approximation guarantee (with respect to a standard LP) is
$O(\ell)$. \end{theorem}
Our focus is upon the subset $k$-connectivity problem.
The followings are our main results:
\begin{theorem}
\label{thm:sskconn-main}
For any set $T$ of terminals, there is an
$\alpha(|T|)$-approximation algorithm for the subset
$k$-connectivity problem where \[ \alpha(|T|) \ \ =\ \ \begin{cases} O(|T|^2) &\ \ \ \mbox{if } |T| < 2k
\quad\mbox{(folklore)} \\ O(k \log^2 k) &\ \ \ \mbox{if } 2k\le |T| < k^2\\ O(k \log k) &\ \ \ \mbox{if } |T| \ge k^2 \end{cases} \] In particular, there is an $O(k^2)$-approximation algorithm for the general case of the subset $k$-connectivity problem, and there is an $O(k\log{k})$-approximation algorithm when $|T| \ge k^2$. \end{theorem}
\begin{proposition} \label{prop:hardest} Consider the subset $k$-connectivity problem. Suppose there is an $\alpha(k)$-approximation algorithm for instances with $|T|=k$. Then there is an $(\alpha(k)+f(k))$-approximation algorithm for any instance with $|T|>k$, where $f(k)$ is the best known approximation guarantee for the rooted subset $k$-connectivity problem. \end{proposition}
\begin{theorem} \label{thm:rooted-to-subset} There is an approximation preserving reduction such that, given an instance of the rooted subset $k$-connectivity problem consisting of a graph $G$, a root vertex $r$ and a set of terminals $T$, outputs an instance of the subset $k$-connectivity problem consisting of a graph $G'$ and a set of terminals $T\cup\{r\}$. \end{theorem}
The hardness result in Theorem~\ref{thm:rooted-to-subset} together with the hardness of the rooted subset $k$-connectivity problem by Cheriyan, Laekhanukit, Naves and Vetta~\cite{CLNV12} implies the hardness of $\Omega(k^\epsilon)$, for the subset $k$-connectivity problem, where $\epsilon>0$ is some fixed constant.
Some results and proofs similar to the ones in this paper have appeared in previous literature; see~\cite{CVV03,KN05,Bundit-Thesis}. In particular, Lemma~\ref{lmm:low-thickness} and Lemma~\ref{lmm:two-cores} appeared in~\cite{KN05} and~\cite{Bundit-Thesis}, respectively. The proofs of Proposition~\ref{prop:hardest} is identical to that of the case $T=V$, which was given in \cite{KR96} and also in \cite{ADNP99}. Our key new contributions are Lemmas~\ref{lmm:bound-1},~\ref{lmm:halo-nbr} and~\ref{lmm:num-cores}, which allow us to extend the result in~\cite{KN05} to the subset $k$-connectivity problem.
We remark that, at the time this paper is written, the approximation guarantee of the subset $k$-connectivity problem was improved by Nutov~\cite{Nutov11} to $O(k\log{k})$ for all $k\leq |T|-o(|T|)$.
\noindent{\bf Organization: } In Section~\ref{sec:algo}, we present an approximation algorithm for the subset $k$-connectivity problem, which is the main result in this paper.
In Section~\ref{sec:below2k}, we give a discussion that our algorithm and analysis can be extended to the case $k<|T|< 2k$. To keep the presentation simple, Section~\ref{sec:below2k} is presented separately from the main result.
In Section~\ref{sec:hardness}, we discuss the hardness of the subset $k$-connectivity problem. To be precise, we show that the hardest instance of the subset $k$-connectivity problem might be when
$|T|\approx{k}$, and we give an approximation preserving reduction from the rooted subset $k$-connectivity problem to the subset $k$-connectivity problem. \\
\section{An approximation algorithm} \label{sec:algo}
Our main result in Theorem \ref{thm:sskconn-main} breaks up into three cases where there are a small number, a moderate number and a large number of terminals, respectively.
Indeed, the first case is a folklore.
When there are a small number of terminals ($|T|<2k$), we apply the following trivial $O(|T|^2)$-approximation algorithm. We find $k$ openly disjoint paths of minimum cost between every pair of terminals by applying a minimum cost flow algorithm. Let $opt$ denote the cost of the optimal solution to the subset $k$-connectivity problem. Since any feasible solution to the subset $k$-connectivity problem has $k$ openly disjoint paths between every pair of terminals, the cost incurred by finding a minimum cost collection of $k$ openly disjoint paths between any pair of terminals is at most $opt$. Since we have at most $|T|^2$ pairs, this incurs a total cost of $O(|T|^2\cdotopt)$.
The remaining two cases are similar. Things are slightly easier, though, when there are large number of terminals ($|T| \ge k^2$), leading to a slightly better guarantee than when there are a moderate number of terminals ($2k \le |T|$). We devote most of this section to presenting an approximation algorithm for the moderate case. (In Section~\ref{sec:very-simple-algo}, we show the improvement for the case of a large number of terminals.)
Our algorithm works by repeatedly increasing the subset connectivity of a graph by one.
We start with a graph that has no edges. Then we apply $k$ {\em outer} iterations. Each outer iteration increases the subset connectivity (of the current graph) by one by adding a set of edges of approximately minimum cost. The analysis of the outer iterations applies linear programming (LP) scaling and incurs a factor of $O(\log{k})$ in the approximation guarantee for the $k$ outer iterations.
The analysis based on LP-scaling can be seen in~\cite{RW97,CV07,KN05,FL08} and also in~\cite{GGPSTW94,Bundit-Thesis}.
The following is a standard LP-relaxation for the subset $k$-connectivity problem. \[ \begin{array}{lll} \min & \displaystyle\sum_{e\in E}c_ex_e \\ \mbox{s.t.}
&
\displaystyle\sum_{e\in\delta(U,W)}x_e \geq k - |V-(U\cup W)|
& \forall (U,W)\in\mathcal{S}\\
& 0 \leq x_e \leq 1 & \forall e\in E \end{array} \] where $\delta(U,W)=\{(u,w)\in E:u\in U, w\in W\}$ is a set of edges with one endpoint in $U$ and the other endpoint in $W$, and $\mathcal{S}=\{(U,W)\in V\times V: U\cap W=\emptyset, U\cap{T}\neq\emptyset, W\cap{T}\neq\emptyset\}$.
\begin{lemma} Suppose there is a $\beta(\ell)$-approximation algorithm for the problem of increasing the subset connectivity of a graph from $\ell$ to $\ell+1$ with respect to a standard LP, where $\beta(\ell)$ is a non-decreasing function.Then there is an $O(\beta(k)\log k)$-approximation algorithm for the subset $k$-connectivity problem. \label{lmm:LP-scaling} \end{lemma}
We are left with the key problem of increasing the subset connectivity (of the current graph) by one by adding a set of edges of approximately minimum cost. Throughout this section, we assume that
the set of terminals $T$ is $\ell$-connected in the current graph, and $|T|\geq 2k \geq 2\ell$.
Also, we assume that no two terminals are adjacent in the input graph $G=(V,E)$.
\\
\noindent{\bf Assumption:}
The set of terminals $T$ is $\ell$-connected in the current graph $\widehat{G}=(V,\widehat{E})$.
Moreover, no two terminals are adjacent in the input graph $G$. \\
Our algorithm solves the problem of increasing the subset connectivity of a graph by one by applying a number of so-called {\em inner} iterations.
To describe our algorithm, we need some definitions and subroutines. Thus, we defer the description of our algorithm to Section~\ref{sec:second-algo}. In Section~\ref{sec:deficient}, we give important definitions and structures of subset $\ell$-connected graphs called ``cores'' and ``halo-families''. Our algorithm requires two subroutines. The first one is the subroutine that employs the rooted subset $(\ell+1)$-connectivity algorithm to cover halo-families. This subroutine is given in Section~\ref{sec:cover-halo}. The second one is the subroutine for decreasing the number of cores to $O(\ell)$, which is given in Section~\ref{sec:reduce-cores}. Then we introduce a notion of ``thickness'' in Section~\ref{sec:thickness}. This notion guides us how to use the rooted subset $(\ell+1)$-connectivity algorithm efficiently. Finally, in Section~\ref{sec:second-algo}, we present an $O(k\log^2{k})$-approximation algorithm for the case $|T|\ge2k$. By slightly modifying the algorithm and analysis, we show in Section~\ref{sec:very-simple-algo} that our algorithm achieves a better approximation guarantee of $O(k\log{k})$ when $|T|\ge k^2$.
\subsection{Subset $\ell$-connected graphs:
deficient sets, cores, halo-families and halo-sets} \label{sec:deficient} \label{sec:core-halo}
In this section, we discuss some key properties of deficient sets that will be exploited by our approximation algorithm.
Assume that the set of terminals $T$ is $\ell$-connected in the graph $G=(V,E)$. Then $G$ has $|\Nbr{U}|\geq\ell$ for all $U\subseteq V$ such that $U\cap T\neq\emptyset$ and $\inv{U}\cap T\neq\emptyset$. Moreover, by Menger Theorem, $G$ is subset $(\ell+1)$-connected if and only if $G$ has no deficient set.
A key property of vertex neighborhoods is that the function $|\Nbr{\cdot}|$ on subsets of $V$ is submodular. In other words, for any subsets of vertices $U,W\subseteq V$, \[
|\Nbr{U\cup W}|+|\Nbr{U\cap W}| \leq |\Nbr{U}| + |\Nbr{W}|. \]
We call a deficient set $U\subseteq V$ {\em small}\footnote{
There is another way to define a {\em small} deficient set. For example, in~\cite{Nutov11}, Nutov defined a small deficient set as a deficient set $U$ such that $|U\capT|\leq\frac{|T|-\ell}{2}$. }
if $|U\capT| \le |\inv{U}\capT|$.
\begin{proposition}
For any small deficient set $U$, $|U\cap T|\leq |T|/2$ and
$|\inv{U}\cap T|\geq (|T|-\ell)/2$. \label{prop:size} \end{proposition}
\begin{proof} The first inequality follows from the definition of small deficient sets. Consider the second inequality. We have \[
|\inv{U}\cap T|\geq
\frac{|U\cap T|+|\inv{U}\cap T|}{2}=
\frac{|T|-|\Nbr{U}\cap T|}{2} \geq \frac{|T|-\ell}{2}
\] \end{proof}
\begin{lemma}[Uncrossing Lemma] \label{lmm:uncross} Consider any two distinct deficient sets $U,W\subseteq V$. If $U\cap W\cap T\neq\emptyset$ and $\inv{U}\cap\inv{W}\cap T\neq\emptyset$, then both $U\cap W$ and $U\cup W$ are deficient sets. Moreover, if $U$ or $W$ is a small deficient set, then $U\cap W$ is a small deficient set. \end{lemma}
\begin{proof}
Suppose $U\cap W\cap T\neq\emptyset$ and $\inv{U}\cap\inv{W}\cap T\neq\emptyset$. Note that \begin{align*} \inv{U}\cap\inv{W}\cap T
&= (V-(U\cup \Nbr{U}))\cap (V-(W\cup \Nbr{W}))\cap T \\
&= (V-(U\cup W\cup \Nbr{U}\cup \Nbr{W})) \cap T \\
&= (V-((U\cup W)\cup \Nbr{U\cup W})) \cap T \\
&= \inv{(U\cup W)}\cap T. \end{align*}
Moreover, $\inv{(U\cup W)}\subseteq\inv{(U\cap W)}$. This means that \[ (U\cap W)\cap T\neq\emptyset,\quad (U\cup W)\cap T\neq\emptyset,\quad \inv{(U\cup W)}\cap T\neq\emptyset\quad \mbox{and} \quad \inv{(U\cap W)}\cap T\neq\emptyset. \]
Hence, by Menger Theorem, we have $|\Nbr{U\cup W}|\geq\ell$
and $|\Nbr{U\cap W}|\geq\ell$. Moreover, since $U,W$ are deficient sets, we have $|\Nbr{U}|=|\Nbr{W}|=\ell$. It then follows by the
submodularity of $|\Nbr{.}|$ that \[
2\ell\leq |\Nbr{U\cup W}|+|\Nbr{U\cap W}| \leq |\Nbr{U}| +
|\Nbr{W}|=2\ell. \]
Thus, $|\Nbr{U\cap W}|=|\Nbr{U\cup W}|=\ell$. This implies that both $U\cup W$ and $U\cap W$ are deficient sets. Moreover, suppose $U$ or $W$ is a small deficient set. Without loss of generality, assume that $U$ is a small deficient set. Then $U\cap W$ is a small deficient set because $U\cap W\subseteq U$. Thus, \[
|U\cap W\cap T|\leq |U\cap T|\leq |\inv{U}\cap T|
\leq |\inv{(U\cap W)}\cap T| \] \end{proof}
By a {\em core}, we mean a small deficient set $C$ that is inclusionwise minimal. In other words, $C$ is a core if it is a small deficient set that does not contain another such set. It can be seen that any small deficient set $U$ contains at least one core.
The {\em halo-family} of a core $C$, denoted by $\mathrm{Halo}(C)$, is the set of all small deficient sets that contain $C$ and contain no other cores; that is, \[ \mathrm{Halo}(C)=\{U: \mbox{$U$ is a small deficient set,
$C\subseteq U$, and
there is no core $D\neq C$ such that $D\subseteq U$}\}. \]
The {\em halo-set} of a core $C$, denoted by $H(C)$, is the union of all the sets in $\mathrm{Halo}(C)$; that is,
\[ H(C) = \bigcup\{U:U\in\mathrm{Halo}(C)\} \]
An example of cores, halo-families and halo-sets is illustrated in Figure~\ref{fig:core-haloset}.
\begin{figure}\label{fig:core-haloset}
\end{figure}
\ \\ {\bf Remark}: We remark that cores and halo-sets of subset $\ell$-connected graphs can be computed in polynomial time. In fact, algorithms for computing cores and halo-sets of the $k$-vertex connected spanning subgraph problem also apply to the subset $k$-connectivity problem. See~\cite{KN05,FL08,Bundit-Thesis}. \\
Some important properties of cores and halo-families that we will require are stated below.
\begin{lemma}[Disjointness Lemma] \label{lmm:sskconn-disjoint} Consider any two distinct cores $C$ and $D$. For any deficient sets $U\in\mathrm{Halo}(C)$ and $W\in\mathrm{Halo}(D)$, either $U\cap W\cap T=\emptyset$ or $\inv{U}\cap\inv{W}\cap T=\emptyset$. \end{lemma}
\begin{proof} Suppose to the contrary that $U\cap W\cap T\neq\emptyset$ and $\inv{U}\cap\inv{W}\cap T \neq\emptyset$. Then, by Lemma~\ref{lmm:uncross}, $U\cap W$ is a small deficient set. Thus, $U\cap W$ contains a core. This core is either $C$ or $D$ or another core distinct from $C$ and $D$. In each case, we have a contradiction. \end{proof}
The next result gives an upper bound on the number of halo-sets that contain a chosen terminal, which is a key for the design of our algorithm.
\begin{lemma}[Upper bound] \label{lmm:bound-1} For any terminal $t\in T$, the number of cores $C$ such that $t\in H(C)$ is at most $\frac{2(|T|-1)}{|T|-\ell}$. \end{lemma}
\begin{proof} Let $C_1,C_2,\ldots,C_q$ be distinct cores such that $t\in H(C_i)$ for all $i=1,2,\ldots,q$. For each $i=1,2,\ldots,q$, since $t$ is in the halo-set $H(C_i)$, there must exist a deficient set $U_i$ in the halo-family $\mathrm{Halo}(C_i)$ that contains $t$. It then follows that $t\in\bigcap_{i=1}^qU_i$. By the Disjointness Lemma (Lemma~\ref{lmm:sskconn-disjoint}), for $i\neq j$, $U_i\cap U_j\cap T\neq\emptyset$ only if $\inv{U_i}\cap\inv{U_j}\cap T=\emptyset$. This is because $U_i$ and $U_j$ are deficient sets of different halo-families. For $i=1,2,\ldots,q$, observe that
$U_i$ is small. Hence, by Proposition~\ref{prop:size}, we have $|\inv{U_i}\cap T|\geq(|T|-\ell)/2$. Thus, the upper bound on the number halo-sets that contain $t$ is \[ \frac{|T|-1}{(|T|-\ell)/2}=\frac{2(|T|-1)}{|T|-\ell}
\] \end{proof}
\subsection{Covering halo-families via rooted subset
$(\ell+1)$-connectivity} \label{sec:cover-halo}
We say that an edge $e=(u,v)$ {\em covers} a deficient set $U$ if $e$ connects $U$ and $\inv{U}$; that is, $u\in U$ and $v\in\inv{U}$. Clearly, $e$ covers $U$ if $e$ covers $\inv{U}$. Observe that if $e$ covers $U$, then after adding the edge $e$ to the current graph, $U$ is no longer a deficient set. Now, consider any core $C$. We say that a set of edges $F$ {\em covers} the halo-family of $C$ if each deficient set $U$ in $\mathrm{Halo}(C)$ is covered by some edge of $F$.
For a terminal $r\in T$, we say that the terminal $r$ {\em hits} the halo-family $\mathrm{Halo}(C)$ if $r$ is in $C$ or $r$ is in the vertex-complement of the halo-set of $C$; that is, $r$ hits $\mathrm{Halo}(C)$ if $r\in C$ or $r\inH(C)^*$.
For a set of terminals $S\subseteqT$, we say that $S$ {\em hits} a halo-family $\mathrm{Halo}(C)$ if there is a terminal $r\inS$ that hits $\mathrm{Halo}(C)$.
The following lemma shows that if $r$ hits the halo-family $\mathrm{Halo}(C)$, then we can find a set of edges $F$ that covers $\mathrm{Halo}(C)$ by applying the rooted subset $(\ell+1)$-connectivity algorithm with $r$ as the root.
\begin{lemma} \label{lmm:cover-halo} Consider a set of edges $F$ whose addition to $\widehat{G}$ makes the resulting graph $\widehat{G}+F$ $(\ell+1)$-connected from a terminal $r$ to $T$. Let $C$ be any core. If $r\in C$ or $r\in\inv{H(C)}$, then $F$ covers all deficient sets in the halo-family of $C$. \end{lemma}
\begin{proof} Consider the graph $\widehat{G}+F$. By the construction, $\widehat{G}+F$ has $(\ell+1)$ openly disjoint paths from $r$ to every terminal of $T$. This means that $F$ covers all deficient sets of $\widehat{G}$ that contains $r$. If $r\in C$, then $r\in U$ for all deficient sets $U\in\mathrm{Halo}(C)$. So, $F$ covers $\mathrm{Halo}(C)$. Similarly, If $r\in \inv{H(C)}$, then $r\in\inv{U}$ for all deficient sets $U\in \mathrm{Halo}(C)$. So, again, $F$ covers $\mathrm{Halo}(C)$ because an edge $e\in F$ covers $\inv{U}$ if and only if $e$ covers $U$, and the lemma follows. \end{proof}
\subsection{Preprocessing to decrease the number of cores} \label{sec:reduce-cores}
In this section, we describe the preprocessing algorithm that decreases the number of cores to $O\left(\frac{\ell |T|}{|T|-\ell}\right)$. We apply the following {\em root padding algorithm} in the preprocessing step.
\ \\{\bf The root padding algorithm: }
The algorithm takes as an input a graph $G=(V,E)$ with the given edge costs, a subset of terminals $R\subseteqT$, and a connectivity parameter $\rho\leq|R|$. We construct a padded graph by adding a new vertex $\widehat{r}$ and new edges of zero cost from $\widehat{r}$ to each terminal of $R$. Then we apply the rooted subset $\rho$-connectivity algorithm to the padded graph with the set of terminals $T$ and the root $\widehat{r}$. We denote a solution subgraph (of the padded graph) by $\widehat{G} = (V\cup\{\widehat{r}\},\,F\cup\{(\widehat{r},t):t\in R\})$, where $F\subseteq E$.
Then the algorithm outputs the subgraph (of the original graph) $\widehat{G}-\widehat{r}=(V,F)$.
The following result shows that, in the resulting graph $\widehat{G}$, every deficient set contains at least one terminal of $R$.
\begin{lemma}[root padding]
\label{lem:rootpad}
Suppose we apply the root padding algorithm as above, and it finds a
subgraph $\widehat{G}-\widehat{r}=(V,F)$.
Then every deficient set of $\widehat{G}-\widehat{r}$ (with respect to
$\rho$-connectivity of $T$ in $\widehat{G}-\widehat{r}$)
contains at least one terminal of $R$. \end{lemma}
\begin{proof}
Observe that $\widehat{G}$ has $\rho$ openly disjoint $\widehat{r},t$-paths,
for all $t\inT$. Suppose $U\subseteq V$ is a deficient set of
$\widehat{G}-\widehat{r}$ that contains none of terminals of $R$. Thus, $U$
contains another terminal $t\in T- R$ and $|\Nbr{U}|<\rho$.
Then $(\widehat{G}-\widehat{r})-\Nbr{U}$ has no path between $t$ and a terminal
of $R$. This also holds for $\widehat{G}-\Nbr{U}$ because adding
$\widehat{r}$ and the edges from $\widehat{r}$ to every terminals of $R$
cannot give a path between $t$ and a terminal of $R$. This is a
contradiction since $\widehat{G}$ should have $\rho$ openly disjoint
$\widehat{r},t$-paths. \end{proof}
\noindent {\bf Remark}: Consider an instance $\Pi_{root}$ of the rooted subset $k$-connectivity problem that is obtained from an instance $\Pi_{subset}$ of the subset $k$-connectivity problem by either ~(1) picking one of the terminals as the root or ~(2) applying the root padding algorithm using any $k$ terminals of $\Pi_{subset}$. Then the cost of the optimal solution to $\Pi_{root}$ is at most the cost of the optimal solution to $\Pi_{subset}$ because any feasible solution to $\Pi_{subset}$ gives a feasible solution to $\Pi_{root}$ (but not vice-versa). \\
Next, recall that $|T|\geq 2\ell$. We apply the root padding algorithm in Lemma~\ref{lem:rootpad} to any subset $R$ of $(\ell+1)$ terminals with $\rho=(\ell+1)$.
By Theorem~\ref{thm:rooted-kconn}, this incurs a cost of at most $O((k\log{k})\cdotopt)$
Moreover, the algorithm adds a set of edges to the current graph such that every deficient set of the resulting graph contains at least one terminal of $R$. Thus, each core of the resulting graph contains at least one terminal of $R$. By Lemma~\ref{lmm:bound-1}, each terminal is in $O\left(\frac{|T|}{|T|-\ell}\right)=O(1)$ halo-sets. Hence, the number of cores in the resulting graph is at most $O(\ell)$. This gives the next result.
\begin{lemma} \label{lmm:num-cores} Given a subset $\ell$-connected graph, where $|T|\geq 2\ell$, there is an $f(k)$-approximation algorithm that decreases the number of cores to $O(\ell)$, where $f(k)$ is the best known approximation guarantee for the rooted subset $k$-connectivity problem. \end{lemma}
\subsection{Thickness of terminals} \label{sec:thickness}
Consider a graph $\widehat{G}$ such that $T$ is $\ell$-connected in $\widehat{G}$. We define the {\em thickness} of a terminal $t\in T$ to be the number of halo-families $\mathrm{Halo}(C)$ such that $t\in \Nbr{H(C)}$. Thus, the thickness of a terminal $t$ is
$|\{\mathrm{Halo}(C):C\mbox{ is a core}, t \in \Nbr{H(C)}\}|$.
The following lemmas show the existence of a terminal with low thickness.
\begin{lemma} \label{lmm:halo-nbr}
For every core $C$, $|\Nbr{H(C)}|\leq \ell$. \end{lemma}
\begin{proof} We use induction on the number of deficient sets in $\mathrm{Halo}(C)$. For the induction basis, $\mathrm{Halo}(C)$ has one deficient set $U$, and $H(C)=U$. Then $\Nbr{H(C)} = \Nbr{U}$ has size $\ell$ since the graph is subset $\ell$-connected.
Suppose that $\mathcal{U}$ is the union of $j$ deficient sets that each contains the core $C$, and suppose that
$|\Nbr{\mathcal{U}}|\le\ell$. Consider another deficient set $W$ that contains $C$. Our goal is to show that $\Nbr{\mathcal{U}\cup{W}}$ has size at most $\ell$.
If $W\subseteq\mathcal{U}$, then we are done. Otherwise, we apply the submodularity of $|\Nbr{\cdot}|$. Observe that $\mathcal{U}\cap{W}$ contains a terminal since $\mathcal{U}\cap{W}\supseteq{C}$, and $\inv{(\mathcal{U}\cap{W})}$ contains a terminal since $\inv{(\mathcal{U}\cap W)}\supseteq\inv{W}$ and $\inv{W}$ contains a terminal. Hence, $\Nbr{\mathcal{U}\cap{W}}$ has size at least $\ell$. Thus, we have
\[ 2\ell \ge |\Nbr{\mathcal{U}}| + |\Nbr{W}| \ge
|\Nbr{\mathcal{U}\cup{W}}| + |\Nbr{\mathcal{U}\cap{W}}| \ge
|\Nbr{\mathcal{U}\cup{W}}| + \ell. \] This implies that $\Nbr{\mathcal{U}\cup{W}}$ has size at most $\ell$, and the lemma follows. \end{proof}
The following lemma shows the existence of a terminal with low thickness.
\begin{lemma} \label{lmm:low-thickness} Consider a subset $\ell$-connected graph. Let $q$ denote the number of halo-families. Then there exists a terminal $t\in T$ with thickness at most $\frac{\ell q}{|T|}$. \end{lemma}
\begin{proof} Consider the following bipartite incidence graph $B$ of terminals and halo-families: $B$ has a vertex for each terminal and each halo-family, and it has an edge between a terminal $t$ and a halo-family $\mathrm{Halo}(C)$ if and only if $t\in\Nbr{H(C)}$. The previous lemma shows that each halo-family is adjacent to at most $\ell$ terminals in $B$. Hence, $B$ has at most $\ell q$ edges. Therefore, $B$ has a terminal that is adjacent to at most $\frac{\ell q}{|T|}$ cores; that is, there exists a terminal with the required thickness. \end{proof}
\subsection{An $O(k\log^2{k})$-approximation algorithm for
$|T|\geq 2k$} \label{sec:second-algo}
In this section, we describe our approximation algorithm for the case of a moderate number of terminals. Recall that we solve the problem by iteratively increasing the subset connectivity of a graph by one. Initially, we apply the algorithm in Section~\ref{sec:reduce-cores} to decrease the number of core to $O(\ell)$. Then we apply inner iterations until all the deficient sets are covered. At the beginning of each inner iteration, we compute the cores and the halo-sets. Then we apply a {\em covering-procedure} to find a set of edges that covers all the computed halo-families. This completes one inner iteration. Note that an inner iteration may not cover all of the deficient sets because deficient sets that contain two or more of the initial cores (those computed at the start of the inner iteration) may not be covered. So, we have to repeatedly apply inner iterations until no core is present.
See Algorithm~\ref{algo:2nd-algo}.
\begin{algorithm} \caption{An approximation algorithm for moderate and large number of terminals} \label{algo:2nd-algo} \begin{algorithmic} \For {$\ell=0,1,\ldots,k-1$} \Comment{(outer iterations)}
\State {\em (* Increase the subset connectivity of a graph by one. *)}
\State Decrease the number of cores to $O(\ell)$.
\While {the number of cores is greater than $0$}
\Comment{(inner iterations)}
\State Compute cores and halo-sets.
\State Apply a {\bf covering-procedure} to cover all the halo-families
\EndWhile \EndFor \end{algorithmic} \end{algorithm}
We now describe the covering-procedure.
The procedure first finds a set of terminals $S\subseteqT$ that hits all the computed halo-families. Then it applies the rooted subset $(\ell+1)$-connectivity algorithm (Theorem~\ref{thm:rooted-kconn}) from each terminal of $S$. Let $F$ be the union of all edges found by the rooted subset $(\ell+1)$-connectivity algorithm. Then, by Lemma~\ref{lmm:cover-halo}, $F$ covers all the halo-families.
The key idea of our algorithm is to pick a terminal $\widehat{r}$ with a minimum thickness. Observe that a halo-family $\mathrm{Halo}(C)$ is not hit by $\widehat{r}$ only if \begin{itemize} \item[(1)] its halo-set $H(C)$ has $\widehat{r}$ as a neighbor (that is,
$\widehat{r}\in \Nbr{H(C)}$) or \item[(2)] its halo-set $H(C)$ contains $\widehat{r}$, but its core $C$ does
not contain $\widehat{r}$. \end{itemize} The number of halo-families $\mathrm{Halo}(C)$ such that $\widehat{r}\in \Nbr{H(C)}$ may be large, but the number of halo-families whose halo-sets contain $\widehat{r}$ is $O(1)$, assuming that $|T|\ge 2\ell$. Hence, we only hit halo-families of the second case by picking one terminal from each core $C$ whose halo-set contains $\widehat{r}$. Thus, the number of terminals picked is $O(1)$. We call this a {\em micro} iteration.
Then the remaining halo-families are the halo-families whose halo-sets have $\widehat{r}$ as a neighbor. We repeatedly apply micro iterations until we hit all of the halo-families computed at the start of the inner iteration.
To be precise, initially let $S=\emptyset$. In each micro iteration, we add to $S$ a terminal $\widehat{r}$ of minimum thickness (with respect to halo-families that are not hit by $S$). Then, for each core $C$ such that $\widehat{r}\in H(C)- C$, we add to $S$ any terminal in $C\capT$. We repeatedly apply micro iterations until $S$ hits all the halo-families. At the termination, we apply the rooted subset $(\ell+1)$-connectivity algorithm (Theorem~\ref{thm:rooted-kconn}) from each terminal of $S$, and we return all the set of edges found by the algorithm as an output.
The covering-procedure is presented in Figure~\ref{algo:cover-proc}.
\begin{algorithm} \caption{Covering-procedure} \label{algo:cover-proc} \begin{algorithmic} \State $S\leftarrow\emptyset$. \While {some halo-family is not hit by $S$}
\Comment{(micro iteration)}
\State Add to $S$ a terminal $\widehat{r}$ with a minimum thickness.
\For {each halo-family $\mathrm{Halo}(C)$ (not hit by $S$) such
that $\widehat{r}\in H(C)- C$}
\State Add to $S$ any terminal $r\in C$.
\EndFor \EndWhile \For {each terminals $r$ in $S$}
\State Apply the rooted subset $(\ell+1)$-connectivity algorithm
from $r$. \EndFor \end{algorithmic} \end{algorithm}
\subsubsection{Analysis} \label{sec:analysis}
The feasibility of a solution directly follows from the condition of the inner iteration; that is, the inner iteration terminates when a current graph has no core. So, at the termination of the inner iteration, the resulting graph has no deficient set. Thus, the subset connectivity of the graph becomes $\ell+1$. Applying the outer iteration $k$ times, the final graph is then subset $k$-connected.
It remains to analyze the cost of the solution subgraph. First, we analyze the number of times that the covering-procedure applies the rooted subset $(\ell+1)$-connectivity algorithm.
Then we analyze the total cost incurred by all inner iterations, which is the cost for increasing the subset connectivity of a graph by one.
Finally, we apply Theorem~\ref{lmm:LP-scaling} to analyze the final approximation guarantee.
Consider any micro iteration of the covering-procedure.
By Lemma~\ref{lmm:bound-1}, $\widehat{r}$ is contained in at most $O(1)$ halo-sets, assuming that $|T|\geq 2\ell$. Hence, we have to apply the rooted subset $(\ell+1)$-connectivity algorithm $O(1)$ times.
We now analyze the number of micro iterations needed to hit all of the halo-families. Let $h_i$ denote the number of halo-families that are not hit by $S$ at the beginning of the $i$-th micro iteration. Recall that the number of cores after the preprocessing step is $O(\ell)$. Thus, $h_1= O(\ell)$. We claim that, at the $i$-th iteration, the number of halo-families that are not hit by $S$ is at most $h_1/2^{i-1}$.
\begin{lemma} Consider the $i$-th micro iteration. The number of halo-families that are not hit by terminals of $S$ at the start of the iteration is $h_1/2^{i-1}$. \label{lmm:uncovered-halo} \end{lemma}
\begin{proof} We proceed by induction on $i$. It is trivial for $i=1$. Suppose that the assertion is true for the $(i-1)$-th micro iteration for some $i>1$. Consider the $(i-1)$-th micro iteration. Since we choose a vertex $\widehat{r}$ with a minimum thickness, by Lemma~\ref{lmm:low-thickness}, the thickness of $\widehat{r}$ is at most $\frac{h_{i-1}\ell}{|T|}$. Note that $\ell/|T|\leq 1/2$ since $|T|\geq 2\ell$. This means that $\widehat{r}$ is a neighbor of at most $h_{i-1}/2$ halo-sets. At the end of the micro iteration, halo-families that are not hit by terminals of $S$ are halo-families whose halo-sets have $\widehat{r}$ as a neighbor. Thus, the number of remaining halo-families is at most $h_{i-1}/2$. Hence, we have \[ h_i \leq h_{i-1}\left(\frac{1}{2}\right) \leq \left(h_1\left(\frac{1}{2}\right)^{i-2}\right)
\left(\frac{1}{2}\right) = h_1\left(\frac{1}{2}\right)^{i-1} \] \end{proof}
Lemma~\ref{lmm:uncovered-halo} implies that the maximum number of micro iterations (within the covering-procedure) is $O(\log{h_1})=O(\log{\ell})$.
So, in each inner iteration, we have to call the rooted subset $(\ell+1)$-connectivity algorithm $O(\log{\ell})$ times.
Lastly, we analyze the total cost incurred by all inner iterations, which is the cost for increasing the subset connectivity of a graph by one.
We may apply Theorem~\ref{thm:rooted-kconn} directly to analyze the cost of the solution. However, this leads to a bound slightly weaker than what we claimed. To get the desired bound, we apply a stronger version of Nutov's theorem~\cite{Nutov12}. In particular, the approximation guarantee of Nutov's algorithm depends on the size of a smallest deficient set. To be precise, {\em the size of a smallest
deficient set} is defined by \[
\min\{|U\capT|: \mbox{$U$ is a deficient set}\}. \]
\begin{lemma}[Nutov 2009~\cite{Nutov12}] \label{lmm:root-size} Consider the problem of increasing the rooted subset connectivity of a graph from $\ell$ to $\ell+1$.
Let $\phi=\min\{|U\capT|:U \mbox{ is a deficient set}\}$. That is, each deficient set of the initial graph contains at least $\phi$ terminals. Then there is an $O(\ell/\phi)$-approximation algorithm. \end{lemma}
Now, we analyze the size of a smallest deficient set of a graph at the beginning of each inner iteration.
Consider the cores at any inner iteration. We call cores at the beginning of the iteration {\em old cores} and call cores at the end of the iteration {\em new cores}. We claim that every new core $\widehat{C}$ contains at least two old cores $C$ and $D$ that are disjoint on $T$. This follows from the following lemma.
\begin{lemma} No small deficient set contains two distinct cores $C$ and $D$ such that $C\cap D\cap T\neq\emptyset$. \label{lmm:two-cores} \end{lemma}
\begin{proof} Suppose to the contrary that there is a small deficient set $U$ that contains two distinct cores $C$ and $D$ such that $C\cap D\cap T\neq\emptyset$. Since $C\cap D\cap T\neq\emptyset$, by Lemma~\ref{lmm:sskconn-disjoint}, $\inv{C}\cap\inv{D}\cap T=\emptyset$; that is, $\inv{C}\cap\inv{D}$ has no terminals. Since $U$ contains both $C$ and $D$, it follows that $\inv{U}$ is contained in both $\inv{C}$ and $\inv{D}$.
Hence, $\inv{U}$ has no terminals. This contradicts the fact that $U$ is a deficient set. \end{proof}
Lemma~\ref{lmm:two-cores} implies that no new cores contain two old cores that are intersecting on $T$. This is because new cores are small deficient sets of the old graph. Moreover, since all small deficient sets that contain only one core have been covered, new cores must contain at least two old cores that are disjoint on $T$.
Thus, the size of a smallest deficient set increases by a factor of $2$. This implies the following lemma.
\begin{lemma} Consider the $j$-th inner iteration. At the beginning of the iteration, the size of a smallest deficient set of the current graph is at least $2^{j-1}$. \label{lmm:size-defi} \end{lemma}
\begin{proof} As in the above discussion, the size of a smallest deficient set increases by a factor of two in each inner iteration.
In more detail, consider the size of a smallest deficient set at the beginning and the end of an inner iteration.
We call the graph at the beginning of the iteration an ``old graph'' and the graph at the end of the iteration a ``new graph''.
Let $U$ and $U'$ denote smallest deficient sets of the old and the new graph, respectively.
By the size argument, we conclude that $U$ and $U'$ are cores of the old and the new graph. At the end of the inner iteration, small deficient sets containing one core are all covered. Thus, $U'$ contains two distinct cores $C$ and $C'$ of the old graph. Moreover, Lemma~\ref{lmm:two-cores} implies that $C$ and $C'$ have no terminals in common. By the minimality of $U$, we have
$|C\capT|\geq|U\capT|$ and $|C'\capT|\geq|U\capT|$.
Thus, $|U'\cap T|\geq |C\cupT|+|C'\capT|\geq 2|U\cap T|$ as claimed.
Now, we prove the lemma by induction. At the first inner iteration, each deficient set contains at least one terminal. Thus, the statement holds for the base case. Assume that the assertion is true for the $(j-1)$-th inner iteration; that is, at the beginning of the $(j-1)$-th inner iteration, any deficient set $U$ has at least $2^{j-2}$ terminals. By the above claim, this number increases by a factor of two at the end of the iteration. Thus, at the beginning of the $j$-th iteration, the size of a smallest deficient sets is $2^{j-1}$, proving the lemma. \end{proof}
By Lemma~\ref{lmm:root-size} and \ref{lmm:size-defi}, at the $j$-th inner iteration, the cost incurred by the rooted subset $(\ell+1)$-connectivity algorithm is $O(\ell/2^{j-1})$.
Combining everything together, the approximation guarantee for the problem of increasing the subset connectivity of a graph by one is \[ O\left(\frac{\ell}{2^0}\log{\ell} + \frac{\ell}{2^1}\log{\ell} +
\ldots \right) = O(\ell\log{\ell}). \]
Thus, by Theorem~\ref{lmm:LP-scaling}, our algorithm achieves an approximation guarantee of $O(k \log^2{k})$, assuming that $|T|\ge 2k$.
\subsection{An $O(k\log{k})$-approximation algorithm for
$|T|\geq k^2$.} \label{sec:very-simple-algo}
To finish, we show that if the number of terminals is large, then we get a slightly better performance guarantee. Observe that if $|T|\ge k^2$, then, by Lemma~\ref{lmm:low-thickness}, there is a terminal $\widehat{r}$ with a thickness of at most $\frac{q\ell}{|T|} \leq \frac{2\ell^2}{\ell^2} = 2$. Moreover, by Lemma~\ref{lmm:bound-1}, each terminal is contained in at most $\frac{2|T|}{|T|-\ell}=O(1)$ halo-sets. Thus, the number of halo-families that are not hit by $\widehat{r}$ is $O(1)$. This means that we can hit all the remaining halo-families by choosing $O(1)$ terminals; that is, for each halo-family, we choose one terminal from its core. So, we can skip the micro iterations of the covering-procedure, and the approximation guarantee becomes $O(k\log{k})$.
\subsection{Analysis for the case $k<|T|<2k$} \label{sec:below2k}
Our algorithm in Section~\ref{sec:second-algo} indeed applies to the case $k<|T|<2k$ with an approximation guarantee of $O\left(\left(\frac{|T|}{|T|-k}\right)^2k\log^2{k}\right)$. To see this, we leave the bounds in Lemma~\ref{lmm:bound-1} and Lemma~\ref{lmm:low-thickness} untouched.
Then we have \begin{itemize} \item Each terminal is contained in at most
$O\left(\frac{|T|}{|T|-k}\right)$ halo-families. \item There is a terminal with a thickness of
$O\left(\frac{\ell q}{|T|}\right)$,
where $q$ is the number of halo-families. \end{itemize}
Recall the micro iterations of the covering-procedure. In each micro iteration, we choose $O\left(\frac{|T|}{|T|-\ell}\right)$ terminals, and the number of halo-families (which are not hit) decreases by a factor of $\frac{|T|}{\ell}$.
Here the number of micro iterations is not logarithmic because $\frac{|T|}{\ell}$ is not constant when $|T|\approx\ell$.
To analyze the upper bound, we write $\frac{\ell}{|T|}$ as $1-\frac{1}{|T|/(|T|-\ell)}$ and apply an equation: \[ \lim_{x\rightarrow\infty}\left(1-\frac{1}{x}\right)^x=\frac{1}{e} \] Thus, we need $O\left(\frac{|T|}{|T|-\ell}\right)$ micro iterations to decrease the number of halo-families (which are not hit) by a factor of $e$.
This means that the covering-procedure terminates in $O\left(\frac{|T|}{|T|-\ell}\log{q}\right)$ iterations, where $q$ is the number of halo-families. (Note that, in this case, we do not need the preprocessing step because the number of halo-families is at most $|T|^2=O(k^2)$.)
So, the covering-procedure has to call the rooted subset $(\ell+1)$-connectivity algorithm for $O\left(\left(\frac{|T|}{|T|-\ell}\right)^2\log{k}\right)$ times.
Following the analysis in Section~\ref{sec:analysis}, we have an approximation guarantee of $O\left(\left(\frac{|T|}{|T|-k}\right)^2k\log^2{k}\right)$ as claimed.
\section{Hardness of the subset $k$-connectivity problem} \label{sec:hardness}
In this section, we discuss the hardness of the subset $k$-connectivity problem. First, we will show in Section~\ref{sec:hardest-instance} that the hardest instance of the subset $k$-connectivity problem might be when
$k\approx |T|$; that is, we prove Proposition~\ref{prop:hardest}. Then we will present in Section~\ref{sec:root2subset} an approximation preserving reduction from the rooted subset $k$-connectivity problem to the subset $k$-connectivity problem; that is, we prove Theorem~\ref{thm:rooted-to-subset}.
\subsection{The hardest instance} \label{sec:hardest-instance}
We will show that an $\alpha(k)$-approximation algorithm for the case $|T|=k$ implies an $(\alpha(k)+f(k))$-approximation algorithm for all instances with $|T|>k$, where $f(k)$ is the best known approximation guarantee for the rooted subset $k$-connectivity problem. In particular, instances with $|T|\approx{k}$ might be the hardest cases of the subset $k$-connectivity problem.
Suppose there is an $\alpha(k)$-approximation algorithm ${\cal A}$ for the subset $k$-connectivity problem for the case $|T|=k$.
We apply ${\cal A}$ to solve an instance of the subset $k$-connectivity problem with $|T|>k$ as follows. Let $G=(V,E)$ be a given graph and $T\subseteq V$ be a set of terminals, where $|T|>k$. First, we take any subset $R$ of $k$ terminals from $T$. Then we apply the algorithm ${\cal A}$ to this instance with $R$ as the set of terminals; this results in a graph $G_{R}=(V,E_{R})$. Clearly, $R$ is $k$-connected in $G_{R}$.
Now, we make the remaining terminals connected to $R$ by applying the rooted subset $k$-connectivity algorithm. To be precise, we construct a padded graph by adding a new vertex $\widehat{r}$ and new edges of zero cost from $\widehat{r}$ to each terminal of $R$. Then we apply the rooted subset $k$-connectivity algorithm to the padded graph with the set of terminals $T$ and the root $\widehat{r}$. Denote a solution subgraph (of the padded graph) by $G_{pad} = (V\cup\{\widehat{r}\},\,E_{root}\cup\{(\widehat{r},t):t\in R\})$, where $E_{root}\subseteq E$.
The algorithm outputs the union of the two subgraphs, namely $\widehat{G}=(V,E_{R}\cup E_{root})$.
We claim that the set of all terminals $T$ is $k$-connected in $\widehat{G}$.
Suppose not. Then there is a set of vertices $X\subseteq V$ of size $k-1$ that separates some terminals $s,t\inT- X$; that is, $s$ and $t$ are not connected in $\widehat{G}- X$.
Consider the padded subgraph $G_{pad}$. By the construction, since $G_{pad}$ is $k$-connected from $\widehat{r}$ to $T$, both $s$ and $t$ have paths to $\widehat{r}$ in $G_{pad}-{X}$. Moreover, each of these two paths must visit some terminals $s'$ and $t'$ in $R$, respectively. If $s'=t'$, then $s$ and $t$ are connected by the union of these paths. So, we have a contradiction. If $s'\neq t'$, then we can join these two paths by an $s',t'$-path in $G_{R}- X$.
Such $s',t'$-path exists because $R$ is $k$-connected in $G_{R}$, meaning that $X$ cannot separates a pair of terminals in $R$.
Thus, $s$ and $t$ are connected, and we again have a contradiction.
Now, consider the cost. The approximation factor incurred by the algorithm ${\cal A}$ is $\alpha(k)$, and the approximation factor incurred by the rooted subset $k$-connectivity algorithm is $f(k)$. Thus, the above algorithm gives an approximation guarantee of $(\alpha(k)+f(k))$ as claimed.
\subsection{A reduction from the rooted subset $k$-connectivity problem} \label{sec:root2subset}
As we showed in the previous section, an approximation algorithm for the rooted subset $k$-connectivity problem implies an approximation algorithm for the subset $k$-connectivity problem. Hence, it is more likely that the rooted problem is easier than the subset problem. Here we show a solid evidence of this statement; that is, we will give an approximation preserving reduction from the rooted subset $k$-connectivity problem to the subset $k$-connectivity problem.
The key idea of the reduction is that a solution $\widehat{G}$ to the rooted subset $k$-connectivity problem is indeed almost subset $k$-connected.
In particular, if the root vertex $r$ is not allowed to be removed, then there is no set of vertices of size less than $k$ that can separate a pair of terminals.
So, we want to prevent the root vertex $r$ from being in a separator. To do this, we replace $r$ by a clique $K_d$ of size at least $k+1$. Thus, removing any set of less than $k$ vertices cannot remove all vertices corresponding to $r$.
Now, we shall realize the above idea.
First, take any instance $\Pi^{root}$ of the rooted subset $k$-connectivity problem consisting of a graph $G=(V,E)$, a set of terminals $T\subseteq V$ and a root vertex $r\in V\setminusT$. Let $d$ be the degree of $r$ in $G$. Clearly, if the instance $\Pi^{root}$ is feasible, then $d\geq k$. We construct an instance $\Pi^{subset}$ of the subset $k$-connectivity problem consisting of a graph $G'=(V',E')$ and a set of terminals $T'$ as follows.
Let $\{v_1,v_2,\ldots,v_d\}$ be a set of neighbors of $r$ in $G$. We remove from $G$ the vertex $r$ and replace it with a clique $K_{d+1}$ on a set of vertices $\{r',v'_1,v'_2,\ldots,v'_d\}$. All edges of $K_{d+1}$ have zero costs. The vertex $r'$ corresponds to the root vertex $r$ of $G$, and each vertex $v'_i$ corresponds to each neighbor $v_i$ of $r$ in $G$. Then we connect $K_{d+1}$ to $G$ by adding to $G'$ an edge $(v'_i,v_i)$ for each edge $(r,v_i)$ in $G$ and setting the cost of $(v'_i,v_i)$ to be the same as the cost of $(r,v_i)$. Thus, each edge $(v'_i,v_i)$ in $G'$ corresponds to an edge $(r,v_i)$ in $G$. The set of terminals of this new instance is $T'=T\cup\{r'\}$, and the connectivity requirements is $k$, the same for both instances.
This completes the construction.
In sum, we have \begin{align*}
G' &= G - \{r\} + K_{d+1} +
\{(v'_1,v_1),(v'_2,v_2),\ldots,(v'_d,v_d)\}\\
T' &= T\cup\{r'\}\\
r' &\leftrightarrow r\\
(v'_i,v_i) &\leftrightarrow (r,v_i)\mbox{ for all $i=1,2,\ldots,d$}\\ \end{align*}
\noindent{\bf Completeness:} First, we show that any feasible solution $H$ of $\Pi^{root}$ maps to a feasible solution $H'$ of $\Pi^{subset}$ with the same cost.
The mapping is as follows. Given a graph $H$, we construct a solution $H'$ to $\Pi^{subset}$ by taking all edges of $K_{d+1}$ and all edges of $G'$ corresponding to edges of $H$. Clearly, the cost of $H'$ and $H$ are the same. It remains to show that $T'$ is $k$-connected in $H'$.
The connectivity between the vertex $r'$ and each terminal $t\in T$ is clearly satisfied. This is because any collection of openly disjoint $r,t$-paths in $H$ maps to a collection of openly disjoint $r',t$-paths in $H'$. In particular, any path $P=(r,v_i,\ldots,t)$ in $H$ maps to a path $P'=(r,v'_i,v_i,\ldots,t)$ in $H'$, and it is easy to see that the mapping preserves vertex-disjointness. By the same argument, we can deduce that every vertex $v'_j\in K_{d+1}$ is $k$-connected to $t$ in $H'$. This is because the path $P$ also maps to a path $P''=(v'_j,v'_i,v_i,\ldots,t)$ or $P''=(v'_j,v_j,\ldots,t)$ in $H'$.
Now, consider the connectivity between a pair of vertices $t,t'\inT$.
Assume a contradiction that $t$ and $t'$ are not $k$-connected. Then there is a subset of vertices $X$ of $G'$ with $|X|\leq k-1$ such that $t$ and $t'$ are not connected in $H'-{X}$.
Since $|X| \leq k-1 \leq d$, there is a vertex $s$ in $K_{d+1}-{X}$. (The vertex $s$ is either the vertex $r'$ or some vertex $v_i$ in $K_{d+1}$.) As we have shown, $s$ is $k$-connected to $t$ and $t'$ in $H'$.
Thus, by Menger's theorem, $H'-{X}$ contains both an $s,t$-path and an $s,t'$-path. So, $t$ and $t'$ are connected in $H'-{X}$, a contradiction. Therefore, $T'$ is $k$-connected in $H'$, implying that $H'$ is feasible to the subset $k$-connectivity problem.
\noindent{\bf Soundness:} Now, we show the converse; that is, any feasible solution $H'$ of $\Pi^{subset}$ maps to a feasible solution $H$ of $\Pi^{root}$ with the same cost.
This direction is easy. We construct $H'$ by taking all edges of $H'$ that correspond to edges of $G$. Clearly, the cost of $H$ and $H'$ are the same. By feasibility, $H'$ has, for each terminal $t\in T$, a collection of $k$ openly disjoint $r',t$-paths, namely $P'_1,P'_2,\ldots,P'_k$, and each path $P'_j$ is of the form $P'_j=(r',v'_i,v_i,\ldots,t)$. The path $P'_j$ maps to a path $P_j=(r,v_i,\ldots,t)$ in $H$. So, we have a collection of paths $P_1,P_2,\ldots,P_k$ in $H$ that are openly disjoint. Therefore, $H$ is feasible to the rooted subset $k$-connectivity problem, finishing the proof.
\section{Conclusions and Discussions} \label{sec:discuss} \label{sec:conclusion}
We studied the structure of the subset $k$-connectivity problem and used this knowledge to design an approximation algorithm for the subset $k$-connectivity problem.
When the number of terminals is moderately large, at least $2k$, our algorithm gives a very good approximation guarantee of $O(k\log^2k)$.
When the number of terminals is tiny, at most $\sqrt{k}$, then the trivial algorithm also gives a very good approximation guarantee of $O(k)$.
However, when the number of terminals is between $\sqrt{k}$ and $2k$, the approximation guarantee can be as large as $\Theta(k^2)$. Interestingly, as we have shown, it does seem that the hardest instances of the subset $k$-connectivity problem are when the number of terminals is close to $k$.
\noindent {\bf Acknowledgments.} We thank Joseph Cheriyan for useful discussions over a year. Also, we thank Adrian Vetta, Parinya Chalermsook, Danupon Nanongkai, Jittat Fakcharoenphol and anonymous referees for useful comments on the preliminary draft.
\end{document} | arXiv |
CONF. NOTES
SEM. NOTES
My research interests lie in number theory, more specifically, in the arithmetic applications of automorphic forms.
I am currently the PI on an NSF grant (DMS #1852001) that will fund an REU at Occidental College during the summers of 2020 and 2021. Check back later for more details!
I am the founder and organizer of the Number Theory Series in Los Angeles. This is a biannual regional number theory conference. The first meeting will October 26-27, 2019. The conference is funded by grants from the NSA and NSF.
During the summers of 2012 and 2013 I was the PI on a grant at Clemson that funded an REU on Combinatorics, Computational Algebraic Geometry, and Number Theory. There is an REU and a preliminary REU as part of the Research Training group at Clemson for which I was the PI from 2016-18. For past research students, please scroll past the publications.
Congruence primes for automorphic forms on symplectic groups (with Huixi Li)
preprint, submitted 2018.
It has been well-established that congruences between automorphic forms have far-reaching applications in arithmetic. In this paper we construct congruences for Siegel Hilbert modular forms defined over a totally real field of class number one. As an application of this general congruence, we produce congruences between paramodular Saito-Kurokawa lifts and non-lifted Siegel modular forms. These congruences are used to produce evidence for the Bloch-Kato conjecture for elliptic newforms of square-free level and odd functional equation.
Eigenform Product Identities for Degree-Two Siegel Modular Forms (with Hugh Geller, Rico Vicente, Alexandra Walsh)
Journal of Number Theory, 16 pages, (2019), https://doi.org/10.1016/j.jnt.2019.03.013.
It is known via work of Duke and Ghate that there are only finiely many pairs of full level, degree one eigenforms $f$ and $g$ whose product $fg$ is also an eigenform. We prove a partial generalization of this theorem for degree two Siegel modular forms. Namely, we show that there is only one pair of eigenforms $F$ and $G$ such that $FG$ is a non-cuspidal eigenform. In the case that $FG$ is a cuspform, we provide necessary conditions for $FG$ to be an eigenform, give one example, and conjecture that is the only example. Part of this work was done during the 2018 REU at Clemson University.
Congruence primes for automorphic forms on unitary groups and applications to the arithmetic of Ikeda lifts (with Krzysztof Klosin)
Kyoto J. Math. (to appear), 37 pages.
In this paper we provide a sufficient condition for a prime to be a congruence prime for an automorphic form $f$ on the unitary group $\textrm{U}(n,n)(\mathbb{A}_F)$ for a large class of totally real fields $F$ via a divisibility of a special value of the standard $L$-function associated to $f$. We also study $p$-adic properties of the Fourier coefficients of an Ikeda lift $I_{\phi}$ (of an elliptic modular form $\phi$) on $\textrm{U}(n,n)(\mathbb{A}_{\mathbb{Q}})$ proving that they are $p$-adic integers which do not all vanish modulo $p$. Finally we combine these results to show that the condition of $p$ being a congruence prime for $I_{\phi}$ is controlled by the $p$-divisibility of a product of special values of the symmetric square $L$-function of $\phi$.
On the action of the $U_p$ operator on Siegel modular forms (with Krzysztof Klosin)
Ramanujan Journal, 44(3), 597-615, (2017).
In this article we study the action of the $U_p$ Hecke operator on the normalized spherical vector $\phi$ in the representation of $\textrm{GSp}_4(\mathbb{Q}_p)$ induced from a character on the Borel subgroup. We compute the Petersson norm of $U_p \phi$ in terms of certain local $L$-values associated with $\phi$.
Amicable pairs and aliquot cycles for elliptic curves over number fields (with David Heras, Kevin James, Rodney Keaton, and Andrew Qian)
Rocky Mountain Journal of Mathematics, 46(6), 1853-1866, (2016).
This paper is a result of one of the projects at the 2012 REU at Clemson University. The notion of amicable pairs and aliquot cycles on elliptic curves was introduced by Silverman and Stange. They provided a very detailed analysis of these concepts for elliptic curves over the rational numbers. In this paper we consider amicable pairs and aliquot cycles over general number fields. We focus on the existence of aliquot cycles in various cases and explore some of the sublteties of dealing with primes of different degrees.
Counting tamely ramified extensions of local fields up to isomorphism (with Robert Cass, Kevin James, Rodney Keaton, Salvatore Parenti, and Daniel Shankman)
Integers, 16 #A53, 1-12, (2016).
This paper is a result of one of the projects at the 2013 REU at Clemson University. Let $p$ be a prime number and let $K$ be a local field of residue characteristic $p$. In this paper we give a formula that counts the number of degree $n$ tamely ramified extensions of $K$ in the case $p$ is of order 2 modulo $n$. This result is achieved via elementary counting methods and simple group theory.
Mixed level Saito-Kurokawa liftings (with Dania Zantout)
The Ramanujan Journal, 39 , 247-257, (2016).
In a 2007 paper R. Schmidt constructed the congruence level and paramodular level Saito-Kurokawa lifts via representation theoretic methods. We use these methods to construct Saito-Kurokawa lifts of more general levels. In particular, we recover a mixed level Saito-Kurokawa lift that was claimed by M. Manickam and B. Ramakrishnan.
Saito-Kurokawa lifts of odd square-free level (with Mahesh Agarwal)
Kyoto Journal of Mathematics, 55(3), 641-662 (2015).
In the paper On the Bloch-Kato conjecture for elliptic modular forms of square-free level the Saito-Kurokawa lift of square-free level and its arithmetic properties are heavily used. However, until Ibukiyama provided a construction of the Maass lifting with level in 2012 there was no correct classical construction of this Saito-Kurokawa lift. In this paper we put together the classical and automorphic construction in one article as well as give the necessary arithmetic properties. We also calculate the norm of the Saito-Kurokawa lift.
Congruence Primes for Ikeda Lifts and the Ikeda ideal (with Rodney Keaton)
Pacific J. Math., 274(1), 27-52 (2015).
Let $f$ be a newform of level 1 and weight $2k-n$ for $k$ and $n$ positive even integers. In this paper we study congruence primes for the Ikeda lift of $f$. In particular, we consider a conjecture of Katsurada stating that primes dividing certain $L$-values of $f$ are congruence primes for the Ikeda lift. Instead of focusing on a congruence to a single eigenform, we deduce a lower bound on the number of all congruences between the Ikeda lift and forms not lying in the space spanned by Ikeda lifts by considering the same primes and slightly relaxed hypotheses. In particular, we define the Ikeda ideal and show how this can be used to study all congruences instead of focusing on a single congruence.
Degree 14 extensions of $\mathbb{Q}_7$ (with Robert Cass, Rodney Keaton, Salvatore Parenti, and Daniel Shankman)
Int. J. of Pure and Appl. Math., 100(2), 337-345 (2015).
This paper is a result of one of the projects at the 2013 REU at Clemson University. We calculate all degree 14 extensions of $\mathbb{Q}_7$ up to isomorphism. We give the Galois group of each extension along with enough information about each Galois group so that one can distinguish between them. The data that accompanies this paper can be found at data.
Classifying extensions of the field of formal Laurent series over $\mathbb{F}_p$ (with Alfeen Hasmani, Lindsey Hiltner, Angela Kraft, Daniel Scofield, and Kirsti Wash)
Rocky Mountain Journal of Mathematics, 45(1), 115-130 (2015).
This paper is a result of one of the projects at the 2012 REU at Clemson university. In previous works, Jones-Roberts and Pauli-Roblot have studied finite exten- sions of the p-adic numbers $\mathbb{Q}_p$. This paper focuses on results for local fields of characteristic $p$. In particular we are able to produce analogous results to Jones-Roberts in the case that the characteristic does not divide the degree of the field extension. Also in this case, following from the work of Pauli-Roblot, we prove that the defning polynomials of these extensions can be written in a simple form amenable to computation. Finally, if $p$ is the degree of the extension, we show there are infinitely many extensions of this degree and thus these cannot be classified in the same manner.
On the Bloch-Kato conjecture for elliptic modular forms of square-free level (with Mahesh Agarwal)
Mathematische Zeitschrift, 276 (3), 889-924 (2014).
This paper contains generalizations of the results of On the cuspidality of pullbacks of Siegel Eisenstein series and applications to the case of square-free level. It also contains a discussion of the relation of the bounds on the Selmer group to the Bloch-Kato conjecture. This discussion is absent in On the cuspidality of pullbacks of Siegel Eisenstein series and applications. The computational evidence will be removed from the final version, but will remain available here: Computational Evidence. One should note that in the main theorem we require the conductor of the Dirichlet character be divisible by the level of the modular form in question. Such an assumption is not necessary; one can take an auxilliary level divisible by the conductor and the level of the modular form and work in that level. That should make it more feasible to expand the computational data provided.
Special values of $L$-functions for Saito-Kurokawa lifts (with Ameya Pitale)
Mathematical Proceedings of the Cambridge Philosophical Society, 155 (2), 237-255 (2013).
In this paper we obtain special value results for $L$-functions associated to classical and paramodular Saito-Kurokawa lifts. In particular, we construct standard L-functions associated to Saito-Kurokawa lifts as well as degree eight L-functions obtained by twisting with an automorphic form defined on $\textrm{GL}(2)$. The results are obtained by combining classical and representation theoretic arguments. One should note here there is a typo in Theorem 4.5 (replace $m-1$ with $\varphi(m)$) and in Corollary 4.6 (replace $m-1$ with $\varphi(m)$ and remove the $m+1$ in $C_{k,m}$.)
Level stripping for Siegel modular forms with reducible Galois representations (with Rodney Keaton)
Journal of Number Theory, 133 (5), 1492-1501 (2013).
In this paper we consider level stripping for genus 2 cuspidal Siegel eigenforms. In particular, we show that it is possible to strip primes from the level of weak endoscopic lifts as well as from Saito-Kurokawa lifts that arise as theta lifts with a mild restriction on the associated character.
Pullbacks of Siegel Eisenstein series and weighted averages of critical $L$-values (with Nadine Amersi, Jeff Beyerl, Allison Proffer, and Larry Rolen)
The Ramanujan Journal, 27(2), 151-162 (2012).
This paper is the result of an REU project I directed at the Clemson REU on Combinatorics and Computational Number Theory during the summer of 2010. We study the pullback of a Siegel Eisenstein series of weight $k$ and full level from $\textrm{Sp}(6)$ to $\textrm{Sp}(4) \times \textrm{Sp}(2)$. By explicitly working out the constants in this pullback formula we are able to produce a weighted average formula for the special values $D(k-1,f)$ where $f$ runs over an orthogonal basis of $S_{k}(\textrm{SL}_2(\mathbb{Z})$.
On the cuspidality of pullbacks of Siegel Eisenstein series and applications
International Mathematics Research Notices, 7, 1706-1756 (2011).
This paper is essentially a follow-up paper to Saito-Kurokawa lifts and applications to the Bloch-Kato conjecture. (One should see the description of that paper for the set-up of this one.) Here we remove some of the ad-hoc arguments used to produce a congruence between a Saito-Kurokawa lift and a cuspidal Siegel eigenform with irreducible Galois representation. The results are phrased in terms of the CAP-ideal (an ideal analogous to the Eisenstein ideal). Some results of E. Urban are generalized which allow us to strengthen the results from earlier work to give essentially one inclusion of the Bloch-Kato conjecture for the k-th twist of the Galois representation associated to f (up to our technical hypotheses needed to produce the congruence). This paper does not include an explicit description of the relation of the main result stated and the Bloch-Kato conjecture. Such a description is included in a forthcoming paper with Mahesh Agarwal that generalizes these results to include newforms of square-free level.
On the cuspidality of pullbacks of Siegel Eisenstein series to $\textrm{Sp}(2m) \times \textrm{Sp}(2n)$
Journal of Number Theory, 131, 106-119 (2011).
This paper studies the conditions under which one can conclude that the pullback of a Siegel Eisenstein series from $\textrm{Sp}(2m) \times \textrm{Sp}(2n)$ is cuspidal in the smaller variable. It was shown by Garrett that if $n=m$, for a certain choice of section the pullback of the associated Eisenstein series is cuspidal in each variable. Here we generalize this to show that if $m$ is not equal to $n$, the pullback of the Eisenstein series is cuspidal in the smaller variable.
$L$-functions on $\textrm{GSp}(4) \times \textrm{GL}(2)$ and the Bloch-Kato conjecture
International Journal of Number Theory, 6(8), 1901-1926 (2010).
Here we use a pullback formula of B. Heim that calculates the inner product of the pullback of a Siegel Eisenstein series on $\textrm{GSp}(10)$ to $\textrm{GSp}(4) \times \textrm{GSp}(4) \times \textrm{GL}(2)$ with the Saito-Kurokawa lift of a newform $f$ in each of the $\textrm{GSp}(4)$ variables and a newform $g$, allowed to vary, in the $\textrm{GL}(2)$ variable. We show how this can be used to give results towards the Bloch-Kato conjecture for $f$. In particular, this gives a different flavor of result than Saito-Kurokawa lifts and applications to the Bloch-Kato conjecture as the freedom in the technical hypotheses there are in varying a character and here we are allowed to vary a modular form.
On the congruence primes of Saito-Kurokawa lifts of odd square-free level
Mathematical Research Letters, 17(5), 977-991 (2010).
In this paper we generalize a conjecture of H. Katsurada about the congruence primes of Saito-Kurokawa lifts to the case of odd square-free level. We also provide evidence for this new conjecture.
The first negative Hecke eigenvalue of genus 2 Siegel cuspforms with level $N \geq 1$
International Journal of Number Theory, 6(4), 857-867 (2010).
In this short note we extend results of W. Kohnen and J. Sengupta on the sign of eigenvalues of Siegel cuspforms. We show that their bound for the first negative Hecke eigenvalue of a genus 2 Siegel cuspform of level 1 extends to the case of level $N \geq 1$. We also discuss the signs of Hecke eigenvalues for CAP forms.
Level lowering for half-integral weight modular forms (with Yingkun Li)
Proceedings of the American Mathematical Society, 138, 1171-1173 (2010).
This is one of two papers that resulted from the Summer Undergraduate Research Fellowship (SURF) of Yingkun Li that I directed at Caltech during the summer of 2008. Here we provide a level stripping result for half-integral weight modular forms that we originally thought we would need in the work contained in Distribution of powers of the partition function modulo $\ell^{j}$.
Distribution of powers of the partition function modulo $\ell^{j}$ (with Yingkun Li)
Journal of Number Theory, 129, 2557-2568 (2009).
This is one of two papers that resulted from the Summer Undergraduate Research Fellowship (SURF) of Yingkun Li that I directed at Caltech during the summer of 2008. In this paper we study Newman's conjecture for powers of the partition for exceptional primes. We settle this conjecture in many cases for small powers of the partition function by generalizing results of Ono and Ahlgren.
Residually reducible representations of algebras over local Artinian rings
Proceedings of the American Mathematical Society, 136(10), 3409-3414 (2008).
In this paper we generalize a result of E. Urban on the structure of residually reducible representations on local Artinian rings from the case the semi-simplification of the residual representation splits into 2 absolutely irreducible representations to the case where it splits into $m >2$ absolutely irreducible representations. In particular, the case of $m=3$ is needed in On the cuspidality of pullbacks of Siegel Eisenstein series and applications.
An inner product relation on Saito-Kurokawa lifts
The Ramanujan Journal, 14(1), 89-105 (2007).
This paper consists of the calculation of the Petersson norm of a Saito-Kurokawa lift of square-free level. One should note this calculation is based on a construction of the Saito-Kurokawa lift that was later shown to be incorrect. The correct construction and calculation can be found in On the Bloch-Kato conjecture for elliptic modular forms of square-free level with Mahesh Agarwal. The paper also has an error in section 7 due to a mistake in factoring the $L$-function in Proposition 7.5.
Saito-Kurokawa lifts and applications to the Bloch-Kato conjecture
Compositio Mathematica, 143(2), 290-322 (2007).
Let $f$ be a normalized eigenform of weight $2k-2$ and level 1. In this paper we provide evidence for the Bloch-Kato conjecture for modular forms. We demonstrate an implication that under suitable hypotheses if $p$ divides the algebraic part of $L(k,f)$, the $p$ divides an appropriate Selmer group. We demonstrate this by establishing a congruence between the Saito-Kurokawa lift of $f$ and a cuspidal Siegel eigenform with irreducible Galois representation. The method here is essentially due to Ribet and his proof of the converse of Herbrand's theorem.
Other publications/project write-ups
Saito-Kurokawa lifts and applications to arithmetic
Conference Proceedings of the 9th Autumn Conference on Number Theory, Hakuba Japan, 1-11 (2007)
These are notes from my plenary address at the 9th Autumn Conference on Number Theory in Hakuba, Japan. The topic of the conference was automorphic forms on $\textrm{GSp}(4)$.
Variation of Hodge Structure (with Kirsten Eisentrager, Krzysztof Klosin, Jorge Pineiro, Mak Trifkovic, and Oliver Watson)
Arizona Winter School on Periods 2002
This is the short write-up from the project supervised by Johan de Jong during the 2002 Arizona Winter School.
Reviews of my articles on MathSciNet (subscription required)
Reviews I have written for MathSciNet (subscription required)
PhD graduates
Huixi Li, Clemson University, 2018.
PhD Thesis: Some conjectures in additive number theory
Initial Employment: 3 year postdoctoral position at University of Nevada - Reno
Rodney Keaton, Clemson University, 2014.
PhD Thesis: Level stripping for genus 2 Siegel modular forms
Initial Employment: 3 year postdoctoral position at University of Oklahoma
Dania Zantout, Clemson University, 2013.
PhD Thesis: On the cuspidality of Maass-Gritsenko and mixed level lifts
Initial Employment: Visiting assistant professor at Clemson University
MS graduates
Hugh Geller, Clemson University, 2016.
Master's thesis: Ramanujan type congruences for the Klingen-Eisenstein series
Rodney Keaton, Clemson University, 2010. (Jointly advised with Kevin James.)
Master's thesis: Explicit level-lowering for 2-dimensional modular Galois representations
Andrew Bell, Creative inquiry student, Clemson University, 2013.
Palak Bhasin, Honors Thesis, Queens College, 2015.
Jarryd Boyle, Creative inquiry student, Clemson University, 2016-17.
Luna Bozeman, Creative inquiry student, Clemson University, 2016-17.
Joel Clingempeel, Clemson University, 2011-13.
Patrick Dynes, Creative inquiry student, independent study student, Clemson University 2013-16.
Rivers Jenkins, Creative inquiry student, Clemson University, 2015 (jointly advised with Felice Manganiello).
Catherine Kenyon, Creative inquiry student, Clemson University 2016-17.
Yingkun Li, California Institute of Technology, 2009.
Sam Mixon, Creative inquiry student, Clemson University, 2015 (jointly advised with Felice Manganiello).
Sloan Neitert, Creative inquiry student, Clemson University 2016.
Debra Parmentola, Creative inquiry student, Clemson University 2013-14.
Kristen Savary, Creative inquiry student, Clemson University, 2015 (jointly advised with Felice Manganiello).
Trevor Squires, Creative inquiry student, Clemson University, 2016-17.
Bo Sun, Creative inquiry student, Clemson University, 2016-17.
Ashley Stanziola, Creative inquiry student, Clemson University, 2015 (jointly advised with Felice Manganiello).
Dalton Randall, Creative inquiry student, Clemson University, 2015 (jointly advised with Felice Manganiello).
Brittany Rosener, Creative inquiry student, Clemson University 2013-14.
REU students
Nadine Amersi, University College London (2010 REU)
Allison Arnold-Roksandich, Harvey Mudd College (2013 REU) (Jointly advised with Kevin James.)
Robert Cass, University of Kentucky (2013 REU) (Jointly advised with Kevin James.)
Beren Gunsolus, University of Minnesota (2018 REU) (Jointly advised with Felice Manganiello)
Alfeen Hasmani, Molloy College (2012 REU)
David Heras, William and Mary University (2012 REU) (Jointly advised with Kevin James.)
Lindsey Hiltner, University of North Dakota (2012 REU)
Angela Kraft, Bethany Lutheran College (2012 REU)
Jeremy Lilly, Oregon State University (2018 REU) (Jointly advised with Felice Manganiello)
Jennifer Loe, Oklahoma Christian University (2013 REU) (Jointly advised with Kirsti Wash.)
Danielle Middlebrooks, Spelman College (2013 REU) (Jointly advised with Kirsti Wash.)
Ashley Morris, Savannah State University (2013 REU) (Jointly advised with Kirsti Wash.)
Salvatore Parenti, University of Michigan (2013 REU) (Jointly advised with Kevin James.)
Allison Proffer, Virginia Commonwealth University (2010 REU)
Andrew Qian, University of California - Berkeley (2012 REU) (Jointly advised with Kevin James.)
Larry Rolen, University of Wisconsin (2010 REU)
Daniel Scofield, Grove City College (2012 REU)
Daniel Shankman, University of Tennessee (2013 REU) (Jointly advised with Kevin James.)
Kimberly Stubbs, UNC - Asheville (2013 REU) (Jointly advised with Kevin James.)
Brandon Tran, MIT (2012 REU) (Jointly advised with Kevin James.)
Minh-Tam Trinh, Princeton University (2012 REU) (Jointly advised with Kevin James.)
Rico Vicente, California State University - Long Beach (2018 REU)
Alexandra Walsh, Brown University (2018 REU)
Philip Wertheimer Johns Hopkins University (2012 REU) (Jointly advised with Kevin James.)
email: [email protected] phone: 323-259-2680 office: Fowler 305 | CommonCrawl |
Steven J. Miller
Steven Joel Miller is a mathematician who specializes in analytic number theory and has also worked in applied fields such as sabermetrics and linear programming.[1] He is a co-author, with Ramin Takloo-Bighash, of An Invitation to Modern Number Theory (Princeton University Press, 2006), with Midge Cozzens of The Mathematics of Encryption: An Elementary Introduction (AMS Mathematical World series 29, Providence, RI, 2013), and with Stephan Ramon Garcia of ``100 Years of Math Milestones: The Pi Mu Epsilon Centennial Collection (American Mathematical Society, 2019). He also edited Theory and Applications of Benford's Law (Princeton University Press, 2015) and wrote The Mathematics of Optimization: How to do things faster (AMS Pure and Applied Undergraduate Texts Volume: 30; 2017) and ``The Probability Lifesaver: All the Tools You Need to Understand Chance (Princeton University Press, 2017). He has written over 100 papers in topics including accounting, Benford's law, computer science, economics, marketing, mathematics, physics, probability, sabermetrics, and statistics, available on the arXiv and his homepage.
Steven Joel Miller
NationalityAmerican
Alma materYale University
Princeton University
Scientific career
FieldsMathematics
InstitutionsWilliams College
Smith College
Mount Holyoke College
Brown University
Boston University
Ohio State University
American Institute of Mathematics
NYU
Princeton University
Thesis1 and 2 Level Densities for Families of Elliptic Curves: Evidence for the Underlying Group Symmetries (2002)
Doctoral advisorsPeter Sarnak
Henryk Iwaniec
Websiteweb.williams.edu/Mathematics/sjmiller/public_html/
Academic career
Miller earned his B.S. in mathematics and physics at Yale University and completed his graduate studies in mathematics at Princeton University in 2002. His Ph.D. thesis, titled "1 and 2 Level Densities for Families of Elliptic Curves: Evidence for the Underlying Group Symmetries," was written under the direction of Peter Sarnak and Henryk Iwaniec.[2] He is currently a professor of mathematics at Williams College, where he has served as the Director of the Williams SMALL REU Program and is currently the faculty president of the Williams Phi Beta Kappa chapter.[3] He's also a faculty fellow at the Erdos Institute.[4]
He was included in the 2019 class of fellows of the American Mathematical Society "for contributions to number theory and service to the mathematical community, particularly in support of mentoring undergraduate research".[5]
Books
Miller has published six books.
• 100 Years of Math Milestones: The Pi Mu Epsilon Centennial Collection (with Stephan Ramon Garcia): https://bookstore.ams.org/mbk-121
• Benford's Law: Theory and Applications (editor): https://press.princeton.edu/books/hardcover/9780691147611/benfords-law
• An Invitation to Modern Number Theory (with Ramin Takloo-Bighash): https://press.princeton.edu/books/hardcover/9780691120607/an-invitation-to-modern-number-theory
• The Mathematics of Encryption: An Elementary Introduction (with Margaret Cozzens): https://bookstore.ams.org/mawrld-29
• Mathematics of Optimization: How to do Things Faster: https://bookstore.ams.org/amstext-30/
• The Probability Lifesaver: All the Tools You Need to Understand Chance: https://press.princeton.edu/books/hardcover/9780691149547/the-probability-lifesaver
Controversies
In the aftermath of the 2020 United States presidential election Miller performed a statistical analysis of the integrity of mail in voting in Pennsylvania. The data underlying the analysis was collected by former Trump staffer Matt Braynard's Voter Integrity Fund. The data was collected by calling 20,000 Republican voters in Pennsylvania who, according to state records, had requested but not returned ballots. Of the 20,000 called 2,684 agreed to take the survey, which found that 463 reported that they actually had mailed in a ballot and 556 reported that they had not requested a ballot in the first place.[6]
In Miller's statement to the court - Exhibit A of Donald J. Trump for President v. Boockvar - he stated: "I estimate that with a reasonable degree of mathematical certainty (based on the data I received being accurate and a representative sample of the population) the number of the 165,412 mail-in ballots requested by someone other than the registered Republican is at least 37,000, and the number of the 165,412 mail-in ballots requested by registered Republicans and returned but not counted is at least 38,910 ... The analysis is based on responses from a data set drawn from 165,412 registered Republican voters who had a mail-in ballot requested in their name but not counted in the election. We estimate on the order of 41,000 of these ballots were requested by someone other than the proper voter. Who made such requests, and why? One possible explanation is that ballots were requested by others. Another possible explanation is that a large number of people requested ballots and forgot they did so later. Again, the conclusions above are based on the data provided being both accurate and a representative sample."[7]
Miller's statement drew sharp criticism from his peers, centered on the low response rate of phone surveys yielding unrepresentive data upon which Miller's estimates were based. Miller apologized for the "lack of clarity and due diligence" in a leaked early draft of his work.[6] Richard D. De Veaux, Vice President of the American Statistical Association and Professor of Statistics at Williams College, commented "any estimates based on unverifiable or biased data are inaccurate, wrong and unfounded. To apply naïve statistical formulas to biased data and publish this is both irresponsible and unethical".[8]
In interviews Miller has gone on the record about being a conservative.[9]
Research Experiences for Undergraduates
Miller has continuously run summer research groups in, among other topics, Benford's law, combinatorics, discrete geometry, number theory, probability, and random matrix theory at Williams College as part of the SMALL REU (Research Experiences for Undergraduates).
In 2020 with several colleagues, in response to the loss of opportunities for student research due to many summer programs being cancelled due to covid, he helped create the Polymath REU. The program has been supported by the National Science Foundation and Elsevier. From its homepage: Our goal is to provide research opportunities to every undergraduate who wishes to explore advanced mathematics. The program consists of research projects in a variety of mathematical topics and runs in the spirit of the Polymath Project. Each project is mentored by an active researcher with experience in undergraduate mentoring. Each project consists of 20-30 undergraduates, a main mentor, and additional mentors (usually graduate students). This group works towards solving a research problem and writing a paper. Each participant decides what they wish to obtain from the program, and participates accordingly.
Students interested in either program should apply through Math Programs.
College Courses
Starting in 2014, and consistently from 2016 onward, Miller has recorded his courses and made them freely available through YouTube. Below is a subset; the complete list, including different iterations of each course, is available at his homepage.
• Advanced Applied Analysis: Math 466: https://web.williams.edu/Mathematics/sjmiller/public_html/466Fa17/index.htm
• Advanced Applied Linear Programming: Math 416: https://web.williams.edu/Mathematics/sjmiller/public_html/416/index.htm
• Advanced Analysis: Math 389: https://web.williams.edu/Mathematics/sjmiller/public_html/389/index.htm
• Applied Analysis: Math 317 (Operations Research): https://web.williams.edu/Mathematics/sjmiller/public_html/317/index.htm
• Complex Analysis: Math 383: https://web.williams.edu/Mathematics/sjmiller/public_html/383Fa21/
• Multivariable Calculus: Math 150: http://web.williams.edu/Mathematics/sjmiller/public_html/150Sp21/
• Number Theory: Math 313: https://web.williams.edu/Mathematics/sjmiller/public_html/313Sp17/index.htm
• Operations Research: Math 317: https://web.williams.edu/Mathematics/sjmiller/public_html/317Fa19/
• Probability: Math/Stat 341: https://web.williams.edu/Mathematics/sjmiller/public_html/341Fa21/
• Problem Solving: Math 331: http://web.williams.edu/Mathematics/sjmiller/public_html/331Fa18/
References
1. "Steven J. Miller". Retrieved 11 May 2020.
2. Steven J. Miller at the Mathematics Genealogy Project
3. "Steven J". Web.williams.edu. Retrieved 2016-01-20.
4. "Steven J. Miller". Williams College. Retrieved 11 May 2020.
5. 2019 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2018-11-07
6. Paris, Francesca (24 November 2020). "Williams prof disavows own finding of mishandled GOP ballots". The Berkshire Eagle.
7. https://williamsrecord.com/wp-content/uploads/2020/11/Attachment-A.pdf
8. De Veaux, Richard D. (25 November 2020). "A rebuttal to Steven Miller's "REPORT ON PA GOP MAIL-IN BALLOT REQUESTS"". The Williams Record.
9. Eagle, Francesca Paris, The Berkshire (2020-11-24). "Williams prof disavows own finding of mishandled GOP ballots". The Berkshire Eagle. In interviews with The Williams Record and the public radio station WAMC, Miller has gone on the record about being a conservative.{{cite web}}: CS1 maint: multiple names: authors list (link) CS1 maint: url-status (link)
External links
• Steven J. Miller at the Mathematics Genealogy Project
Authority control
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| Wikipedia |
Grace Alele-Williams
Grace Alele-Williams OON, FMAN, FNAE (16 December 1932 – 25 March 2022) was a Nigerian professor of mathematics education,[1][2] who made history as the first Nigerian woman to receive a doctorate,[3][4] and the first Nigerian female vice-chancellor at the University of Benin.[5][6][7]
Grace Alele-Williams
OON, FMAN, FNAE
Born
Grace Alele
(1932-12-16)16 December 1932
Warri, Western Region, Nigeria
Died25 March 2022(2022-03-25) (aged 89)
Lagos, Nigeria
EducationPhD (mathematics)
Alma materUniversity College of Ibadan
University of Chicago
Spouse
Babatunde Abraham Williams
(m. 1963; died 2010)
Children5
Early life and education
Grace Awani Alele was born to Itsekiri parents in Warri, Western Region (present-day Delta State), Nigeria on 16 December 1932.[8][9][10][11] She attended Government School, Warri, Queen's College, Lagos and the University College of Ibadan[12] (now University of Ibadan). She obtained a master's degree in mathematics while teaching at Queen's School, Ede in Osun State in 1957 and her PhD degree in mathematics education at the University of Chicago (U.S.) in 1963,[12] thereby making her the first Nigerian woman to be awarded a doctorate. Grace Alele was married later that year and became known as Grace Alele-Williams.[11] She returned to Nigeria for a couple of years' postdoctoral work at the University of Ibadan before joining the University of Lagos in 1965.[13]
Career
Alele-Williams's teaching career started at Queen's School, Ede, Osun State, where she was a mathematics teacher from 1954 to 1957.[3] She left for the University of Vermont to become a graduate assistant and later assistant professor. From 1963 to 1965, Alele-Williams was a postdoctoral research fellow, department (and institute) of education, University of Ibadan from where she was appointed a professor of mathematics at the University of Lagos in 1976.[6]
She had a special interest in women's education. While spending a decade directing the institute of education, she introduced innovative non-degree programmes, allowing older women working as elementary school teachers to receive certificates. Alele-Williams has always demonstrated concern for the access of female African students to scientific and technological subjects.[1] Her interest in mathematics education was originally sparked by her stay in the US, which coincided with the Sputnik phenomenon. Working with the African Mathematics Program in Newton, Massachusetts, under the leadership of MIT professor Ted Martins, she participated in mathematics workshops held in various African cities from 1963 to 1975.[13] Highlights included writing texts and correspondence courses covering basic concepts in mathematics working in concert with leading mathematicians and educators. such as the book Modern Mathematics Handbook for Teachers published in 1974. She taught at the University of Lagos from 1965 to 1985, and spent a decade directing the institute of education, which introduced innovative non-degree programmes, with many of the certificate recipients older women working as elementary school teachers.[6] By serving in various committees and boards, Alele-Williams had made useful contributions in the development of education in Nigeria. She was chairman of the curriculum review committee, former Bendel State 1973–1979.[14][15] From 1979 to 1985, she served as chairman of the Lagos State curriculum review committee and Lagos State examinations boards.[13]
Alele-Williams was appointed vice-chancellor of the University of Benin in 1985, becoming the first female vice-chancellor of a Nigerian university,[16] and she believes her appointment, which ended in 1992, was "a test case to demonstrate a woman's executive capability".[6]
Alele-Williams was a force for reform in the dark age for Nigeria's higher education in the 1980s. Then, the activities of secret cults, confraternities and societies had spread within the Nigerian universities, especially in University of Benin. A task which many men had failed, she was able to make notable contributions.[1]
After serving as the vice-chancellor of the University of Benin, she joined the board of directors of Chevron-Texaco Nigeria. She was also on the board of HIP asset management company limited, an asset management company in Lagos, Nigeria.[11]
Alele-Williams was a member of governing council, UNESCO Institute of Education.[1] She was also a consultant to the UNESCO and Institute of International Education Planning.[17][13] For a decade (1963–73), she was a member of the African Mathematics Programme, located in Newton, Massachusetts, United States.[6] She was vice-president of the World Organisation for Early Childhood Education and later president of the Nigeria chapter,[15] and the first president of the African Mathematical Union Commission on Women in Mathematics.[18] She also served ten years (1993–2004) as regional vice-president for Africa of the Third World Organization for Women in Science.[19]
Personal life and death
Alele-Williams married Babatunde Abraham Williams (1932–2010) in December 1963, not long after returning to Nigeria from the United States. Williams was a political scientist who, at the time of their marriage, was a senior lecturer at the University of Ife, Osun State.[11]
Alele-Williams had five children, and, as of 2017, ten grandchildren.[2] She died on 25 March 2022 at the age of 89.[20][21][22] The next day, the University of Benin flew the institution's flag at half-mast to mourn her death.[8][10]
Awards
Alele-Williams received several awards and honours. She received the Order of the Niger in 1987, and was elected a Fellow of the Mathematical Association of Nigeria and a Fellow of the Nigerian Academy of Education.[23]
On 28 February 2014, she was one of 100 people to receive the Centenary Award, "a special recognition of unique contributions of Nigerians to the socio-cultural, economic and political development of the country in the last 100 years".[24][25][26]
She is included in a deck of playing cards featuring notable women mathematicians published by the Association of Women in Mathematics.[27]
Publications
• "Dynamics of Curriculum Change in Mathematics" – Lagos State Modern Mathematics Project[28]
• "Education of Women for National Development"[29]
• "Report: The Entebbe Mathematics Project"[30]
• "The development of a modern mathematics curriculum in Africa"[31]
• "Education and Government in Northern Nigeria"[32]
• "Education and Status of Nigerian Women"[33]
• "Major Constraints to Women's Access to Higher Education in Africa"[34]
• "The Politics of Administering a Nigerian University"[35]
• "The Political Dilemma of Popular Education: An African Case"[36]
References
1. Taire, Morenike (14 April 2018). "Grace Alele-Williams: Mathematician who dealt with cultism at UNIBEN". Vanguard News. Archived from the original on 23 May 2018. Retrieved 23 May 2018.
2. "Grace Alele, Role Model, Teacher, Professor, Doctor, Vice-chancellor, Warrior, Prominent Nigerian, Nigeria Personality Profiles". www.nigeriagalleria.com. Archived from the original on 29 April 2021. Retrieved 20 May 2021.
3. Riddle, Larry (16 February 2022). "Grace Alele Williams". Biographies of Women Mathematicians. Agnes Scott College. Archived from the original on 22 February 2022. Retrieved 27 March 2022.
4. "5 women who have made their marks in education". www.pulse.ng. 8 March 2018. Archived from the original on 1 July 2022. Retrieved 20 March 2019.
5. Nkechi Nwankwo (2006), Women Leadership in Nigeria: Stories of Four Women Role Models, Lagos: Deutchetz Publishers. Review Archived 26 June 2012 at the Wayback Machine by Theresa Onwughalu in the Daily Sun, 25 July 2006.
6. "Biographical sketches of famous African mathematicians: Grace Alele Williams", AMUCHMA Newsletter #12, African Mathematical Union, Commission on the History of Mathematics in Africa, 27 March 1994, archived from the original on 24 February 2020, retrieved 23 January 2010.
7. "Prof. Mrs Grace Alele Williams OFR, HLR". Hallmarks of Labour Foundation. 28 November 2011. Archived from the original on 16 December 2013. Retrieved 9 December 2013.
8. "UNIBEN flies flag at half-mast to mourn Alele-Williams". Vanguard News. 26 March 2022. Archived from the original on 26 March 2022. Retrieved 27 March 2022. The University of Benin (UNIBEN) on Saturday ordered the flying of the institution's flag at half-mast to mourn the death of the first female vice-chancellor in Nigeria, Prof. Grace Alele-Williams. Alele-Williams, who was appointed vice-chancellor of the University of Benin in 1985 died on Friday in Lagos at the age of 89 years. ... Prof. Alele-Williams, Nigeria's first female professor of Mathematics Education was born on Dec. 16, 1932 in Delta to an Itsekiri mother and Owan father from Sobe, Edo.
9. Howell, Karin-Therese; Neudauer, Nancy Ann (2022). "Grace Alele-Williams Nigerian Mathematician of Many Firsts - Breaking Down Barriers and Opening Paths" (PDF). Notices of the American Mathematical Society. 69: 439–455.
10. Ahon, Festus; Ojiego, Nnamdi; Aliu, Ozioruva (27 March 2022). "ALELE – WILLIAMS: More tributes as Okowa, Obaseki mourn". Vanguard News. Archived from the original on 27 March 2022. Retrieved 27 March 2022. the UNIBEN have ordered that the institution's flag be mounted at half-mast to mourn the death of the late professor. Alele-Williams, who passed on Friday evening in Lagos at the age of 89 years, was born in Warri, Delta State.
11. O'Connor, J. J.; Robertson, E. F. "Grace Alele-Williams – Biography". MacTutor History of Mathematics archive. Archived from the original on 27 April 2021. Retrieved 27 April 2021.
12. "Personality of The Week – Grace Alele williams". SilverbirdTV. 20 November 2014. Archived from the original on 1 July 2022. Retrieved 9 May 2019.
13. "Grace Alele Williams - Black Women in Mathematics". www.math.buffalo.edu. Archived from the original on 4 March 2021. Retrieved 9 May 2019.
14. "Grace Alele-Williams". Heels of Influence. Archived from the original on 30 October 2020. Retrieved 29 May 2020.
15. admin. "Grace awani ALELE-WILLIAMS – Legacy Way". Archived from the original on 28 November 2020. Retrieved 2 May 2020.
16. "First Female Vice Chancellor in Nigeria". Hintnaija. 12 April 2018. Archived from the original on 27 April 2021. Retrieved 27 April 2021.
17. "Women in Higher Education Management" (PDF). Unesco: 7. Archived (PDF) from the original on 19 March 2015. Retrieved 13 January 2014. {{cite journal}}: Cite journal requires |journal= (help)
18. Ouedraogo, Pr Marie Françoise (30 May 2015). AWMA: une association au service des femmes mathématiciennes africaines (PDF) (Speech). Femmes et Mathematiques: Mathématiciennes africaines (in French). Institut Henri Poincaré. Archived (PDF) from the original on 21 January 2021. Retrieved 15 January 2021.
19. "Executive Board". owsd.net. OWSD. 6 February 2015. Archived from the original on 20 June 2020. Retrieved 28 March 2022.
20. Egbejule, Michael (26 March 2022). "First female VC, Prof. Grace Alele-Williams, dies at 89". The Guardian Nigeria. Archived from the original on 27 March 2022. Retrieved 27 March 2022.
21. Millz, Bayo (25 March 2022). "Just In: First female VC, Prof Alele-Williams allegedly dies at 89". TheNewsGuru. Archived from the original on 25 March 2022. Retrieved 25 March 2022.
22. Akintade, Adefemola (25 March 2022). "Nigeria's first female Vice-Chancellor Grace Alele-Williams is dead". Peoples Gazette. Archived from the original on 25 March 2022. Retrieved 25 March 2022.
23. "Buhari felicitates first woman Nigerian Professor, Alele-Willaims, at 89". The Nation Newspaper. 15 December 2021. Archived from the original on 7 March 2022. Retrieved 7 March 2022.
24. "100 Nigerians get Centenary Awards Friday (tonight) [Full List]". Premium Times Nigeria. 28 February 2014. Archived from the original on 27 March 2022. Retrieved 27 March 2022. The full list of the awardees, including those who have rejected the honour is reproduced below along with the categories: ... C. PIONEERS IN PROFESSIONAL CALLINGS/CAREERS ... 30. Professor Grace Alele-Williams
25. Akpan, Mike (24 March 2014). "Demeaning Centenary Awards". Realnews Magazine. Archived from the original on 17 January 2022. Retrieved 27 March 2022.
26. "Grace Alele Williams receives deafening ovation". Encomium Magazine. 28 February 2014. Archived from the original on 27 April 2021. Retrieved 27 April 2021.
27. "Mathematicians of EvenQuads Deck 1". awm-math.org. Archived from the original on 19 June 2022. Retrieved 18 June 2022.
28. Williams, Grace Alele (1974). "Dynamics of Curriculum Change in Mathematics—Lagos State Modern Mathematics Project". West African Journal of Education. Archived from the original on 7 May 2021. Retrieved 9 May 2019.
29. Alele-Williams, G. (1986). "Education of Women for National Development". Archived from the original on 19 March 2021. Retrieved 9 May 2019. {{cite journal}}: Cite journal requires |journal= (help)
30. Williams, Grace Alele (1 June 1971). "Report: The Entebbe mathematics project". International Review of Education. 17 (2): 210–214. Bibcode:1971IREdu..17..210W. doi:10.1007/BF01421114. ISSN 1573-0638. S2CID 144062711.
31. WILLIAMS, GRACE A. ALELE (1976). "The development of a modern mathematics curriculum in Africa". The Arithmetic Teacher. 23 (4): 254–261. doi:10.5951/AT.23.4.0254. ISSN 0004-136X. JSTOR 41188955.
32. Williams, Grace Alele (1973). "Education and Government in Northern Nigeria". Présence Africaine. 87 (3): 156. doi:10.3917/presa.087.0156.. In: Lema, Anza A; Williams, Grace Alele; Simiyu, Vincent G. (1973). Presence Africaine. Revue Culturelle du Monde Noir / Cultural Review of the Negro World. Nouvelle Serie Bilingual / New Bilingual Series. No. 87, 3e Trimestre / 3rd Quarterly, 1973. Archived from the original on 16 April 2021. Retrieved 27 March 2022.
33. In Nigerian women and development (1988), edited by Ogunṣhẹyẹ, F. Adetowun; Domenico, Catherine D.; Dennis, Carolyne; Awosika, Keziah; Akinkoye, Olu. Ibadan, Nigeria: Ibadan University Press. ISBN 9789781212192, pages 171–179
34. Alele-Williams, G. (1992). "Major Constraints to Women's Access to Higher Education": 71–76. Archived from the original on 7 May 2021. Retrieved 23 January 2021. {{cite journal}}: Cite journal requires |journal= (help)
35. "Celebrating Prof Grace Alele Williams, Nigeria's first female Vice Chancellor". TheDailyNG. 11 February 2020. Archived from the original on 31 January 2021. Retrieved 23 January 2021.
36. Chukunta, N. K. Onuoha (1978). "Education and National Integration in Africa: A Case Study of Nigeria". African Studies Review. 21 (2): 67–76. doi:10.2307/523662. ISSN 0002-0206. JSTOR 523662. S2CID 143632871. Archived from the original on 30 January 2021. Retrieved 23 January 2021.
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\begin{document}
\title{Two-photon interference with thermal light} \author{Giuliano Scarcelli, Alejandra Valencia, and Yanhua Shih} \address{Department of Physics, University of Maryland, Baltimore County, Baltimore, Maryland 21250}\maketitle
\begin{abstract} The study of entangled states has greatly improved the basic understanding about two-photon interferometry. Two-photon interference is not the interference of two photons but the result of superposition among indistinguishable two-photon amplitudes. The concept of two-photon amplitude, however, has generally been restricted to the case of entangled photons. In this letter we report an experimental study that may extend this concept to the general case of independent photons. The experiment also shows interesting practical applications regarding the possibility of obtaining high resolution interference patterns with thermal sources. \end{abstract}
The superposition principle is probably the most mysterious and fascinating concept of the theory of quantum mechanics \cite{feynman}. In Young's double-slit experiment, a light quantum has two indistinguishable alternative amplitudes that result in a photo-electron event at space-time point ($\mathbf{r}, t$). The superposition of the two indistinguishable amplitudes produces the interference of the light quantum itself \cite{Dirac}. Quantum theory may never identify through which slit (or both slits) the light quantum passed, however, it accurately predicts the counting rate as a function of the relative delay between the two amplitudes.
The experimental observations of Hanbury-Brown and Twiss introduced the concept of second order coherence \cite{hanbury}. Quantum theory of the second order interferometry describes the physical process of a joint photo-electron event at space-time points ($\mathbf{r_{1}}, t_{1}$) and ($\mathbf{r_{2}}, t_{2}$), produced by two light quanta with distinguishable and/or indistinguishable alternative amplitudes \cite{fano}. Two-photon optics is a complex subject involving optical coherence, photon statistics, and, the nonlocal physics associated with the Einstein-Podolsky-Rosen two-particle system. Besides probing the fundamental issues of quantum theory, the massive study of entangled states, especially the experimental and theoretical research on the entangled two-photon state of Spontaneous Parametric Down Conversion (SPDC) has provided great insights on two-photon interferometry \cite{zeilinger}. In particular, we have a better understanding of the troubling statement of Dirac: ``Each photon interferes only with itself. Interference between two different photons never occurs." The question whether two individual photons can or cannot interfere with each other has been answered experimentally based on the study of two-photon interferomerty of SPDC: two-photon interference cannot simply be described in terms of interference of two independent photons but must be envisioned as an actual two-photon phenomenon in which the indistinguishable alternatives are two-photon amplitudes contributing to the final joint photo-electron events \cite{pittman,yoonho,mandel}. The concept of two-photon amplitude is somehow troubling, probably because of the nonlocality that it implies, and so, if accepted, it has generally been considered peculiar of entangled photons.
In this letter we wish to report an experimental study that may extend the concept of two-photon interference as the result of superposition of indistinguishable two-photon amplitudes to the general case of two independent photons. The result is intriguing also in a practical sense because $N$-photon interferometry with entangled states has been proven to represent a great potential for imaging and metrological applications\cite{dowlingmetrolo,zeilingerfour}. In particular it has been proposed \cite{dowling,scully} and experimentally shown with two-photon entangled states from SPDC \cite{milena} that it is possible to do quantum lithography beyond the classical diffraction limit. In this experiment we simulated the experiment of D'Angelo et al. \cite{milena} with a pseudo-thermal source of light \cite{thermaltheory}. The experiment involves the measurement of second order interference of pseudo-thermal light through a standard Young's double slit interferometer. In order to provide a clear physical picture of the phenomenon, we will discuss similar and different aspects between the thermal state and the entangled state of SPDC in this regard. It must be noted that a similar source, with a similar setup has been used for different purposes in a historical experiment\cite{haner}.
The experimental setup is schematically shown in Fig.~\ref{setup}: the pseudo-thermal light source (or entangled two-photon light source of SPDC for comparison) illuminates a double slit of slit-width $a$ with slit-distance $d$; the joint photo-detection occurs in the far field plane with two photon counting detectors.
Let's start by noticing that in both cases of pseudo-thermal light and SPDC radiation there is no observable first order interference effect in this experimental setup. The absence of the first order interference precludes the possibility of the second order interference to be a consequence of interference of the first order or the result of ``partial coherence" between the fields at slit A and slit B. In this particular experiment any observable interference is not the result of \textit{each photon interfering with itself}.
The quantity that governs the probability of joint photodetection and therefore the rate of coincidence counts is the second order Glauber correlation function \cite{glauber}: \begin{eqnarray}\label{G2} G^{(2)}(t_{1},r_{1}; t_{2},r_{2}) \equiv Tr[\hat{\rho} E_{1}^{(-)}(t_{1}, r_{1})E_{2}^{(-)}(t_{2}, r_{2}) \nonumber \\ E_{2}^{(+)}(t_{2}, r_{2})E_{1}^{(+)}(t_{1}, r_{1}) ] \end{eqnarray} where $E_{1,2}^{(\pm )}(r_{j}, t_{j})$, $j=1,2$, are positive-frequency and the negative-frequency components of the field at detectors $D_{1}$ and $D_{2}$, and $\hat{\rho}$ represents the density matrix of the quantum state under consideration. The field operators, in both thermal light and SPDC cases, can be written as the superposition of earlier fields at slit A and B: \begin{eqnarray} \label{efield} E_{1}^{(+)}(r_{1},t_{1}) & = & E_{A}^{(+)}(r_A, t_{1}-\frac{r_{A1}}{c})+ E_{B}^{(+)}(r_B, t_{1}-\frac{r_{B1}}{c})\\ \nonumber E_{2}^{(+)}(r_{1},t_{1}) & = & E_{A}^{(+)}(r_A, t_{2}-\frac{r_{A2}}{c})+ E_{B}^{(+)}(r_B, t_{2}-\frac{r_{B2}}{c}) \end{eqnarray} with $r_{Aj}$ ($r_{Bj}$) defining the optical path length from slit $A$ ($B$) to the $j^{th}$ detector. If $G^{(2)}$ is different in the case of thermal light and SPDC, the difference must come from the intrinsic property of the light, as expected.
Let's first briefly review the known physics behind the experiment that uses SPDC as the light source \cite{milena}. SPDC is a nonlinear process in which an entangled pair of photons, signal and idler, are simultaneously created. Therefore a joint photodetection is almost always the result of the detection of the signal-idler pair. In ref. \cite{milena}, the double slit was placed very close to the SPDC crystal so that the source is divided into two regions: upper slit ($A$) and lower slit ($B$). Due to the entangled nature, a signal-idler pair is generated either from slit $A$ or slit $B$, but never from different ones. It is very intuitive, then, to write the two-photon state that would lead to a joint detection measurement in the following way: \begin{equation}\label{spdcstate}
|\Psi \rangle \simeq [a^{\dag}_{s}a^{\dag}_{i}e^{i\phi_{A}}+
b^{\dag}_{s}b^{\dag}_{i}e^{i\phi_{B}}]| 0\rangle \end{equation} here $a^{\dag}$ and $b^{\dag}$ stand for the photon creation operators at the upper and lower slit respectively, and $\phi_{A}$ and $\phi_{B}$ are the phases of the two-photon modes in correspondence to the upper slit ($A$) and the lower slit ($B$). The spatial coherence of the pump beam of the SPDC justifies the assumption: $\phi_{A}-\phi_{B}=constant$. Under these conditions the expected second order correlation function is calculated as: \begin{eqnarray} \label{coinspdc-1} G^{(2)} & = & \mid e^{i k (r_{A1}+r_{A2})}+ e^{i k (r_{B1}+r_{B2})}\mid ^{2}\\ \nonumber & \propto & 1+cos[k(r_{A1}+r_{A2}-r_{B1}-r_{B2})] \end{eqnarray} In the far field zone, taking into account the finite size of the slits, the final interference-diffraction pattern is expected as: \begin{eqnarray} \label{coinspdc} G^{(2)} \propto sinc^{2}[\frac{\pi a (x_{1}+x_{2})}{\lambda z }]cos^{2}[\frac{\pi d (x_{1}+x_{2})}{\lambda z}] \end{eqnarray} where $x_{1}$ and $x_{2}$ are the horizontal displacement of $D_{1}$ and $D_{2}$, respectively, and $z$ is the common distance from the detectors to the double slit. Besides opening the road towards quantum lithography, this experiment clearly demonstrated that the second order interference is the result of the superposition between the ``upper-upper" ($A \rightarrow D_{1}$ with $A \rightarrow D_{2}$) and the ``lower-lower" ($B \rightarrow D_{1}$ with $B \rightarrow D_{2}$) two-photon amplitudes.
In the present experiment, we substituted the SPDC light with a pseudo-thermal source \cite{martienssen}. The source is basically composed by a He-Ne laser beam focused on a rotating ground glass diffuser disk: the radiation is randomly scattered in all possible directions. It has been shown theoretically and experimentally \cite{arecchibook} that the scattered radiation has the same statistical and optical properties as standard thermal sources. During the experiment, the intensity of the light was operated in a very low counting rate regime to achieve the condition in which only two photons were present in the setup within the joint detection time window. It is reasonable then to restrict our analysis at the level of two photons. Let's use a simple physical model for the process: it can be shown that one possible basis of the physical state space that describes the two-photon system may be composed by the following three normalized states \cite{cohen}: \begin{eqnarray}\label{basis}
|\alpha \rangle & = & a^{\dag}_{k}a^{\dag}_{k'} |0 \rangle;\\
\nonumber | \beta \rangle & = & b^{\dag}_{k}b^{\dag}_{k'}|0
\rangle;\\ \nonumber |\gamma \rangle & = & \frac{1}{\sqrt{2}}(
a^{\dag}_{k}b^{\dag}_{k'}+b^{\dag}_{k}a^{\dag}_{k'}) |0\rangle. \end{eqnarray} Here $a^{\dag}$ and $b^{\dag}$ stand for creation operators of the photons generated at the upper and lower slit respectively; while $k$, and $k'$ corresponds to the modes of the radiation leading to detectors $D_{1}$ and $D_{2}$ respectively. Notice that the three basis vectors correspond to the three intuitive alternatives of joint photodetection, i.e. ($\alpha$) the two photons both come from the upper slit $A$; ($\beta$) both come from the lower slit $B$; or ($\gamma$) one comes from $A$ and the other from $B$. It is important to emphasize that ($\gamma$) will lead to the interference feature. We will treat our system as a statistical mixture of the three basis vectors: \begin{equation}\label{combination}
\hat{\rho}= |\alpha|^{2} |\alpha \rangle \langle \alpha | +| \beta
|^{2} |\beta \rangle \langle \beta | + |\gamma |^{2} |\gamma
\rangle \langle \gamma | \end{equation}
where $|\alpha|^{2}$, $|\beta|^{2}$, and $|\gamma|^{2}$, are the probabilities of having the system in one of the basis vectors. In the thermal light case, the three probabilities are equal ($1/3$). Thus, the second order correlation function can be expressed as follows: \begin{eqnarray}\label{G2dens1}
G^{(2)} \propto \langle \alpha
|E_{1}^{(-)}E_{2}^{(-)}E_{2}^{(+)}E_{1}^{(+)}|\alpha \rangle+\\ \nonumber \langle \beta
|E_{1}^{(-)}E_{2}^{(-)}E_{2}^{(+)}E_{1}^{(+)}|\beta \rangle+\\ \nonumber \langle \gamma
|E_{1}^{(-)}E_{2}^{(-)}E_{2}^{(+)}E_{1}^{(+)}|\gamma \rangle. \end{eqnarray} Substituting the field operators of Eq. (\ref{efield}) into Eq. (\ref{G2dens1}) we obtain: \begin{eqnarray} \label{coin3}
G^{(2)} \propto |e^{i k (r_{A1}+r_{A2})}|^{2}+
|e^{i k (r_{B1}+r_{B2})}|^{2}+ \\
\nonumber \frac{1}{2}| e^{i k (r_{A1}+r_{B2})}+ e^{i k
(r_{B1}+r_{A2})}| ^{2}. \end{eqnarray} The superposition of the indistinguishable two-photon amplitudes ``upper-lower" ($A \rightarrow D_{1}$ with $B \rightarrow D_{2}$) and ``lower-upper" ($A \rightarrow D_{2}$ with $B \rightarrow D_{1}$) are responsible for the interference. In the far field zone and considering the finite size of the slits, the interference-diffraction pattern is thus: \begin{eqnarray} \label{resultfar} G^{(2)} \propto 1+ sinc^{2}(\frac{ \pi a (x_{1}-x_{2})}{\lambda z} )cos^{2}(\frac{\pi d (x_{1}-x_{2})}{\lambda z}). \end{eqnarray} Comparing with the SPDC case of (Eq.~\ref{coinspdc}), we obtain a similar interference-diffraction pattern, hinting to the similar two-photon physics behind the two effects.
In the actual experiment we had an attenuated He-Ne laser beam impinging on a double slit, $10cm$ after the slit we put a converging lens ($f=25mm$) and placed the rotating ground glass disk at $33.5mm$ from the lens. Basically we imaged the double slit onto the ground glass in order to produce an effective double slit illumination ($a=0.043mm$ and $d=0.135mm$) on the disk, that is our source, as indicated in Fig.~\ref{setup}. The radiation scattered by the ground glass was then divided by a beam splitter and sent to two horizontally displaceable fibers, connected to single photon counting modules.
Fig.~\ref{coin} reports the measured two-photon interference-diffraction pattern. The solid line represents the theoretical fit using Eq.~\ref{resultfar}. It is interesting to see that, similarly to the SPDC case, the interference-diffraction pattern is twice as narrow as the standard interference-diffraction pattern of He-Ne light (shown in the lower plot of Fig.~\ref{coin} for comparison) and with interference modulation twice as large as the standard pattern, as if it was produced by a source of light with half the wavelength of the He-Ne laser. The visibility of the pattern, however, is about $28 \%$ whereas SPDC allows $100 \%$ visibility. Fig.~\ref{single} reports the single counts of detector $D_{1}$ and $D_{2}$ when the detectors are scanned in the horizontal direction. It is apparent that the single counts are flat over the entire analyzed range. This demonstrates the absence of any first order interference phenomena, i.e. no ``partial coherence" existing in this experiment. In order to ensure that the source we used was indeed ``thermal", before proceeding to the actual measurement, we repeated the historical experiment performed by Arecchi et al \cite{arecchi}. Fig.~\ref{mca} reports the result of the second order correlation measurements. Basically we plot a histogram of number of coincidence counts versus the time difference of the clicks from the two detectors. This measurement is the principal evidence of thermal light statistics of a pseudothermal source. This result was also used to ``calibrate" the coincidence time window. In all the measurements of this experiment, we set a time window of about $600$ ns around the peak of Fig.~\ref{mca} and measured the number of coincidences within that window.
The experimental results therefore confirm the expected similarities and differences between the thermal light and the entangled state regarding to the two-photon interference. As was shown in the theoretical derivation, the analogy between the thermal case and the SPDC case is due to the similar physics: the superposition of the two-photon alternative amplitudes leads to the interference. However there are two main differences: (1) the joint detection counting rate is a function of $x_{1}-x_{2}$ in the thermal case instead of $x_{1}+x_{2}$. The reason is that in the SPDC case, the two-photon amplitudes leading to the interference are the ``upper-upper" ($A \rightarrow D_{1}$ with $A \rightarrow D_{2}$) and the ``lower-lower" ($B \rightarrow D_{1}$ with $B \rightarrow D_{2}$) alternatives; in the case of thermal light, instead, the interference is produced by the superposition of the ``upper-lower" ($A \rightarrow D_{1}$ with $B \rightarrow D_{2}$) and the ``lower-upper" ($A \rightarrow D_{2}$ with $B \rightarrow D_{1}$) alternatives; (2) the visibility in the thermal light case is limited to $1/3$, in fact in the case analyzed in this experiment the ``upper-upper" and ``lower-lower" alternatives do not contribute to the interference, but they do contribute to the constant background of the pattern. In the SPDC case, instead, due to entanglement, the only existing two-photon amplitudes are the indistinguishable ``upper-upper" and ``lower-lower" alternatives.
From a practical point of view it is important to notice that with a pseudo-thermal source we achieved the same doubling in spatial resolution that was obtained with entangled two-photon states. Recently it has been shown that with entangled four-photon states it is possible to achieve resolutions four times better than the standard limit\cite{zeilingerfour}. However notice that since in our experiment the source of light does not involve any nonlinear process (i.e.: the wavelength of the two photons is the same of the pump, while in SPDC the entangled photons have twice the wavelength of the laser pump), the increase in spatial resolution is twice as large as the increase obtained in the analogous (same $N$, number of entangled photons) case with entangled photons. Moreover notice that in the historical experiment similar to ours \cite{haner} the increase in spatial resolution was not obtained, however the similarities between the two experiments and the fact that in \cite{haner} the intensity of light was much higher, lead us to think that with this source of light it can be overcome the main limitation presented by entangled photon sources at the moment, i.e.: the low counting rates. The practical applicability of the method shown here could be precluded by the low visibility of the pattern. For certain measurements, however, it may be possible to implement a detection scheme insensitive to the constant background noise that could restore high visibilities.
In conclusion, we have experimentally studied a second order interference phenomenon with thermal light. By comparing this experiment with the analogous one performed with entangled photons, we have justified the physical interpretation of the phenomenon in terms of interference between indistinguishable two-photon amplitudes. Paraphrasing Dirac, we may summarize the physics as follows: although \textit{each photon did not interfere with itself} in this experiment, the observed interference is the result of each \textit{pair} of independent photons interfering with itself.
The authors would like to thank S. Thanvanthri, J. Wen, M.D'angelo, D.Hudson and M.H. Rubin for helpful discussions. This research was supported in part by NSF, ONR and NASA-CASPR program.
\begin{references}
\bibitem{feynman} R.P. Feynman, R.B. Leighton, and M. Sands, {\em The Feynman Lectures on Physics, Vol. III} , (Addison-Wesley Publishing Co., Reading, MA, 1965).
\bibitem{Dirac} P.A.M. Dirac, {\em The Principles of Quantum Mechanics}, {Clarendon, Oxford 1930}.
\bibitem{hanbury} R. Hanbury-Brown and R.Q. Twiss, Nature \textbf{177}, 28 (1956); \textbf{178}, 1046, 1447 (1956).
\bibitem{fano} M.O. Scully and M.S. Zubairy, {\em Quantum Optics} {Cambridge University Press, 1997}; H. Paul, Rev. Mod. Phys. \textbf{58}, 209 (1986).
\bibitem{zeilinger} A. Zeilinger, Rev. Mod. Phys. \textbf{71}, s288 (1999); Y.H. Shih, Rep. on Progress in Physics, Vol. 66, No. 6, (2003).
\bibitem{pittman}T.B. Pittman, D.V. Strekalov, A. Migdall, M.H. Rubin, A.V. Sergienko, and Y.H. Shih, Phys. Rev. Lett. \textbf{77}, 1917 (1996).
\bibitem{yoonho} Y.H. Kim, M.V. Chekhova, S.P. Kulik, and Y. Shih, Phys. Rev. A \textbf{60}, R37 (1999).
\bibitem{mandel} L. Mandel, Rev. Mod. Phys. \textbf{71}, s274 (1999).
\bibitem{dowlingmetrolo} H. Lee, P. Kok and J.P. Dowling, quant-ph/0306113 (2003).
\bibitem{zeilingerfour} P. Walther, J-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni and A. Zeilinger, Nature \textbf{429}, 158 (2004).
\bibitem{dowling} A.N. Boto et al., Phys. Rev. Lett. \textbf{85}, 2733 (2000).
\bibitem{scully} M.O. Scully, in \textit{Proceedings of the Conference on Effects of Atomic Coherence and Interference in Quantum Optics, Crested Butte, Colorado, 1993} (IOP, Bristol 1994); U. Rathe and M.O. Scully, Lett.Math. Phys. \textbf{34}, 297 (1995).
\bibitem{milena} M. D'Angelo, M.V. Chekhova, and Y. Shih, Phys. Rev. Lett. \textbf{87}, 013602 (2001).
\bibitem{thermaltheory} Gatti et al. first proposed to use thermal radiation to simulate entangled sources for imaging purposes (quant-ph/0307187). At the same time we obtained the final results of this experiment, the possibility of obtaining subwavelength interference with thermal sources was theoretically proposed by K. Wang and D. Cao (quant-ph/0404078) and A. Gatti, E. Brambilla, M. Bache, and L.A. Lugiato, (quant-ph/0405056).
\bibitem{haner} A.B. Haner and N.R. Isenor, Am. J. of Phys. \textbf{38}, 748 (1970).
\bibitem{glauber} R.J. Glauber, Phys. Rev. {\bf 130}, 2529 (1963); {\bf 131}, 2766 (1963).
\bibitem{martienssen} W. Martienssen and E. Spiller, Am. J. Phys. \textbf{32}, 919 (1964).
\bibitem{arecchibook} F.T. Arecchi in {\em Proceedings of the international school of physics Enrico Fermi, Course XLII} edited by R.J. Glauber (Academic Press, 1969).
\bibitem{cohen}C. Cohen-Tannoudji, B. Diu, F. Laloe {\em Quantum Mechanics} {Hermann, Paris, France 1977}.
\bibitem{arecchi} F.T. Arecchi, E. Gatti, and A. Sona, Phys. Lett. \textbf{20} 27 (1966).
\end{references}
\begin{figure}
\caption{Sketch of the experimental setup. An attenuated He-Ne laser beam illuminates a double slit, $10cm$ after the slit there is a converging lens ($f=25mm$) and a the rotating ground glass disk is placed at $33.5mm$ from the lens. Basically the double slit is imaged onto the ground glass producing an effective double slit illumination ($a=0.043mm$ and $d=0.135mm$). The radiation scattered by the ground glass is then divided by a beam splitter and sent to two horizontally displaceable fibers, connected to single photon counting modules.}
\label{setup}
\end{figure}
\begin{figure}
\caption{(a) Normalized second order interference diffraction pattern vs position of the detectors. The dots are the experimental data while the solid line is a theoretical fit from Eq.~\ref{resultfar}. The actual counting rate corresponds to about $1000$ coincidence counts per second in the peak. The single counts are about $45000$ per second in $D_{1}$ and $25000$ per second in $D_{2}$.(b) Equivalent first-order interference diffraction pattern.}
\label{coin}
\end{figure}
\begin{figure}
\caption{Single detector counts vs positions of the detector $D_{1}$ (filled circles) and $D_{2}$ (hollow circles). The low level counting rate shows that the experiment was performed in the two-photon regime. The flatness of the graphs shows the absence of first order interference.}
\label{single}
\end{figure}
\begin{figure}
\caption{Histogram of number of joint detection counts vs time difference of the two photo-electron events. The size of each channel is $0.3$ ns. The graph is useful to ``calibrate" the coincidence time window: in all the measurements of this experiment, the coincidence were counted in a time window of about $600$ ns around the peak of the figure while the noise background was verified by shifting the coincidence window of $4000$ ns towards the region of the accidental coincidences.}
\label{mca}
\end{figure}
\centerline{\epsfxsize=2in \epsffile{Fig1.eps}}
Figure \ref{setup}. Giuliano Scarcelli, Alejandra Valencia, and Yanhua Shih.
\centerline{\epsfxsize=2.5in \epsffile{Fig2.eps}}
Figure \ref{coin}. Giuliano Scarcelli, Alejandra Valencia, and Yanhua Shih.
\centerline{\epsfxsize=2.5in \epsffile{Fig3.eps}}
Figure \ref{single}. Giuliano Scarcelli, Alejandra Valencia, and Yanhua Shih.
\centerline{\epsfxsize=2.5in \epsffile{Fig4.eps}}
Figure \ref{mca}. Giuliano Scarcelli, Alejandra Valencia, and Yanhua Shih.
\end{document} | arXiv |
I-spline
In the mathematical subfield of numerical analysis, an I-spline[1][2] is a monotone spline function.
Definition
A family of I-spline functions of degree k with n free parameters is defined in terms of the M-splines Mi(x|k, t)
$I_{i}(x|k,t)=\int _{L}^{x}M_{i}(u|k,t)du,$
where L is the lower limit of the domain of the splines.
Since M-splines are non-negative, I-splines are monotonically non-decreasing.
Computation
Let j be the index such that tj ≤ x < tj+1. Then Ii(x|k, t) is zero if i > j, and equals one if j − k + 1 > i. Otherwise,
$I_{i}(x|k,t)=\sum _{m=i}^{j}(t_{m+k+1}-t_{m})M_{m}(x|k+1,t)/(k+1).$
Applications
I-splines can be used as basis splines for regression analysis and data transformation when monotonicity is desired (constraining the regression coefficients to be non-negative for a non-decreasing fit, and non-positive for a non-increasing fit).
References
1. Curry, H.B.; Schoenberg, I.J. (1966). "On Polya frequency functions. IV. The fundamental spline functions and their limits". Journal d'Analyse Mathématique. 17: 71–107. doi:10.1007/BF02788653.
2. Ramsay, J.O. (1988). "Monotone Regression Splines in Action". Statistical Science. 3 (4): 425–441. doi:10.1214/ss/1177012761. JSTOR 2245395.
| Wikipedia |
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Suppose $M$ is a smooth manifold of dimension $n$ and $\omega$ is a smooth $k$-form on $M$. I am trying to show that $\omega$ is exact if and only if $d\omega=0$.
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This is where I am stuck. How can I show that $df \wedge \omega=0$? I know that we can show this by first showing that $df \wedge \omega$ is exact and then that $df \wedge \omega=0$, but I can't figure out how to do this.
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But when $U$ is
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\begin{document}
\begin{abstract} We show how to reconstruct a finite directed graph $E$ from its Toeplitz algebra, its gauge action, and the canonical finite-dimensional abelian subalgebra generated by the vertex projections. We also show that if $E$ has no sinks, then we can recover $E$ from its Toeplitz algebra and the generalised gauge action that has, for each vertex, an independent copy of the circle acting on the generators corresponding to edges emanating from that vertex. We show by example that it is not possible to recover $E$ from its Toeplitz algebra and gauge action alone. \end{abstract}
\maketitle
\section{Introduction}
In recent years, there has been an enormous amount of work, led by Eilers and his collaborators ({see,} for example, \cite{EilersEtAl:OAD2013, EilersEtAl:xx16, EilersEtAl:MA2017, EilersEtAl:CJM2018, EilersEtAl:MJM2012, EilersTomforde:MA10, Sorensen:ETDS2013}) on determining which moves on finite directed graphs generate the equivalence relations determined by various types of isomorphism of the associated $C^*$-algebras. One spectacular example of this is \cite[Theorem~3.1]{EilersEtAl:xx16}: if $E$ and $F$ are graphs with finitely many vertices, then the graph $C^*$-algebras $C^*(E)$ and $C^*(F)$ are stably isomorphic if and only if $E$ can be transformed into $F$ using a finite sequence of in-splittings, out-splittings, reductions, additions of sinks, Cuntz splices, Pulelehua moves, and the inverses of these moves.
By contrast, relatively little attention has been paid to the Toeplitz algebras of directed graphs, until the recent interest in KMS-theory (see, for example, \cite{CarlsenLarsen:JFA2016, aHLRS3, KajiwaraWatatani:KJM2013, LacaNeshveyev:JFA04, Thomsen:AM2017}) brought them to the fore. It has been known for some time \cite{KatsoulisKribs:MA04, Solel:JAMS04} that the \emph{non-selfadjoint} Toeplitz algebra (also called the tensor algebra or the quiver operator algebra) of a directed graph $E$ contains all of the information about $E$---if $E$ and $F$ are directed graphs with isomorphic tensor algebras, then they are themselves isomorphic. But there are no results in this direction for the Toeplitz $C^*$-algebras of directed graphs.
Here we consider the extent to which a finite directed graph can be recovered from its Toeplitz algebra and gauge action. We show that at least one additional piece of information is needed (see Examples \ref{ex:not iso}~and~\ref{eg:sinks}) and identify two pieces of information, either of which suffices for finite graphs with no sinks. Our key tool is the KMS structure of $\mathcal{T} C^*(E)$ for the dynamics arising from its gauge action; we show that using this we can recover the rank-one projections in $\mathcal{T} C^*(E)$ that correspond to the vertices of $E$. From this, using the spectral subspaces of the gauge action, it is straightforward to count the number of edges (indeed, the number of paths of length $n$ for any $n$) emanating from a given vertex. However, additional information is required to determine which of these paths have the same ranges. We show that the subalgebra $M_E = \lsp\{q_v : v \in E^0\} \subseteq \mathcal{T} C^*(E)$ generated by the vertex projections is sufficient to recover this information, and that if $E$ has no sinks then the action $\kappa^E$ of the torus $\mathbb{T}^{E^0}$ such that $\kappa^{{E}}_z(t_e) = z_{s(e)} t_e$ for each $e \in E^1$ also suffices.
\textbf{Conventions.} We use the conventions of \cite{Raeburn} for graphs and their $C^*$-algebras; so the Toeplitz algebra of a directed graph $E$ is the universal $C^*$-algebra generated by projections $\{q_v : v \in E^0\}$ and partial isometries $\{t_e : e \in E^1\}$ such that $t_e^* t_e = q_{s(e)}$ and $q_v \ge \sum_{r(e) = v} t_e t^*_e$. We use the notational convention in which, for example, $vE^1 = r^{-1}(v)$ and $E^1 v = s^{-1}(v)$.
\section{An example}
We started this project by asking whether it is possible to recover a directed graph $E$ from its Toeplitz algebra and gauge action. The following example shows that the answer is no, even for the particularly well-behaved class of strongly connected finite graphs in which every cycle has an entrance. We thank S{\o}ren Eilers for very helpful conversations that led to the construction of this example. For a simpler example involving graphs that are not strongly connected, and have sinks and sources, see Example~\ref{eg:sinks}.
\begin{exm}\label{ex:not iso} Consider the following directed graphs $E$ and $F$, that differ only in the range of the edge $e$: \[ \tikzset{->-/.default=0.5, ->-/.style={decoration={
markings,
mark=at position #1 with {\arrow{stealth}}},postaction={decorate}}} \vbox to 5cm{\vskip-1.6cm\begin{tikzpicture}
\node at (-2.2, 2) {$E$};
\node[circle, inner sep=1pt, fill=black] (a) at (0,0) {};
\node[circle, inner sep=0.2pt] (b) at (-1,2) {$u$};
\node[circle, inner sep=0.2pt] (c) at (1,2) {$v$};
\node[circle, inner sep=1pt, fill=black] (d) at (-2,4) {};
\node[circle, inner sep=1pt, fill=black] (e) at (0,4) {};
\node[circle, inner sep=1pt, fill=black] (f) at (2,4) {};
\draw[->-] (b)--(a) node [pos=0.5, anchor=south west, inner sep=1pt] {$f$};
\draw[->-] (c)--(a) node [pos=0.5, anchor=south east, inner sep=1pt] {$g$};
\draw[->-] (d)--(b);
\draw[->-] (e) to (b);
\draw[->-, in=33, out= 273] (e) to node [pos=0.5, anchor=north west, inner sep=1pt] {$e$} (b);
\draw[->-] (f)--(c);
\draw[->-, in=135, out=135] (a) to (d);
\draw[->-] (a) .. controls +(4, 1) and +(4,2.5) .. (e);
\draw[->-, in=45, out=45] (a) to (f); \end{tikzpicture} \begin{tikzpicture}
\node at (-2.2, 2) {$F$};
\node[circle, inner sep=1pt, fill=black] (a) at (0,0) {};
\node[circle, inner sep=0.2pt] (b) at (-1,2) {$u$};
\node[circle, inner sep=0.2pt] (c) at (1,2) {$v$};
\node[circle, inner sep=1pt, fill=black] (d) at (-2,4) {};
\node[circle, inner sep=1pt, fill=black] (e) at (0,4) {};
\node[circle, inner sep=1pt, fill=black] (f) at (2,4) {};
\draw[->-] (b)--(a) node [pos=0.5, anchor=south west, inner sep=1pt] {$f$};
\draw[->-] (c)--(a) node [pos=0.5, anchor=south east, inner sep=1pt] {$g$};
\draw[->-] (d)--(b);
\draw[->-] (e) to (b);
\draw[->-] (e) to node [pos=0.5, anchor=north east, inner sep=1pt] {$e$} (c);
\draw[->-] (f)--(c);
\draw[->-, in=135, out=135] (a) to (d);
\draw[->-] (a) .. controls +(4, 1) and +(4,2.5) .. (e);
\draw[->-, in=45, out=45] (a) to (f); \end{tikzpicture}} \] Let $(t, q)$ be the universal generating Toeplitz--Cuntz--Krieger $F$-family in $\mathcal{T} C^*(F)$. Define elements $\{Q_w : w \in E^0\}$ and $\{T_h : h \in E^1\}$ in $\mathcal{T} C^*(F)$ as follows: \begin{gather*} Q_u = q_u + t_e t^*_e,\quad Q_v = q_v - t_et_e^*,\quad T_f = t_f + t_g t_e t^*_e,\quad T_g = t_g (q_v - t_et_e^*),\quad\text{and}\\ Q_w = q_w \text{ for $w \in E^0 \setminus \{u,v\}$}\quad\text{ and }\quad T_h = t_h \text{ for $h \in E^1 \setminus \{f, g\}$.} \end{gather*} It is routine to check that $(Q, T)$ is a Toeplitz--Cuntz--Krieger $E$-family that generates $\mathcal{T} C^*(F)$, and that the elements $Q_w - \sum_{h \in wE^1} T_h T^*_h$ are all nonzero. So the universal property of $\mathcal{T} C^*(E)$ yields a surjective homomorphism $\pi_{Q,T} : \mathcal{T} C^*(E) \to \mathcal{T} C^*(F)$ such that $\pi_{Q,T}(q_w) = Q_w$ and $\pi_{Q,T}(t_h) = T_h$, and \cite[Theorem~4.1]{FR} implies that $\pi_{Q,T}$ is injective. It is immediate from the definitions of the $T_h$ and $Q_w$ that $\pi_{Q,T}$ is gauge-equivariant. So $(\mathcal{T} C^*(E), \gamma^E) \cong (\mathcal{T} C^*(F), \gamma^F)$, but there is no graph-isomorphism from $E$ to $F$ because, for example, $E$ has a pair of parallel edges, whereas $F$ does not.
In fact, since the canonical diagonals $D_E = \clsp\{t_\mu t^*_\mu : \mu \in E^*\}$ and $D_F = \clsp\{t_\mu t^*_\mu : \mu \in F^*\}$ are maximal abelian in $\mathcal{T} C^*(E)$ and $\mathcal{T} C^*(F)$, we see that $\pi_{Q,T}(D_E)$ is a maximal abelian subalgebra of $\mathcal{T} C^*(F)$. Since this maximal abelian subalgebra is contained in the maximal abelian subalgebra $D_F$ of $\mathcal{T} C^*(F)$, we deduce that $\pi_{Q,T}(D_E) = D_F$. So the triples $(\mathcal{T} C^*(E), \gamma^E, D_E)$ and $(\mathcal{T} C^*(F), \gamma^F, D_F)$ are isomorphic even though $E$ and $F$ are not. \end{exm}
\section{The main theorem}
Example~\ref{ex:not iso} shows that recovering a directed graph from its Toeplitz algebra requires more information than just the gauge action. Our main result identifies two additional bits of data, either one of which bridges the gap. The first one is the C*-subalgebra generated by the vertex projections inside the Toeplitz algebra. The second one is a higher dimensional generalisation of the gauge action.
\begin{dfn}\label{dfn:kappa} When $E$ is a directed graph, the {\em generalised gauge action} on $\mathcal{T} C^*(E)$ is the action $\kappa^E : \mathbb{T}^{E^0} \to \operatorname{Aut} \mathcal{T} C^*(E)$ determined by $\kappa^E_z(t_e) = z_{s(e)} t_e$ for all $e \in E^1$ and $z\in \mathbb{T}^{E^0}$. When $E$ and $F$ are two directed graphs, we say that an isomorphism $\rho : \mathcal{T} C^*(E) \to \mathcal{T} C^*(F)$ {\em intertwines the generalised gauge actions} $\kappa^E$ and $\kappa^F$ if there is a bijection $\varphi : E^0 \to F^0$ such that the induced homomorphism $\varphi^*: \mathbb{T}^{E^0} \to \mathbb{T}^{F^0}$ satisfies $\rho \circ \kappa^E_z = \kappa^F_{\varphi^*(z)} \circ \rho$ for all $z \in \mathbb{T}^{E^0}$. \end{dfn}
\begin{thm}\label{thm:main} Let $E$ and $F$ be finite directed graphs. As before, let $\gamma^E$ be the gauge action of $\mathbb{T}$ on $\mathcal{T} C^*(E)$, and let $M_E := \lsp\{q_v : v \in E^0\} \subseteq \mathcal{T} C^*(E)$. Let $\kappa^E $ be the generalised gauge action of $ \mathbb{T}^{E^0}$ on $ \mathcal{T} C^*(E)$ given by $\kappa^E_z(t_e) = z_{s(e)} t_e$ for all $e \in E^1$. Denote by $\gamma^F$, $M_F$, and $\kappa^{{F}}$ the corresponding concepts for $\mathcal{T} C^*(F)$. \begin{enumerate}
\item\label{it:M} There is an isomorphism $\mathcal{T} C^*(E) \cong \mathcal{T} C^*(F)$ that intertwines
$\gamma^E$ and $\gamma^F$ and carries $M_E$ to $M_F$ if and only if $E \cong F$.
\item\label{it:kappa} Suppose that $E$ and $F$ have no sinks. Then there is an isomorphism
$\mathcal{T} C^*(E) \cong \mathcal{T} C^*(F)$ that intertwines the generalised gauge actions $\kappa^E$ and
$\kappa^F$ if and only if $E \cong F$. \end{enumerate} \end{thm}
\begin{rmk} In both parts of Theorem~\ref{thm:main}, the additional data required beyond the gauge actions includes the number of vertices in the graphs. We point out, however, that this number is already available as an isomorphism invariant of the $C^*$-algebra $\mathcal{T} C^*(E)$ alone: by
\cite[Theorem~4.1]{FR} combined with \cite[Theorem~4.4]{Pimsner}, the Toeplitz algebra $\mathcal{T} C^*(E)$ is $KK$-equivalent to $\mathbb{C}^{E^0}$, and in particular $K_0(\mathcal{T} C^*(E)) \cong \mathbb{Z} E^0$. So if $\mathcal{T} C^*(E) \cong \mathcal{T} C^*(F)$, we already know that $|E^0| = |F^0|$. \end{rmk}
The proof of the ``if" implication is easy in both cases. If $\varphi^0 : E^0 \to F^0$ and $\varphi^1 : E^1 \to F^1$ constitute an isomorphism of graphs, then the isomorphism $\rho : \mathcal{T} C^*(E) \to \mathcal{T} C^*(F)$ given by $\rho(t_e) = t_{\varphi^1(e)}$ and $\rho(q_v) = q_{\varphi^0(v)}$ carries $M^E$ to $M^F$
and intertwines $\kappa^E$ and $\kappa^F$ (and, by restriction, $\gamma^E$ and $\gamma^F$), via the isomorphism $\mathbb{T}^{E^0} \cong \mathbb{T}^{F^0}$ induced by $\varphi^0$.
To prove the reverse implications we shall use the results of \cite{aHLRS, aHLRS3} on the KMS structure of the Toeplitz algebra $\mathcal{T} C^*(E)$ for the dynamics $\alpha:\mathbb{R} \to \operatorname{Aut}(\mathcal{T} C^*(E))$, where $\alpha_t = \gamma^E_{e^{it}}$ is the lift of the gauge action; that is \begin{equation}\label{eq:alpha def}
\alpha_t(q_v) = q_v\quad\text{ and }\quad \alpha_t(t_e) = e^{it} t_e\quad\text{ for all }v \in E^0, e \in E^1,\text{ and }t \in \mathbb{R}. \end{equation} We write \[ \operatorname{Ext}_\beta(\alpha) := \{\phi : \phi\text{ is an extremal KMS$_\beta$ state of $(\mathcal{T} C^*(E), \alpha)$}\}. \]
We first need to be able to recognise, using the data $(\mathcal{T} C^*(E), \alpha)$, when a real number $\beta$ is strictly greater than the natural logarithm of the spectral radius of the adjacency matrix $A_E$ of the directed graph $E$. For this, as in \cite{aHLRS3}, we write $\sim$ for the equivalence relation on $E^0$ given by $v \sim w$ if both $v E^* w \not= \emptyset$ and $w E^* v \not= \emptyset$. We call the equivalence classes $C \in E^0/{\sim}$ the \emph{strongly connected components} of $E$. For $C \in E^0/{\sim}$, we write $A_C$ for the $C \times C$ submatrix of $A_E$, which is the adjacency matrix of the subgraph of $E$ with vertices $C$ and edges $C E^1 C$.
\begin{lem}\label{lem:critical beta} Let $E$ be a finite directed graph. If $\beta < \log\rho(A_E) < \beta'$, then
$|\operatorname{Ext}_\beta(\alpha)| < |\operatorname{Ext}_{\beta'}(\alpha)|$. \end{lem} \begin{proof} If $E$ has no cycles, then \cite[Lemma~A.1(b)]{aHLRS} shows that $\log \rho(A_E) = -\infty$, and so the result is vacuous. So suppose that $E$ has at least one cycle. Then $\rho(A_E) = \max\{\rho(A_C) : C\text{ is a nontrivial strongly connected component of $E$}\}$, as discussed at the beginning of \cite[Section~4]{aHLRS3}. Let $H_\beta := \{s(\mu) : \mu \in E^*\text{ and }r(\mu) \in \bigcup_{\log\rho(A_C) > \beta} C\}$. Theorem~3.1 of \cite{aHLRS} shows that
$|\operatorname{Ext}_{\beta'}(\alpha)| = |E^0|$, and Theorem~5.3 of \cite{aHLRS3} implies that
$|\operatorname{Ext}_{\beta}(\alpha)| \le |E^0 \setminus H_\beta|$. Since $\beta < \log\rho(A_E) =
\max\{\log\rho(A_C) : C \in E^0/{\sim}\}$, we have $H_\beta \not= \emptyset$. Hence $|E^0
\setminus H_\beta| < |E^0|$, which proves the result. \end{proof}
{\begin{lem}\label{cor:beta property} The interior in $\mathbb{R}$ of the set \begin{equation}\label{eq:half-line}
\big\{\beta \in (0,\infty) : |\operatorname{Ext}_{\beta'}(\alpha)| = |\operatorname{Ext}_{\beta}(\alpha)|\text{ for all }\beta' \ge \beta\big\} \end{equation} is the open half-line $\big(\log\rho(A_E), \infty\big)$. \end{lem}} \begin{proof} Theorem~3.1 of \cite{aHLRS} shows that if $\beta > \log\rho(A_E)$, then we have
$|\operatorname{Ext}_{\beta'}(\alpha)| = |\operatorname{Ext}_{\beta}(\alpha)|\text{ for all }\beta' \ge \beta$, and Lemma~\ref{lem:critical beta} shows that if $\beta < \log\rho(A_E)$, then $|\operatorname{Ext}_{\beta'}(\alpha)|
> |\operatorname{Ext}_{\beta}(\alpha)|\text{ for some }\beta' > \beta$. \end{proof}
{Throughout the rest of this note, we shall let $\pi : \mathcal{T} C^*(E) \to \mathcal{B}(\ell^2(E^*))$ be the canonical (faithful) path-space representation of $\mathcal{T} C^*(E)$. We will need to show that the minimal projections in $\mathcal{T} C^*(E)$ corresponding to vertices of $E$ can be recovered using the gauge action $\gamma^E$. For each $\mu \in E^*$, we define \begin{equation}\label{oldnotation4}\textstyle \Delta_\mu := t_\mu \big(q_{s(\mu)} - \sum_{e \in s(\mu) E^1} t_e t^*_e\big) t^*_\mu \in \mathcal{T} C^*(E). \end{equation} The $\Delta_\mu$ are minimal projections in the canonical copy of $\bigoplus_{v \in E^0} \mathcal{K}(\ell^2(E^* v))$ in $\mathcal{T} C^*(E)$; indeed, each $\pi(\Delta_\mu)$ is the rank-1 projection $\theta_{\delta_\mu, \delta_\mu}$ onto the span of the basis vector $\delta_\mu \in \ell^2(E^*)$. }
\begin{lem}\label{lem:get vertices} Let $E$ be a finite directed graph. Let $\alpha$ be the dynamics~\eqref{eq:alpha def}. Let $\beta$ be any real number greater than $\max\{0, \log\rho(A_E)\}$. Let $\phi$ be an extremal KMS$_\beta$ state of $(\mathcal{T} C^*(E), \alpha)$. Let $P_{\min}$ denote the collection of minimal projections on $\mathcal{T} C^*(E)$. There is a unique $p_\phi \in P_{\min}$ such that $\phi(p_\phi) = \max\{\phi(q) : q \in P_{\min}\}$. Moreover, with $\Delta_v$ as in \eqref{oldnotation4}, we have $p_\phi = \Delta_{v_\phi}$ for some $v_\phi \in E^0$. \end{lem} \begin{proof} For each $v\in E^0$, let $\varepsilon^v_{(\cdot)}$ denote the measure $\big(\sum_{\mu \in E^* v}
e^{-\beta|\mu|}\big)^{-1} \delta_v {(\cdot)}$ on $E^0$. Since $\beta > \log\rho(A_E)$, \cite[Theorem~3.1]{aHLRS} implies that there is a unique $v_\phi \in E^0$ such that $\phi$ satisfies \[
\phi(t_\mu t^*_\nu) = \delta_{\mu,\nu} e^{-\beta|\mu|} \varepsilon^{v_\phi}_{s(\mu)},\quad \text{for all $\mu,\nu\in E^*$}. \] By the proof of \cite[Theorem~3.1(b)]{aHLRS}, we know that $\phi$ satisfies \[ \phi(a)
= \sum_{\mu \in E^*v_\phi} e^{-\beta|\mu|} \big(\pi(a) \delta_\mu | \delta_\mu\big) \varepsilon^{v_\phi}_{s(\mu)}\quad \text{for all $a\in \mathcal{T} C^*(E)$}. \] We have \[\textstyle
\phi(\Delta_{v_\phi}) = \varepsilon^{v_\phi}_{v_\phi} = \big(\sum_{\mu \in E^*v_\phi} e^{-\beta|\mu|}\big)^{-1}. \] Fix $q \in P_{\min} \setminus \Delta_{v_\phi}$. It suffices to show that $\phi(q) < \phi(\Delta_{v_\phi})$. Let $\pi_{v_\phi} : \mathcal{T} C^*(E) \to \mathcal{B}(\ell^2(E^* v_\phi))$ be the direct summand in $\pi$ corresponding to ${v_\phi}$. Then $\phi$ factors through $\pi_{v_\phi}$. If $\phi(q) = 0$ then we certainly have $\phi(q) < \phi(\Delta_{v_\phi})$, so suppose that $\phi(q) \not= 0$. Then $\pi_{v_\phi}(q) \not= 0$, and so $\pi_{v_\phi}(q)$ is a minimal projection in $\pi_{v_\phi}(\mathcal{T} C^*(E))$. Since $\pi_{v_\phi}(\mathcal{T} C^*(E))$ contains all of $\mathcal{K}(\ell^2(E^*{v_\phi}))$, it follows that $\pi(q)$ is the rank-one projection $\theta_{\xi,\xi }$projection corresponding to a unit vector $\xi \in \ell^2(E^*{v_\phi})$. Hence \begin{align*}
\phi(q) &= \sum_{\mu \in E^*{v_\phi}} e^{-\beta|\mu|} \big(\pi(q) \delta_\mu | \delta_\mu\big) \varepsilon^{v_\phi}_{s(\mu)}
= \sum_{\mu \in E^*{v_\phi}} e^{-\beta|\mu|} \big(\theta_{\xi,\xi}(\delta_\mu) | \delta_\mu\big) \varepsilon^{v_\phi}_{s(\mu)} \\
&= \sum_{\mu \in E^*{v_\phi}} e^{-\beta|\mu|} \big(\big(\xi \mid \delta_\mu\big)\xi | \delta_\mu\big) \varepsilon^{v_\phi}_{s(\mu)}.
\end{align*}
Since $\beta > 0$, we have $e^{-\beta |\mu |} =1$ when
$\mu = v_\phi$ and $e^{-\beta |\mu | } \leq e^{-\beta} $ when $\mu \neq v_\phi$, and so we deduce that
\[
\phi(q)
\le \Big(|\xi_{v_\phi}|^2 + e^{-\beta} \sum_{\mu \not= {v_\phi}} |\xi_\mu|^2\Big) \varepsilon^{v_\phi}_{s(\mu)}.
\]
Since $q \not= \Delta_{v_\phi}$, we have $\xi \not= \delta_{v_\phi}$, and so $|\xi_{v_\phi}| < 1$. We have $\sum |\xi_\mu|^2 = \|\xi\|^2 = 1$, and so $e^{-\beta} \sum_{\mu \not= v_\phi} |\xi_\mu|^2
= e^{-\beta} (1 - |\xi_{v_\phi}|^2) < 1 - |\xi_{v_\phi}|^2$. Hence $\phi(q) < \varepsilon^v_{s(\mu)} = \phi(\Delta_{v_\phi})$ as claimed. \end{proof}
Lemma~\ref{lem:get vertices} allows us to recover the projections $\Delta_v$ of $\mathcal{T} C^*(E)$ from $\mathcal{T} C^*(E)$ together with its simplex of KMS states. Since the KMS states are intrinsic to the pair $(\mathcal{T} C^*(E), \gamma^E)$, it follows that we can recover the $\Delta_v$ from the Toeplitz algebra and its gauge action. We show next how to recover the cardinalities of the sets $E^n v$ as well. We start with some notation.
\begin{ntn} For each $\mu,\nu \in E^*$ with $s(\mu)=s(\nu)$ we define $\Theta_{\mu,\nu} := t_\mu \Delta_{s(\mu)} t^*_\nu$. Recall that the path-space representation $\pi$ carries each $\Theta_{\mu,\nu}$ to the canonical matrix unit $\theta_{\delta_\mu, \delta_\nu}$. Recall also that for $n \in \mathbb{Z}$, the \emph{$n$\textsuperscript{th} spectral subspace} $\mathcal{T} C^*(E)_n$ of $\mathcal{T} C^*(E)$ with respect to $\gamma$ is \[
\mathcal{T} C^*(E)_n := \{a \in \mathcal{T} C^*(E) : \gamma^E_z(a) = z^n a\text{ for all }z \in \mathbb{T}\}. \] \end{ntn}
\begin{lem}\label{lem:get edges} Let $E$ be a finite directed graph. For $n \ge 0$, we have $\mathcal{T} C^*(E)_n \Delta_v =
\lsp\{\Theta_{\mu, v} : \mu \in E^n v\}$; in particular, $|E^n v| = \dim(\mathcal{T} C^*(E)_n \Delta_v)$. \end{lem} \begin{proof}
It is standard that $\mathcal{T} C^*(E)_n = \clsp\{t_\mu t^*_\nu : \mu,\nu \in E^*, |\mu| - |\nu| = n, s(\mu) = s(\nu)\}$. The path-space representation $\pi$ carries $\Delta_v$ to $\theta_{\delta_v, \delta_v}$, and carries each $t_\mu t^*_\nu$ to the strong-operator sum $\sum_{\lambda \in s(\nu)E^*} \theta_{\delta_{\mu\lambda},\delta_{\nu\lambda}}$. The latter is nonzero at $\delta_v$ only if $v = \nu\lambda$ for some $\lambda \in s(\mu)E^*$, which forces $\nu = v = \lambda = s(\mu)$. So if $a \in \mathcal{T} C^*(E)_n$ and $a \Delta_v \not= 0$, then $a \Delta_v \in \lsp\{t_\mu \Delta_v : \mu \in E^n v\} = \lsp\{\Theta_{\mu,v} : \mu \in E^n v\}$. Since each $\Theta_{\mu, v} = \Theta_{\mu, v} \Delta_v$, the reverse containment is clear. \end{proof}
We can now prove the first part of the main theorem.
\begin{proof}[Proof of Theorem~\ref{thm:main}(\ref{it:M})] {By Lemma~\ref{cor:beta property} we may determine the value of $\log\rho(A_E)$ from the KMS state structure of $\alpha$,} and then choose $\beta > \log\rho(A_E)$. For $\phi \in \operatorname{Ext}_\beta(\alpha)$, Lemma~\ref{lem:get vertices} yields a unique minimal projection $p_\phi$ of $\mathcal{T} C^*(E)$ such that $\phi(p_\phi) = \max\{\phi(q) : q\text{ is a minimal projection of }\mathcal{T} C^*(E)\}$, and we have $p_\phi = \Delta_{v_\phi}$ for some $v_\phi \in E^0$. We have $q_{v_\phi} \ge \Delta_{v_\phi}$, and then for $w \not= v_\phi$ in $E^0$ we have $q_w \Delta_{v_\phi} = q_w q_{v_\phi} \Delta_{v_\phi} = 0$. So there is a unique minimal projection $P_\phi \in M_E$ that dominates $p_\phi$, namely $P_\phi = q_{v_\phi}$.
For $\phi,\psi \in \operatorname{Ext}_\beta(\alpha)$, let \[ N(\phi,\psi) := \dim P_\phi \mathcal{T} C^*(E)_1 p_\psi. \]
Let $\widetilde{E}$ be the directed graph with vertices $\operatorname{Ext}_\beta(\alpha)$ and with $|\phi
\widetilde{E}^1\psi| = N(\phi,\psi)$ for all $\phi,\psi \in \operatorname{Ext}_\beta(\alpha)$. By construction, the graph $\widetilde{E}$ is an isomorphism invariant of the triple $(\mathcal{T} C^*(E), \gamma^E, M_E)$. We claim that $\widetilde{E} \cong E$.
We know from Lemma~\ref{lem:get vertices} that $\phi \mapsto v_\phi$ from $\widetilde{E}^0$ to
$E^0$ is a bijection, so it suffices to show that $N(\phi,\psi) = |v_\phi E^1 v_\psi|$ for all $\phi,\psi$. Lemma~\ref{lem:get edges} shows that $\mathcal{T} C^*(E)_1 p_\psi = \lsp\{\Theta_{e,v_\psi} : e \in E^1 v_\psi\}$. Since for each $e \in E^1 v_\psi$ we have $\Theta_{e,v_\psi} \Theta_{e,v_\psi}^* = \Theta_{e,e} \le q_{r(e)}$, it follows that $P_\phi \mathcal{T} C^*(E)_1 p_\psi = \lsp\{\Theta_{e,v_\psi} : e \in v_\phi E^1 v_\psi\}$. Hence \[
|v_\phi E^1 v_\psi| = \dim P_\phi \mathcal{T} C^*(E)_1 p_\psi = N(\phi,\psi). \]
So $\widetilde{E} \cong E$, as claimed. Applying the process of the preceding three paragraphs to the system $(\mathcal{T} C^*(F), \gamma^F, M_F)$ we obtain a graph $\widetilde{F} \cong F$. Since the systems $(\mathcal{T} C^*(E), \gamma^E, M_E)$ and $(\mathcal{T} C^*(F), \gamma^F, M_F)$ are isomorphic, we see that $\widetilde{E} \cong \widetilde{F}$, and therefore $E \cong F$. \end{proof}
To prove statement~(\ref{it:kappa}) of Theorem~\ref{thm:main} we first show how to determine which coordinate of the generalised gauge action $\kappa^E$ corresponds to the minimal projection $p_\phi$ obtained from $\phi \in \operatorname{Ext}_{\beta}(\alpha)$ as in Lemma~\ref{lem:get vertices}.
\begin{lem}\label{lem:identify vertices} Let $E$ be a finite directed graph with no sinks, and let $\kappa^E$ and $\alpha$ be as in Definition~\ref{dfn:kappa} and~\eqref{eq:alpha def}. Fix $\beta > \ln\rho(A_E)$ and let $\phi$ be an extremal KMS$_\beta$ state of $(\mathcal{T} C^*(E), \alpha)$. Let $p_\phi$ be the projection of Lemma~\ref{lem:get vertices}. Then the vertex $v_\phi$ such that $p_\phi = \Delta_{v_\phi}$ is the unique vertex such that $\kappa^E_z(a) = z_{v_\phi} a$ for all $a \in \mathcal{T} C^*(E)_1 p_\phi$ and $z\in \mathbb{T}^{E^0}$. \end{lem} \begin{proof} For $w \in E^0$, Lemma~\ref{lem:get edges} gives $\mathcal{T} C^*(E)_1\Delta_w = \lsp\{\Theta_{e, w} : e \in E^1 w\} = \lsp\{t_e \Delta_w : e \in E^1\}$, and so it follows from the definition of $\kappa^E$ that $\kappa^E_z(a) = z_w a$ for all $a \in \mathcal{T} C^*(E)_1\Delta_w$ and $z \in \mathbb{T}^{E^0}$. Since $E$ has no sinks, each $\lsp\{\Theta_{e, w} : e \in E^1 w\}$ is nontrivial, which proves uniqueness. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}(\ref{it:kappa})] First observe that the dynamics $\alpha$ of $\mathcal{T} C^*(E)$ defined in~\eqref{eq:alpha def} is determined by $\kappa^E$ via $\alpha_t = \kappa^E_{(e^{it}, \dots, e^{it})}$. Using Lemma~\ref{cor:beta property} as in the proof of Theorem~\ref{thm:main}(\ref{it:M}), fix $\beta > \ln\rho(A_E)$. For each extremal KMS$_\beta$ state $\phi \in \operatorname{Ext}_\beta(\alpha)$, Lemma~\ref{lem:get vertices} yields a unique minimal projection $p_\phi$ of $\mathcal{T} C^*(E)$ such that \[\phi(p_\phi) = \max\{\phi(q) : q\text{ is a minimal projection of } \mathcal{T} C^*(E)\}.\] Lemma~\ref{lem:identify vertices} shows that $p_\phi = \Delta_{v_\phi}$ where $v_\phi \in E^0$ is the unique vertex such that $\kappa^E_z(a) = z_{v_\phi} a$ for all $a \in \mathcal{T} C^*(E)_1 p_\phi$.
Suppose that $\phi, \psi \in \operatorname{Ext}_\beta(\alpha)$ are distinct. For $z \in \mathbb{T}$ let $\omega(\phi,\psi,z) \in \mathbb{T}^{E^0}$ be the element such that \[ \omega(\phi,\psi,z)_u = \begin{cases}
\overline{z} &\text{ if $u = v_\phi$}\\
z &\text{ if $u = v_\psi$}\\
1 &\text{ otherwise.}
\end{cases} \] Define an action $\gamma^{\phi,\psi} : \mathbb{T} \to \operatorname{Aut}(\mathcal{T} C^*(E))$ by $\gamma^{\phi,\psi}_z = \kappa^E_{\omega(\phi,\psi,z)}$. Note that this action fixes the partial isometry $t_{ef}$ associated to $ef \in E^2 v_\psi$ if and only if $r(f) = s(e) = v_\phi$. Combining the fixed point algebra $\mathcal{T} C^*(E)^{\gamma^{\phi,\psi}}$ of $\gamma^{\phi,\psi}$ with the second spectral subspace of the gauge action $\gamma^E$, we define \[ N(\phi,\psi) :=
\dim\big(\mathcal{T} C^*(E)^{\gamma^{\phi,\psi}} \cap \mathcal{T} C^*(E)_2 p_\psi\big) / \dim(\mathcal{T} C^*(E)_1 p_\phi). \] We extend the definition of $N$ to the case $\phi = \psi \in \operatorname{Ext}_\beta(\alpha)$ by setting \[
N(\psi, \psi) := \dim(\mathcal{T} C^*(E)_1 p_\psi) - \sum_{\phi \not= \psi} N(\phi,\psi). \]
We claim that $N(\phi,\psi)\in \mathbb{N}$ for all $\phi,\psi \in \operatorname{Ext}_\beta(\alpha)$, and that $E$ is isomorphic to the directed graph $\widetilde{E}$ with vertices $\widetilde{E}^0 :=
\operatorname{Ext}_\beta(\alpha)$, and such that $|\phi \widetilde{E}^1 \psi| = N(\phi, \psi)$ for all
$\phi,\psi \in \operatorname{Ext}_\beta(\alpha)$. Since we already have a bijection $\phi \mapsto v_\phi$ from $\widetilde{E}^0$ to $E^0$, to prove the claim, we just have to show that $N(\phi,\psi) = |v_\phi E^1 v_\psi|$ for all $\phi,\psi$.
For this, fix $\phi,\psi \in \operatorname{Ext}_\beta(\alpha)$ and let $ ef \in E^2$. Then \[ \gamma^{\phi,\psi}_z(t_{ef} p_\psi) = \begin{cases}
t_{ef} p_\psi &\text{ if $f \in v_\phi E^1 v_\psi$}\\
z^2 t_{ef} p_\psi &\text{ if $f \in v_\psi E^1 v_\psi$}\\
zt_{ef} p_\psi &\text{ if $f \in E^1v_\psi \setminus (v_\phi E^1 v_\psi \cup v_\psi E^1 v_\psi)$}\\
0 &\text{ if $f \not\in E^1 v_\psi$.}
\end{cases} \]
So Lemma~\ref{lem:get edges} implies that $\mathcal{T} C^*(E)^{\gamma^{\phi,\psi}} \cap \mathcal{T} C^*(E)_2 p_\psi = \lsp\{\Theta_{ef, v_\psi} : ef \in E^1 v_\phi E^1 v_\psi\}$. Hence, $|E^1 v_\phi| \cdot
|v_\phi E^1 v_\psi| = |E^1 v_\phi E^1 v_\psi| = \dim(\mathcal{T} C^*(E)^{\gamma^{\phi,\psi}} \cap \mathcal{T} C^*(E)_2 p_\psi)$. By Lemma~\ref{lem:get edges}, we have $|E^1 v_\phi| = \dim(\mathcal{T} C^*(E)_1 p_\phi)$. Since, by hypothesis, $E$ has no sinks, we have $|E^1 v_\phi| \not= 0$, and so we deduce that \[
|v_\phi E^1 v_\psi| = \dim\big(\mathcal{T} C^*(E)^{\gamma^{\phi,\psi}} \cap \mathcal{T} C^*(E)_2 p_\psi\big) / \dim(\mathcal{T} C^*(E)_1 p_\phi)
= N(\phi, \psi). \] Now for each $\psi \in \operatorname{Ext}_\beta(\alpha)$, we see that \begin{align*}
|v_\psi E^1 v_\psi| &= |E^1 v_\psi| - \sum_{\phi\not= \psi} |v_\phi E^1 v_\psi|\\
&= \dim(\mathcal{T} C^*(E)_1 p_\psi) - \sum_{\phi \not= \psi} \dim\big(\mathcal{T} C^*(E)^{\gamma^{\phi,\psi}} \cap \mathcal{T} C^*(E)_2 p_\psi\big) / \dim(\mathcal{T} C^*(E)_1 p_\phi)\\
&= N(\psi, \psi).
\end{align*} {This shows that $E\cong \widetilde E$ and concludes the proof of the claim.}
{ To finish the proof of the ``only if" assertion in Theorem~\ref{thm:main}(\ref{it:kappa}) assume now there exist an isomorphism $\rho: \mathcal{T} C^*(E) \to \mathcal{T} C^*(F)$ and a bijection $\varphi:E^0 \to F^0$ intertwining the generalised gauge actions $\kappa^E$ and $\kappa^F$. Then $\varphi^*: \mathbb{T}^{E^0} \to \mathbb{T}^{F^0}$ maps constant functions to constant functions, that is, $\varphi^*$ respects the diagonal embeddings of $\mathbb{T}$. Hence $\rho$ intertwines the gauge actions $\gamma^E$ and $\gamma^F$, and also the dynamics $\alpha^E$ and $\alpha^F$ obtained from them on setting $z = e^{it}$. Passing to extremal KMS$_\beta$ states, we get a bijection $\widetilde{E}^0 :=\operatorname{Ext}_\beta(\alpha^E) \cong \operatorname{Ext}_\beta(\alpha^F) =: \widetilde{F}^0$ in which $\phi \mapsto \phi' := \phi\circ \rho^{-1}$. The isomorphism $\rho$ also intertwines the action $\gamma^{\phi,\psi} : \mathbb{T} \to \operatorname{Aut}(\mathcal{T} C^*(E))$ with the action $\gamma^{\phi',\psi'} : \mathbb{T} \to \operatorname{Aut}(\mathcal{T} C^*(F))$ and thus $N(\phi,\psi) = N(\phi',\psi')$. Thus, much like in the final paragraph of the proof of the ``only if" assertion in Theorem~\ref{thm:main}(\ref{it:M}), we conclude that $\widetilde E \cong \widetilde F$ and hence that $E\cong F$.} \end{proof}
\begin{exm}\label{eg:sinks} As compared to statement~(\ref{it:M}), statement~(\ref{it:kappa}) of our main theorem has the additional hypothesis that $E$ and $F$ have no sinks. Here we present an example---first shown to the fourth author in the context of Cohn path algebras by Gene Abrams, and then independently by S{\o}ren Eilers---that shows that the additional hypothesis in statement~(\ref{it:kappa}) is necessary. Consider the graphs \[ \tikzset{->-/.default=0.5, ->-/.style={decoration={
markings,
mark=at position #1 with {\arrow{stealth}}},postaction={decorate}}} \begin{tikzpicture}[scale=2]
\node at (0,0) {$E$};
\node[circle, inner sep=1pt, fill=black] (u) at (0.5,0) {};
\node[circle, inner sep=1pt, fill=black] (v) at (0.5,0.3) {};
\node[circle, inner sep=1pt, fill=black] (w) at (1.5,0) {};
\node[inner sep=1pt, anchor=east] at (u.west) {$u$};
\node[inner sep=1pt, anchor=east] at (v.west) {$v$};
\node[inner sep=1pt, anchor=west] at (w.east) {$w$};
\draw[->-, out=210, in=330] (w) to node [pos=0.6, anchor=north, inner sep=1pt] {$e$} (u);
\draw[->-, out=150, in=30] (w) to node [pos=0.6, anchor=south, inner sep=1pt] {$f$} (u);
\node at (3,0) {$F$};
\node[circle, inner sep=1pt, fill=black] (u2) at (3.5,0) {};
\node[circle, inner sep=1pt, fill=black] (v2) at (3.5,0.3) {};
\node[circle, inner sep=1pt, fill=black] (w2) at (4.5,0) {};
\node[inner sep=1pt, anchor=east] at (u2.west) {$u$};
\node[inner sep=1pt, anchor=east] at (v2.west) {$v$};
\node[inner sep=1pt, anchor=west] at (w2.east) {$w$};
\draw[->-] (w2) to node [pos=0.6, anchor=south west, inner sep=1pt] {$e$} (v2);
\draw[->-] (w2) to node [pos=0.6, anchor=north, inner sep=1pt] {$f$} (u2); \end{tikzpicture} \] There is an isomorphism $\mathcal{T} C^*(E) \to \mathcal{T} C^*(F)$ that carries $q_v$ to $q_v - t_e t^*_e$, carries $q_u$ to $q_u + t_e t^*_e$ and takes each of the remaining generators of $\mathcal{T} C^*(E)$ to the generator of $\mathcal{T} C^*(F)$ with the same label. This isomorphism intertwines $\kappa^E$ and $\kappa^F$ because in both graphs every edge has source $w$. It does not, however, carry $M_E$ to $M_F$ since, for example, $q_v - t_e t^*_e \not\in M_F$. \end{exm}
\end{document} | arXiv |
Time series analysis of hemorrhagic fever with renal syndrome in mainland China by using an XGBoost forecasting model
Cai-Xia Lv1,
Shu-Yi An2,
Bao-Jun Qiao2 &
Wei Wu1
Hemorrhagic fever with renal syndrome (HFRS) is still attracting public attention because of its outbreak in various cities in China. Predicting future outbreaks or epidemics disease based on past incidence data can help health departments take targeted measures to prevent diseases in advance. In this study, we propose a multistep prediction strategy based on extreme gradient boosting (XGBoost) for HFRS as an extension of the one-step prediction model. Moreover, the fitting and prediction accuracy of the XGBoost model will be compared with the autoregressive integrated moving average (ARIMA) model by different evaluation indicators.
We collected HFRS incidence data from 2004 to 2018 of mainland China. The data from 2004 to 2017 were divided into training sets to establish the seasonal ARIMA model and XGBoost model, while the 2018 data were used to test the prediction performance. In the multistep XGBoost forecasting model, one-hot encoding was used to handle seasonal features. Furthermore, a series of evaluation indices were performed to evaluate the accuracy of the multistep forecast XGBoost model.
There were 200,237 HFRS cases in China from 2004 to 2018. A long-term downward trend and bimodal seasonality were identified in the original time series. According to the minimum corrected akaike information criterion (CAIC) value, the optimal ARIMA (3, 1, 0) × (1, 1, 0)12 model is selected. The index ME, RMSE, MAE, MPE, MAPE, and MASE indices of the XGBoost model were higher than those of the ARIMA model in the fitting part, whereas the RMSE of the XGBoost model was lower. The prediction performance evaluation indicators (MAE, MPE, MAPE, RMSE and MASE) of the one-step prediction and multistep prediction XGBoost model were all notably lower than those of the ARIMA model.
The multistep XGBoost prediction model showed a much better prediction accuracy and model stability than the multistep ARIMA prediction model. The XGBoost model performed better in predicting complicated and nonlinear data like HFRS. Additionally, Multistep prediction models are more practical than one-step prediction models in forecasting infectious diseases.
Hemorrhagic fever with renal syndrome (HFRS) is a zoonotic disease caused by hantaviruses that cause a high degree of harm to humans. To date, more than 28 hantaviruses resulting in human diseases have been identified worldwide. Most HFRS cases occur in Asian and European countries, such as China, South Korea and Russia. More than 100,000 cases of HFRS occur every year worldwide, and China accounts for more than 90 % of them [1, 2]. In recent years, the number of HFRS cases in mainland China has shown an overall downward trend [3], but it is still prevalent in some regions, such as Heilongjiang, Liaoning, Jilin, Shandong, Shanxi and Hebei provinces [4]. It should be pointed out that epidemic areas for rodent have a tendency to spread towards cities, as hantavirus is carried and spread by rodents. The main transmission routes from rodents to humans are aerosolized excreta inhalation and contact infection. Person-to-person spread may occur but is extremely rare [3,4,5]. The clinical symptoms of HFRS are mainly characterized by fever, hemorrhaging and kidney damage with a 4 to 46 day incubation period [5]. HFRS can lead to death if the patient is not treated in time. The Chinese Center for Disease Control (CDC) established a surveillance system for HFRS in 2004 and classified it as a class II infectious disease. The surveillance system requires newly confirmed cases of HFRS to be reported within 12 h, which ensures the accuracy and timeliness of the data [6]. Although the government and health departments have taken on many control measures, such as active rodents control, vaccination implementation, health education implementation, environmental management of the epidemic areas, and disease surveillance strengthening, HFRS still severely affects people's health with approximately 9,000–30,000 cases annually in China [7].
To delineate the changing trend in the incidence of infectious diseases, domestic and foreign researchers have applied various statistical and mathematical models to the prediction of infectious diseases, such as random forest [8], gradient boosting machine (GBM) [9] and support vector machine models [10]. At present, some models have been used in predicting HFRS, including neural networks [11] and generalized additive models (GAMs) [12]. Most of these methods are based on one-step forecasting. The autoregressive integrated moving average (ARIMA) model, as a fundamental method in time series analysis that regresses the lag value of the time series and random items to build a model, has been applied in many fields [13]. Although an ARIMA model can capture the linear characteristics of infectious disease series well, such as the autoregressive (AR) term and moving average(MA) term, some information may be lost when it analyzes the residuals consisting of non-linear information [14]. XGBoost is a boosting algorithm based on the evolution of gradient boosting decision tree (GBDT) algorithm, which has achieved remarkable results in practical applications due to its high accuracy, fast speed and unique information processing scheme. Compared with traditional statistical models, it has advantages in predicting nonlinear data [15,16,17,18,19]. Previous studies usually applied one-step predictive statistical models to characterize and predict epidemic trends in infectious diseases. Currently, a multistep XGBoost model has not been used to forecast infectious diseases such as HFRS.
In this study, we aim to develop a prediction model for HFRS in mainland China by using one-step and multistep XGBoost models and comparing them with an ARIMA model.
We collected HFRS incidence data from 2004 to 2018 from the official website of the National Health Commission of the People's Republic of China (http://www.nhc.gov.cn). Based on the requirements of China's Infectious Disease Control Law, hospital physicians must report every HFRS case within 12 h to the local health authority. Once the patient is diagnosed with a suspected case based on clinical symptoms, patient blood samples are collected and sent to local CDC laboratories for serological confirmation; if the result is positive, it is considered as a confirmed case. Local health authorities later report monthly HFRS cases to the national health department for surveillance purposes. However, the monitoring system relies on hospitals passively monitoring the occurrence of infectious diseases, and there will be a certain time delay in information collection. If the patient's symptoms are mild and not require hospitalization, underreporting may occur [20]. The dataset analyzed during the study is included in Supplementary Material 1. The HFRS data from 2004 to 2017 were adopted to establish the seasonal ARIMA model and XGBoost model, while the 2018 data were used for model verification.
ARIMA model
An ARIMA model is a time series forecasting method that was first proposed by Box and Jenkins in 1976 [21]. The principle of the ARIMA model is to adopt appropriate data conversion to transform nonstationary time series into stationary time series and then adjust the parameters to find the optimal model. Finally, the changes in past trends are quantitatively described and simulated to predict future outcomes [13, 22]. The specific procedures for establishing the seasonal ARIMA model were as follows: first, we performed a Box-Cox transformation to smooth the variance of the original HFRS time series. Simultaneously, long-term trends and seasonal differences were stabilized through first-order differences and seasonal differences. Then, we preliminarily judge the possible parameter values of the ARIMA model based on the truncation and tailing properties of the autocorrelation function (ACF) and partial autocorrelation function (PACF) diagrams. The advantages and disadvantages of the model fit were evaluated by the corrected Akaike information criterion (CAIC) value, and the model with the smallest CAIC value was considered the optimal model. After the order of the specific parameters was determined, a parameter test was performed through maximum likelihood estimation (MLE). Finally, the Ljung-Box test judges whether the residual sequence is white noise.
Building the XGBoost model
XGBoost, a kind of boosting algorithm, which assembles multiple learning algorithms to achieve a better predictive performance than any of the constituent learning algorithms alone, has excelled in many fields. Compared with the traditional GBDT algorithm, XGBoost applies a second-order Taylor expansion to the loss function and simultaneously implements the first derivative and the second derivative. In addition, a regularization term is added to the objective function, which improves the generalizability of a single tree and reduces the complexity of the objective function. In short, XGBoost has attracted the attention of researchers due to its fast speed, excellent classification effect, and ability to allow custom loss functions.
The classification and regression tree (CART) algorithm, first proposed by Breiman et al., refers to the general term of a classification tree and regression tree. The CART classification tree introduces the Gini coefficient to replace the information gain or information gain rate. The regression tree adopts different methods to evaluate the effect, including the prediction error (mean squared error, log error, etc.). Therefore, the node is no longer a category but a numerical value. In a CART model, for any feature j, there is a corresponding segment point s. If j is less than s, it is divided into the left-hand subtree. Otherwise, it is divided into the right-hand tree, as in formula (1).
$${R}_{1}\left(j,s\right)=\left\{x|{x}^{\left(j\right)}\le s\right\} \,and \,{R}_{2}\left(j,s\right)=\left\{x|{x}^{\left(j\right)}>s\right\}$$
The objective function of a typical CART regression tree is defined in formula (2):
$$\sum _{xi\in Rm}({yi-f\left(xi\right))}^{2}$$
As shown in formula (3), find the corresponding j and s that minimize the MSE of c1 and c2, respectively, and minimize the sum of the MSE between the two parts of c1 and c2. When we traverse all the segment points s of all features j, we can find the optimal j and s, and finally obtain a regression tree.
$$\underset{\text{j},\text{s}}{\text{min}}\left[\underset{{\text{x}}_{\text{i}}\in {\text{R}}_{1}(\text{j},\text{s})}{\text{min}}{(\text{y}\text{i}-\text{c}1)}^{2}+\underset{{\text{x}}_{\text{i}}\in {\text{R}}_{1} (\text{j},\text{s})}{\text{min}}{(\text{y}\text{i}-\text{c}2)}^{2}\right]$$
$${\widehat{ \text{c}}}_{1}=\text{a}\text{v}\text{e}\left({\text{y}}_{\text{i}}\mid {\text{x}}_{\text{i}}\in {\text{R}}_{1}(\text{j},\text{s})\right)$$
$${\widehat{\text{c}}}_{2}=\text{a}\text{v}\text{e}\left({\text{y}}_{\text{i}}\mid {\text{x}}_{\text{i}}\in {\text{R}}_{2}(\text{j},\text{s})\right)$$
The CART regression tree applies the mean or median of the final leaves to predict the output. To avoid overfitting, cost complexity pruning (CCP) is used to prune the non-leaf node with the smallest error gain and delete the child nodes with the non-leaf node.
The XGBoost algorithm is mainly composed of two parts: the decision tree algorithm and gradient boosting algorithm. Gradient boosting is an excellent technique for constructing prediction models and a representative algorithm for boosting. The theory of boosting is to establish weak evaluators individually and iteratively integrate multiple weak evaluators. The gradient boosting tree uses the CART algorithm as the main structure. Therefore, the steps of the XGBoost algorithm can be expressed as follows (formular (5)):
$$\widehat{{\text{y}}} = \phi \left( {{\text{x}}_{{\text{i}}} } \right) = \sum\limits_{{{\text{k}} = 1}}^{{\text{K}}} {{\text{f}}_{{\text{k}}} } \left( {{\text{x}}_{{\text{i}}} } \right)$$
In the XGBoost model, every leaf node has a forecasting score, called the leaf weight. \({f}_{k}\left({x}_{i}\right)\) is the value of all samples on this leaf node, where represents the th decision tree and represents the feature vector of sample. Each tree was added iteratively to keep the predicted value \(\hat {{y}}_{i}\) as close as possible to the actual value yi. Therefore, the following function reaches the minimum after t iterations:
$$O\text{b}{\text{j}}^{\left(\text{t}\right)}=\sum _{\text{i}=1}^{\text{n}} \text{l}\left({\text{y}}_{\text{i}},{\widehat{\text{y}}}_{\text{i}}^{(\text{t}-1)}+{\text{f}}_{\text{t}}\left({\text{x}}_{\text{i}}\right)\right)+{\Omega }\left({\text{f}}_{\text{t}}\right)+\text{ constant }$$
As shown in formula (6), the objective function consists of two parts: a loss function and a regularization term. The loss function assesses the forecasting function of the XGBoost model on the training data, and the regularization term \({\Omega }\left({f}_{t}\right)\) prevents the model from being too complicated. \(\hat {{\mathbf{y}}}^{(t-1)}\) is the predicted value of the last iteration and \({\text{f} }_{t}\)is a new function that the model learns. Next, a second-order Taylor development of the error term was performed on the objective function. Then the first derivative and the second derivative are defined as follows:
$${\text{Obj}}^{({\rm t})}\simeq \sum _{{\text{i}}=1}^{{\text{n}}} \left[{\text{l}} \left({\text{y}}_{\text{i}}, {\widehat{\text{y}}}_{\text{i}}^{({\text{t}}-1)}\right)+{{\text{g}}_{\text{i}}}{{\text{f}}_{\text{t}}}\left({\text{x}}_{\text{i}}\right)+\frac{1}{2}{{\text{h}}_{\text{i}}}{{\text{f}}_{\text{t}}^{2}}\left({\text{x}}_{\text{i}}\right)\right]+{\Omega }\left({\text{f}}_{\text{t}}\right)+{\text{constant}}$$
$${\text{g}}_{\text{i}}={\partial }_{{\widehat{\text{y}}}^{(\text{t}-1)}}\text{l}\left({\text{y}}_{\text{i}},{\widehat{\text{y}}}^{(\text{t}-1)}\right), {\text{h}}_{\text{i}}={\partial }_{{\widehat{\text{y}}}^{(\text{t}-1)}}^{2}\text{l}\left({\text{y}}_{\text{i}},{\widehat{\text{y}}}^{(\text{t}-1)}\right)$$
First, we define the mapping function of the decision tree: q indicates the structure of the tree, and w is the leaf node weight vector (the value of the sample predicted by the model).
$${f}_{t}\left(x\right)={w}_{q\left(x\right)},w\in {\text{R}}^{T},q:{\text{R}}^{d}\to \{\text{1,2},\cdots ,T\}$$
The complexity of the XGBoost tree is shown in formula (10). T is the quantitative complexity of leaf nodes in the tree, and the sum of squares term represents the L2 regularization term of the leaf node.
$${\Omega }\left({f}_{t}\right)=\gamma T+\frac{1}{2}\lambda \sum _{j=1}^{T} {w}_{j}^{2}$$
After combining the defined loss function and complexity of the tree, the objective function can be expressed by formula (13).
$$Ob{j}^{\left(t\right)}=\sum _{j=1}^{T} \left[\left(\sum _{i\in {I}_{j}} {g}_{i}\right){w}_{j}+\frac{1}{2}\left(\sum _{i\in {I}_{j}} {h}_{i}+\lambda \right){w}_{j}^{2}\right]+\gamma T$$
$${G}_{j}=\sum _{i\in {I}_{j}} {g}_{i}, {H}_{j}=\sum _{i\in {I}_{j}} {h}_{i}$$
$$Obj=-\frac{1}{2}\sum _{j=1}^{T} \frac{{G}_{j}^{2}}{{H}_{j}+\lambda }+\gamma T$$
Because it is not possible to traverse all the tree structures, constructing a decision tree based on space division is an NP problem. XGBoost uses a greedy algorithm to traverse the segmentation points of all features in the CART regression tree and calculates the gain before and after the segmentation point to determine whether a node continues to grow. The node will split when the value of the objective function after splitting is higher than the gain of the single-leaf node. At the same time, the maximum depth of the tree and a threshold should be set to limit its growth. The gain formula is shown in formula (14):
$$\text{Gain }=\frac{1}{2}\left[\frac{{G}_{L}^{2}}{{H}_{L}+\lambda }+\frac{{G}_{R}^{2}}{{H}_{R}+\lambda }-\frac{{\left({G}_{L}+{G}_{R}\right)}^{2}}{{H}_{L}+{H}_{R}+\lambda }\right]-\gamma$$
One-hot encoding was used to address the seasonality. Three types parameters should be set when building the XGBoost model: general parameters, booster parameters and task parameters. The XGBoost model also draws on the idea of random forest, introducing row sampling and column sampling that can reduce the amount of calculation and prevent overfitting. Moreover, it introduces the early-stopping mechanism to prevent overfitting. In this study, the booster parameter is gbtree; early_stopping_round was set to 5; subsample and colsample_bytree were set from 0.3 to 0.7; max_depth was set to 2 and 3; min_child set to 1 and 2, the learning rates of XGBoost were set to 0.04, 0.05 and 0.06; and eval_metric was set to 'rmse'. A grid search was conducted to exhaustively search for specified parameter values when the potential parameter values were ordered and combined. Notably, the performance of the XGBoost was evaluated by tenfold cross-validation and the RMSE. Additionally, XGBoost can rank the importance of variables by the frequency functions used to split the feature. After the XGBoost model was built, the accuracies of the one-step forecast and multistep forecast were compared by the RMSE, MAE and MAPE.
One-step forecasting and multistep forecasting
Generally, a one-step time series uses actual historical data, such as data at time t-n, time t-(n-1), time t to predict the value at time t + 1 in the next step. In contrast, when performing multistep prediction, single-step prediction is performed and the predicted value is used (instead of the actual value) as an input variable for the second step of prediction. Then, the process was repeated until all the predicted values were obtained [23, 24]. There are four multistep forecasting strategies: direct forecasting, recursive forecasting, direct recursive hybrid forecasting and multioutput forecasting. One-step forecasting is more accurate, but it will prevent the model from simulating the trends in the next month. When the forecast cycle is long, a multistep forecast is prone to face larger error accumulation. When the forecasted value is used as input, the error will inevitably accumulate with the input value in the next step. In this study, one-step forecasting and multistep forecasting were carried out.
Model comparison and data analysis
Model evaluation and comparison are mainly judged by the accuracy of the model. The accuracy refers to the degree to which the predicted result matches with the actual result, so the error can be used to evaluate the accuracy of the prediction model. The smaller the error is, the better the fitting effect. Model evaluation generally includes two parts: training sample evaluation and prediction sample evaluation. To better compare the accuracy of the ARIMA and XGBoost models, a series of evaluation indices were applied in this study. mean error (ME), root mean squared error (RMSE), mean absolute error (MAE), mean percentage error (MPE), mean absolute percentage error (MAPE), mean absolute scaled error (MASE) and autocorrelation of errors at lag 1 (ACF1). Generally, the larger the criteria are, the greater the error size is. Theil's U statistic measures the accuracy by comparing the predicted results with the prediction results using minimal historical data. It tends to place more weight on large errors by squaring the deviations and overstating errors, which can help eliminate methods with large errors. Theil's U < 1 indicates that the predicted results are better than the expected results.
The HFRS data analysis process was completed in R version 3.6.2. Packages like TSstudio, forecast, xgboost were included to achieve different functions. In addition, we set the statistical significance level at 0.05.
As shown in Fig. 1, the original time series graph showed a slight downward trend and seasonal variation. The number of HFRS cases had a bimodal seasonal distribution throughout the year (Fig. 2), one from October to January of the following year and the other from March to June, which means that the time series was not stationary. Therefore, logarithmic or square root conversion was used to transform the time series variance. The time series diagram after applying a Box-Cox transformation is shown in Fig. 3. The small gray blocks of different sizes show the proportion of each component. The additive time series decompositions subjected to Box-Cox transformation were arranged in order of magnitude, including the original data, season, trend and noise element. The seasonal component showed obvious periodicity, while the trend showed an overall decrease from 2004 to 2010 but increased briefly in 2010–2013. In addition, there was no noticeable form of noise.
Time series plot for cases of HFRS in mainland China from January 2004 to December 2018
Monthly chart of HFRS cases from 2004 to 2017
Seasonal decomposition of the Box-Cox-transformed HFRS cases
To eliminate seasonal characteristics and long-term trends in the time series, the first difference (d = 1) and seasonal difference (D = 1) were used (Fig. 4). The ADF test demonstrated that the time series after the difference was stable (t =− 6.4674, p < 0.01). Consequently, from d = 1 and s = 12, the seasonal ARIMA model can be preliminarily denoted by ARIMA (p, 1, q) × (P, 1, Q)12.
Plot of the Box-Cox-transformed HFRS cases
As seen in the graphs of the ACF and PACF (Fig. 5). The ACF had obvious peaks at lags 3 and 12, indicating respectively nonseasonal MA (3) components and seasonal MA (1) components respectively. In addition, in the PACF graph, the obvious lag peaks at 3 and 12 indicate a nonseasonal AR (3) element and a seasonal AR (1) element. Therefore, the parameters were set as follows: p from 0 to 3, q from 0 to 3, P from 0 to 1 and Q from 0 to 1. By assembling all possible values of each parameter, multiple candidate models are generated. Nine models remained after the residual and parameter test was implemented, and the ARIMA (3, 1, 0) × (1, 1, 0)12 model had the smallest CAIC (427.1528) (Table 1). The Ljung–Box test (Q = 7.5588, p = 0.9944) indicated that the sequence residual was white noise, which means that the final fitted data sequence was stationary. The estimated parameters of the ARIMA (3, 1, 0) × (1, 1, 0)12 model are listed in Table 2. The curves of training, forecasting and the actual HFRS incidence by ARIMA model are pictured in Fig. 6.
Autocorrelation and partial autocorrelation plots of the differenced HFRS incidence series
Table 1 CAIC value and Ljung-Box Q value of the candidate seasonal ARIMA models
Table 2 Estimated parameters of the seasonal ARIMA (3,1,0) × (1,1,0)12 model
The curves of the fitted ARIMA model, forecasted ARIMA model and actual HFRS incidence series
XGBoost model
The grid search algorithm was used in the XGBoost model to realize the automatic optimization of the parameters. In this research, we realized automatic optimization of max_depth, n_estimators and min_child_weight. According to the grid search and tenfold cross-validation, the possible parameters are shown in Table 3. Among all six combined parameters, the first had the lowest test RMSE (238.3084). The optimal parameters of the XGBoost model were listed in Table 4. The importance of a feature is determined by whether the forecasting capability changes significantly when the feature is replaced by random noise. In the XGBoost algorithm, we input several features to calculate the feature importance and determine how each feature contributes to the prediction performance in the training step (Fig. 7). Characteristic variables such as x_lag12 and x_lag1 had a significant impact on the prediction of the number of HFRS cases. Finally, based on the hyperparameter optimization results, the final one-step forecasting model was built. The curves of training, forecasting and the actual HFRS incidence by the XGBoost model are showed in Fig. 8.
Table 3 Possible parameters of the XGBoost model
Table 4 List of the optimal parameters and description of the XGBoost model
Importance of the XGBoost characteristic variables
The curves of the fitted XGBoost model, forecasted XGBoost model and actual HFRS incidence series
Comparison of the models
Table 5 shows the one-step and multistep forecasting accuracies of the two models. In the training sample, the ME, MAE, MAPE, MPE and MASE of XGBoost were higher than those of the ARIMA model, whereas the RMSE of XGBoost was lower than that of the ARIMA model. In the test sample, the ME, RMSE, MAE, MPE, MAPE and MASE of XGBoost model were obviously lower than those of the ARIMA model in both one-step forecasting and multistep forecasting. Therefore, the XGBoost model had a better forecasting performance in the prediction of the number of HFRS cases.
Table 5 The one-step and multistep forecasting accuracy of the ARIMA and XGBoost models
This study showed the seasonal distribution of HFRS cases. The main incidence peaks were concentrated from October to January, especially in November of the following year. The second incidence peak occurred from March to June. The overall shape was bimodal, which was consistent with the literatures reported in the South Korean army and different regions of China [4, 25]. In addition, the incidence of HFRS in mainland China from 2004 to 2018 showed an overall downward trend, but there was a clear upward trend in 2010 that continued until 2013. The periodicity of HFRS incidence may be related to climate factors, the number of rodents in the wild, and the accumulation speed of susceptible people. As an important climate factor, the monsoon phenomenon may affect periodic trends in HFRS, which can change annually. Since the data collected in this study were not from a sufficiently long period, the periodicity was not obvious in this study. The influence of meteorological factors and monsoon phenomena on HFRS can be considered in the future.
Therefore, understanding the changing trend in HFRS is particularly important for exploring the influencing factors. It is also crucial for predicting epidemics and formulating corresponding preventive and early-warning measures. The accuracy of infectious disease forecasting has drawn the attention of a number of scholars [9, 26, 27]. Many mathematical methods and statistical models have been applied to predict HFRS incidence. The ARIMA model is developed based on a linear regression model, combining the advantages of autoregressive and moving average models, which can explain the data well. We can obtain the coefficient of each variable and know whether each coefficient is statistically significant. Stationary data are a prerequisite for establishing an ARIMA model; thus, the seasonal ARIMA model needs to transform nonlinear data into linear data after differencing and transformation. According to the characteristics of HFRS, we decomposed the infectious disease time series into trend components, seasonal components and random fluctuation components. The more differences use, the more data are lost. In this study, the first-order and 12th-order differences were used, so 13 months of data were lost. When forecasting, the ARIMA model considers only historical data to understand the disease trend and obtain a more accurate prediction effect instead of requiring specific influencing factors. Therefore, the ARIMA method is easy to master and widely used. However, the nonlinear mapping performance of ARIMA models is weak, and its accuracy is unsatisfactory when it tries to fit and predict nonlinear and complex infectious disease time series. For example, in this study, the fitting effect was not perfect when the disease trend changed suddenly, and the error between the fitted value and the actual value in May 2010 and 2013 was relatively large (Fig. 6). Many factors can affect HFRS, including meteorological factors and human-made control measures, most of which have a nonlinear relationship with the number of cases, so when the number of HFRS suddenly increases or decreases, these nonlinear factors may affect the fitting accuracy of the ARIMA model. In addition, the ARIMA method is more suitable to predict a short-term time series. Thus, it is necessary to constantly collect data and obtain the longest time series possible. Based on the characteristic of the ARIMA model and HFRS, this study used the monthly incidence data of HFRS from 2004 to 2017 to establish a seasonal ARIMA model. The results showed that the ARIMA (3, 1, 0) × (1, 1, 0)12 model can better fit and predict the monthly incidence than other forms.
The XGBoost model is a powerful machine learning algorithm, especially in terms of the speed and accuracy are concerned. It is good at dealing with nonlinear data but has poor interpretability. From studies in other fields, the XGBoost model performed well in predicting nonlinear time series [28,29,30,31]. By integrating multiple CART models, XGBoost model can achieve a better generalizability than a single model, which means that the XGBoost has a larger postpruning penalty than a GBDT model and makes the learned model less prone to overfitting. Moreover, a regularization term is added to control the complexity reduce the variance of the model. Moreover, XGBoost model is a hyperparameter model [32], that can control more parameters than other models and is flexible to tune parameters. Compared with the complexity of the conditions that the ARIMA model needs to meet, the modeling process of the XGBoost is very simple. In this study, a grid search was conducted to exhaustively search for specified parameters, and tenfold cross-validation was used to evaluate the performance of the XGBoost. The grid search made XGBoost achieve a good generalizability but also consumed more calculation resources and storage space. In addition, XGBoost model fit the range of normal values more stably, but the ARIMA model was slightly better than the XGBoost model when fitting outliers (Fig. 8). This finding is mainly due to the following reasons: during ARIMA modeling, the best parameters were determined by the minimum CAIC and residual white noise of the training set, and the problem of overfitting was not considered. For the XGBoost model, to prevent overfitting, tenfold cross-validation and an early-stopping mechanism were used to select the best parameters. These factors increased the prediction performance of the XGBoost model but reduced the fitting effect of outliers. With these characteristics, our study applied it in prediction of the incidence of HFRS. We tried one-step forecasting and multistep XGBoost forecasting models to predict HFRS cases in mainland China. The results showed that the MAEs of the one-step and multistep XGBoost models were 132.055 and 173.403 respectively, which were 28.76 and 33.27 % lower than that of ARIMA model. The MAPE values were 12.353 and 15.615, which were 33.45 and 48.16 % lower than that of the ARIMA model. The RMSEs were 178.547 and 223.187, which were 28.37 and 26.29 % lower than that of ARIMA model.
As predicted, the one-step prediction accuracy of the two models was better than the multistep prediction accuracy. From the perspective of predicting infectious diseases, each predicted value of one-step prediction is obtained from the actual value, and it is unrealistic to predict diseases that have not occurred. Multistep prediction uses the previous prediction value as input to predict the next value, which will produce cumulative errors, but it has practical significance for predicting infectious diseases that have not occurred. The results indicated that the proposed one-step and multistep XGBoost model can significantly improve the accuracy of the overall prediction. The value of Theil's U also proved this finding. From the perspective of the prediction accuracy and prediction stability, the XGBoost model is suitable for HFRS prediction tasks. In other words, by integrating the prediction results of multiple regression trees, the XGBoost model can achieve better prediction results than the ARIMA model in the one-step forecasting and multistep forecasting.
In this paper, we built a seasonal ARIMA model and XGBoost model to conduct one-step and multistep prediction of the number of HFRS cases in mainland China for 2004 to 2018. The multistep XGBoost prediction model showed a much better prediction accuracy and model stability than the multistep ARIMA prediction model. The XGBoost model performed better in predicting complicated and nonlinear HFRS data. Additionally, a multistep prediction model has more practical significance than one-step prediction for forecasting infectious diseases.
The datasets generated and/or analysed during the current study are available in Additional file 1 (http://www.nhc.gov.cn). The datasets used and/or analysed during the current study are also available from the corresponding author on reasonable request.
HFRS:
Hemorrhagic fever with renal syndrome
ARIMA:
Autoregressive integrated moving average
XGBoost:
Extreme gradient boosting
Classification and regression tree
ADF:
Augmented Dickey-Fuller
ACF:
Autocorrelation function
PACF:
Partial autocorrelation function
Conditional sum of squares
CAIC:
Corrected Akaike Information Criteria
Mean error
RMSE:
Root mean squared error
MAE:
Mean absolute error
MPE:
Mean percentage error
MAPE:
Mean absolute percentage error
MASE:
Mean absolute scaled error
ACF1:
Autocorrelation of errors at lag 1
Jiang H, Du H, Wang LM, Wang PZ, Bai XF. Hemorrhagic fever with renal syndrome: pathogenesis and clinical picture. Front Cell Infect Microbiol. 2016;6:1.
Avšič-Županc T, Saksida A, Korva M. Hantavirus infections. Clin Microbiol Infect. 2019;21:e6–16.
Zhao Y, Ge L, Zhou Y, Sun Z, Zheng E, Wang X, Huang Y, Cheng H. A new Seasonal Difference Space-Time Autoregressive Integrated Moving Average (SD-STARIMA) model and spatiotemporal trend prediction analysis for Hemorrhagic Fever with Renal Syndrome (HFRS). PLoS ONE. 2018;13(11):e0207518–e0207518.
Sun L, Zou L-X. Spatiotemporal analysis and forecasting model of hemorrhagic fever with renal syndrome in mainland China. Epidemiol Infect. 2018;146(13):1680–8.
Du H, Wang PZ, Li J, Bai L, Li H, Yu HT, Jiang W, Zhang Y, Wang JN, Bai XF. Clinical characteristics and outcomes in critical patients with hemorrhagic fever with renal syndrome. BMC Infect Dis. 2014;14:191.
Liu X, Jiang B, Bi P, Yang W, Liu Q. Prevalence of haemorrhagic fever with renal syndrome in mainland China: analysis of National Surveillance Data, 2004–2009. Epidemiol Infect. 2012;140(5):851–7.
Fang L, Yan L, Liang S, de Vlas SJ, Feng D, Han X, Zhao W, Xu B, Bian L, Yang H, et al. Spatial analysis of hemorrhagic fever with renal syndrome in China. BMC Infect Dis. 2006;6:77.
Cheng HY, Wu YC, Lin MH, Liu YL, Tsai YY, Wu JH, Pan KH, Ke CJ, Chen CM, Liu DP, et al. Applying machine learning models with an ensemble approach for accurate real-time influenza forecasting in taiwan: development and validation study. J Med Intern Res. 2020;22(8):e15394.
Guo P, Liu T, Zhang Q, Wang L, Xiao J, Zhang Q, Luo G, Li Z, He J, Zhang Y, et al. Developing a dengue forecast model using machine learning: A case study in China. PLoS Negl Trop Dis. 2017;11(10):e0005973.
Gu J, Liang L, Song H, Kong Y, Ma R, Hou Y, Zhao J, Liu J, He N, Zhang Y. A method for hand-foot-mouth disease prediction using GeoDetector and LSTM model in Guangxi, China. Sci Rep. 2019;9(1):17928.
Wang YW, Shen ZZ, Jiang Y. Comparison of autoregressive integrated moving average model and generalised regression neural network model for prediction of haemorrhagic fever with renal syndrome in China: a time-series study. BMJ Open. 2019;9(6):e025773.
Zhang C, Fu X, Zhang Y, Nie C, Li L, Cao H, Wang J, Wang B, Yi S, Ye Z. Epidemiological and time series analysis of haemorrhagic fever with renal syndrome from 2004 to 2017 in Shandong Province, China. Sci Rep. 2019;9(1):14644.
Giraka O, Selvaraj VK. Short-term prediction of intersection turning volume using seasonal ARIMA model. Transport Lett. 2020;12(7):483–90.
Tian CW, Wang H, Luo XM. Time-series modelling and forecasting of hand, foot and mouth disease cases in China from 2008 to 2018. Epidemiol Infect. 2019;147:e82.
Ho CS. Application of XGBoost ensemble method on nurse turnover prediction. Basic Clin Pharmacol Toxicol. 2019;125:134–134.
Ji XJ, Tong WD, Liu ZC, Shi TL. Five-Feature Model for Developing the Classifier for Synergistic vs. Antagonistic Drug Combinations Built by XGBoost. Fron Genet. 2019;10:1.
Li W, Yin YB, Quan XW, Zhang H. Gene expression value prediction based on XGBoost algorithm. Front Genet. 2019;10:1.
Zhang XL, Nguyen H, Bui XN, Tran QH, Nguyen DA, Bui DT, Moayedi H. Novel soft computing model for predicting blast-induced ground vibration in open-pit mines based on particle swarm optimization and XGBoost. Nat Resour Res. 2020;29(2):711–21.
Zheng H, Wu YH. A XGBoost model with weather similarity analysis and feature engineering for short-term wind power forecasting. Appl Sci -Basel. 2019;9:15.
Liu Q, Liu X, Jiang B, Yang W. Forecasting incidence of hemorrhagic fever with renal syndrome in China using ARIMA model. BMC Infect Dis. 2011;11:218.
Helfenstein U. Box-Jenkins modelling in medical research. Stat Methods Med Res. 1996;5(1):3–22. https://doi.org/10.1177/096228029600500102.
Zhang G, Huang S, Duan Q, Shu W, Hou Y, Zhu S, Miao X, Nie S, Wei S, Guo N, et al. Application of a hybrid model for predicting the incidence of tuberculosis in Hubei, China. PLoS ONE. 2013;8(11):e80969.
Sorjamaa A, Hao J, Reyhani N, Ji Y, Lendasse A. Methodology for long-term prediction of time series. Neurocomputing. 2007;70(16):2861–9.
Zhang J, Nawata K. Multistep prediction for influenza outbreak by an adjusted long short-term memory. Epidemiol Infect. 2018;146(7):809–16.
Gauld RL, Craig JP. Epidemiological pattern of localized outbreaks of epidemic Hemorr-hagic Fever. Am J Hyg. 1954;59(1):32–8.
Liao Z, Zhang X, Zhang Y, Peng D. Seasonality and Trend Forecasting of Tuberculosis Incidence in Chongqing, China. Interdiscipl Sci Comput Life Sci. 2019;11(1):77–85.
Singh RK, Rani M, Bhagavathula AS, Sah R, Rodriguez-Morales AJ, Kalita H, Nanda C, Sharma S, Sharma YD, Rabaan AA, et al. Prediction of the COVID-19 Pandemic for the Top 15 Affected Countries: Advanced Autoregressive Integrated Moving Average (ARIMA) Model. JMIR Public Health Surveill. 2020;6(2):e19115.
Zhou Y, Li T, Shi J, Qian Z. A CEEMDAN and XGBOOST-based approach to forecast crude oil prices. Complexity. 2019;2019:1–15.
Ma J, Ding Y, Cheng JCP, Tan Y, Gan VJL, Zhang J. Analyzing the leading causes of traffic fatalities using XGBoost and grid-based analysis: a city management perspective. IEEE Access. 2019;7:148059–72.
Zheng H, Yuan J, Chen L. Short-term load forecasting using EMD-LSTM neural networks with a Xgboost algorithm for feature importance evaluation. Energies. 2017;10:8.
Alim M, Ye GH, Guan P, Huang DS, Zhou BS, Wu W. Comparison of ARIMA model and XGBoost model for prediction of human brucellosis in mainland China: a time-series study. BMJ Open. 2020;10(12): https://doi.org/10.1136/bmjopen-2020-039676.
Putatunda S, Rama K. A modified bayesian optimization based hyper-parameter tuning approach for extreme gradient boosting. Fifteenth Int Conf Inform Process. 2019;2019:1–6.
We would especially like to thank the reviewers for critically reading the manuscript and providing comments for improving the presentation.
This study was supported by the National Natural Science Foundation of China (Grant No. 81202254) and the Science Foundation of Liaoning Provincial Department of Education (LK201644). They have not participated in the design of the study, the collection, analysis, interpretation of data nor in writing the manuscript.
Department of Epidemiology, School of Public Health, China Medical University, Shenyang, Liaoning, China
Cai-Xia Lv & Wei Wu
Liaoning Provincial Center for Disease Control and Prevention, Shenyang, Liaoning, China
Shu-Yi An & Bao-Jun Qiao
Cai-Xia Lv
Shu-Yi An
Bao-Jun Qiao
Wei Wu
CXL designed and wrote the manuscript. SYA and BJQ analyzed and collected the data. WW proposed the original idea and modified and refined the manuscript. All authors read and approved the final manuscript.
Correspondence to Wei Wu.
Monthly cases of hemorrhagic fever with renal syndrome in Mainland China from 2004 to 2018.
Lv, CX., An, SY., Qiao, BJ. et al. Time series analysis of hemorrhagic fever with renal syndrome in mainland China by using an XGBoost forecasting model. BMC Infect Dis 21, 839 (2021). https://doi.org/10.1186/s12879-021-06503-y
DOI: https://doi.org/10.1186/s12879-021-06503-y
Hemorrhagic fever with renal syndrome (HFRS)
Multistep prediction | CommonCrawl |
History of Science and Mathematics
History of Science and Mathematics Meta
History of Science and Mathematics Beta
When was the earliest use of log-log plots to demonstrate power-law behavior?
After reading this answer and writing this comment, I decided to ask this question: When and where was the earliest known use of a log-log plot to demonstrate power-law behavior?
Napier introduced the logarithms in 1614, and Gunter invented the slide rule somewhere in 1620s. One might think that log log plots came soon after, but no. According to The Age of Graphical Computing: "In 1844 Leon Lalanne succeeded in linearizing the curves $y=x^p$ by plotting the first log-log plot in history, thereby creating his Universal Calculator, chock-full of lines for common engineering calculations and capable of graphically computing formulas in powers or roots of x ( or of trigonometric functions in x) with ease... Lalanne envisioned copies of his Universal Calculator posted in public squares and business meeting places for popular use". Lallane's successor, d'Ocagne, also credits him with the invention of logarithmic graph paper. The first link has a nice image of Lalanne's Universal Calculator, which looks like a fanciful version of it.
Lalanne's approach was not specific to logarithms, he came up with a general idea of transforming curved graphs into straight line ones by modifying the scales on the axes, which he called "geometrical anamorphosis". d'Ocagne developed Lalanne's ideas after 1880 under the name of nomography, and nomograms remained a popular computing tool, especially among engineers, until the onset of pocket calculators. Smith's Source Book in Mathematics has large excerpts from d'Ocagne's Treatise on Nomography (1899), which describes his and Lalanne's methods of graphical computing. The slide rule, logarithmic paper and nomograms, those nostalgic pieces of the old lore of scientific and technical computing wiped out by the advent of electronic calculators and computers...
ConifoldConifold
$\begingroup$ Wow, thank you for your thorough answer! I'm quite surprised by the 200+ year gap. I have fond memories of nomograms, some were quite complex and beautiful. A lot of work went into them. I wonder if anything like "The Beauty of Nomograms" or "Nomogram Hall of Fame" exists. $\endgroup$ – uhoh Mar 27 '16 at 1:56
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Origin of the "law of quadratic reciprocity" | CommonCrawl |
Module of covariants
In algebra, given an algebraic group G, a G-module M and a G-algebra A, all over a field k, the module of covariants of type M is the $A^{G}$-module
$(M\otimes _{k}A)^{G}.$
where $-^{G}$ refers to taking the elements fixed by the action of G; thus, $A^{G}$ is the ring of invariants of A.
See also
• Local cohomology
References
• M. Brion, Sur les modules de covariants, Ann. Sci. École Norm. Sup. (4) 26 (1993), 1 21.
• M. Van den Bergh, Modules of covariants, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zurich, 1994), Birkhauser, Basel, pp. 352–362, 1995.
| Wikipedia |
Theoretical aspects
Algorithm implementations in the supercell program
Comparison with existing solutions
Exploring supercell configurations: supercell application examples
Availability and requirements
Supercell program: a combinatorial structure-generation approach for the local-level modeling of atomic substitutions and partial occupancies in crystals
Kirill Okhotnikov1, 2Email author,
Thibault Charpentier1 and
Sylvian Cadars2, 3
© Okhotnikov et al. 2016
Disordered compounds are crucially important for fundamental science and industrial applications. Yet most available methods to explore solid-state material properties require ideal periodicity, which, strictly speaking, does not exist in this type of materials. The supercell approximation is a way to imply periodicity to disordered systems while preserving "disordered" properties at the local level. Although this approach is very common, most of the reported research still uses supercells that are constructed "by hand" and ad-hoc.
This paper describes a software named supercell, which has been designed to facilitate the construction of structural models for the description of vacancy or substitution defects in otherwise periodically-ordered (crystalline) materials. The presented software allows to apply the supercell approximation systematically with an all-in-one implementation of algorithms for structure manipulation, supercell generation, permutations of atoms and vacancies, charge balancing, detecting symmetry-equivalent structures, Coulomb energy calculations and sampling output configurations. The mathematical and physical backgrounds of the program are presented, along with an explanation of the main algorithms and relevant technical details of their implementation. Practical applications of the program to different types of solid-state materials are given to illustrate some of its potential fields of application. Comparisons of the various algorithms implemented within supercell with similar solutions are presented where possible.
The all-in-one approach to process point disordered structures, powerful command line interface, excellent performance, flexibility and GNU GPL license make the supercell program a versatile set of tools for disordered structures manipulations.
Disordered compounds
Quantum calculations
The structure and properties of disordered condensed systems have always attracted the attention of scientists and engineers [1]. Many unique properties of solid-state materials appear only in a disordered and/or defected state. Of the various types of disorder that exist at different length scales in real compounds [2, 3], local atomic impurities, substitutions and/or vacancies are among the most important. Such point defects are in particular responsible for the unique properties of a number of semiconductors, high temperature superconductors, metallic alloys, ceramics (including piezoelectric), zeolite catalysts and many other types of technologically important materials.
A wide range of experimental techniques exist to explore the local disorder in otherwise crystalline solids, complementary to diffraction techniques which provide information on the average long-range structure, including in favorable cases fractional occupancies an/or mixed atomic compositions on the different crystallographic sites. Because they do not rely on long-range atomic periodicity, local spectroscopies such as solid-state nuclear magnetic resonance (NMR) [4], Raman [5], infra-red or X-ray adsorption near-edge structure (XANES) are particularly relevant to reveal and characterize point defects to then understand how they affect the materials properties [6, 7]. Solid-state NMR, in particular, has successfully been used, often in combination with density functional theory (DFT) calculations of NMR parameters [4, 6, 8, 9], to unravel the effects of substitution disorder in systems as diverse as clays [10–13], layered and microporous silicate catalysts [14–16], ceramics [17, 18], Li-battery [19, 20] and other inorganic oxides [21, 22], chalcogenide semiconductors [23, 24] and doped graphene derivatives [25]. The data that such local probes provide (in the form of chemical shift and electric field gradient tensors or vibrational frequencies, for example) are however often difficult to interpret in terms of local structure around the defected sites. This makes molecular modelling absolutely crucial for the fine understanding of such systems.
The same distinction between local structure and long-range atomic periodicity exists in the theory of solid-state physics, which primarily deals with ideal crystalline systems. One of the most commonly-used approach to connect "periodic" theories with the molecular-level structure probed by local spectroscopies in disordered solids is called the long-range or supercell approximation. The idea is to create a large periodic cell that, within its boundaries, reflects as closely as possible the local structural properties of a disordered system: composition, coordination sequences, etc. This local-level similarity between the real and model (albeit still periodic) systems gives a hope that calculated physical properties will also reflect the real materials properties [26]. While supercells are easy to construct when point defects are in small concentrations and can be considered as isolated, this is more challenging at high defect concentrations, where the relative positions of defects and their interactions become critical. Many factors should be taken into account when implementing the long-range approximation in such systems: the supercell size, its total charge, but also the distribution of local charges, the probability of the supercell configurations, etc.
Different strategies have been described to explore the various atomic supercell configurations that can be generated from a crystal unit cell containing one or several disordered sites (including partially-occupied and/or sites of mixed composition). A particularly successful method called the special quasi-random structures (SQS) [27], consists in constructing one or a small number of supercells with atomic configurations most representative of a random distribution of atom types among the disordered sites. This paper focuses instead on a more systematic and thereby more general strategy, which applies not only to random (as the SQS approach) but also to other types of disorder. It consists in performing a comprehensive search among all possible configurations and thereby offers extended possibilities to explore the local interactions that govern the overall local to long-range compositional (dis)order in such materials. Such exhaustive explorations are however obviously limited by the number of atomic configurations, which very quickly explodes with the number of disordered sites, the concentration of defect atoms and the supercell size. Several excellent theoretical papers [28–30] have therefore described optimized procedures to generate complete sets of unique (i.e., symmetry-independent) configurations, which have been implemented in open [28, 29] and commercial [31] software.
Most of these theoretical developments, however, were initially made for metallic alloys and semi-conductors and have not yet reached certain research areas (and in particular the solid-state NMR community) where molecular modeling is now commonly used to interpret experimental spectroscopic and/or crystallographic data. Possible reasons for this may include the technicality of the corresponding articles [29, 30, 32] and/or discrepancies in the vocabulary employed to describe compositional disorder and its modelling in different communities. In addition, some of the existing implementations suffer from performance issues, limitations on the types of disorder (number of substituents or of disordered crystallographic sites) and/or rather inconvenient input and output formats. It is therefore our opinion that a new software emphasizing the ease of access and use to a broad community of materials scientists, while offering good versatility and performance, was needed.
The new program called supercell, which is presented in this paper, aims to fulfill this need. It implements within a single executable multiple tools specifically designed to apply the supercell approximation easily and systematically to a wide range of crystalline solids with compositional disorder. This article describes in very general terms and for a broad audience the basic principles, functionalities, and corresponding algorithms intergrated in the program, and presents its application to a few materials representative of different fields of research where this general approach to the modelling of compositional disorder in crystalline solids could become an essential tool.
Supercell approximation
In many types of materials, point defects consist in atomic substitutions or vacancies, meaning that the nature of the atoms occupying one or several crystallographic sites is changed, while preserving the crystallographic positions and hence the overall (average) periodicity of the system. In this case, the disordered structure can be described as a regular periodic structure with well-defined cell parameters, a space-group and a set of crystallographic sites. But contrarily to "ideal" crystals where each crystallographic site is strictly (and fully) occupied by a single type of atom, disordered structures, in their long-range average vision derived from diffraction data, have some of their crystallographic sites partially occupied. This long-range vision, however, has no physical relevance at the local level, where two (or more) atoms cannot of course occupy the same physical position, or one type of atom occupy it partially. What these mixed compositions and partial occupancies mean is that the same crystallographic site in the different periodic images of the original cell can be occupied by different atom types or stay unoccupied (i.e., occupied by a vacancy). A non-periodic occupation of these crystallographic sites will in turn break the periodicity of the system and lead to properties, which, strictly speaking, will depend on the distribution of the different atoms onto these disordered sites in the whole system. In practice, however, many physical properties can be satisfactorily approximated by performing an average (or a summation) over a finite number of small parts of the sample. The essence of the supercell approximation is based on this fact.
Within a supercell numerous atomic configurations can be compatible with the partial occupancies and mixed compositions of the disordered long-range average structure, and the important question is how to explore them efficiently. The approach implemented in our program is an exhaustive exploration where all individual configurations satisfying the conditions are generated and processed. This approach is particularly suitable for small numbers of configurations, but special strategies have also been implemented in the program to treat relatively large cells and/or complex disorder, as will be discussed below.
Atom combinatorics
The exhaustive search for possible atomic configurations in a given supercell is based on methods of enumerative combinatorics. The problem consisting in distributing atoms among one or several sets of disordered crystallographic sites was reformulated in terms of multinomial permutations.1 Let us consider a system with one disordered crystallographic site with multiplicity \(K\) and \(N\) different types of atoms occupying this site. Vacancies are treated in the same way, as a special "null" atom type. The number of atoms of each type \(k_i\), with \(i=1\ldots N\), verify \(K=\sum _{i=1}^N k_i\). The main two tasks are (1) to calculate the total number of possible permutations and (2) to loop through all of them.
The total number of combinations \(P\) can be calculated by a multi-set permutation formula:
$$\begin{aligned} P\left( k_1,k_2,\ldots ,k_N\right) =\frac{\left( k_1+k_2+\cdots +k_N\right) !}{k_1!k_2!\ldots k_N!}=\frac{\left( \sum _{i=1}^N k_i\right) !}{\prod _{i=1}^N k_i!} \end{aligned}$$
Equation (1) is a generalization of the well-known binomial distribution formula, obtained for \(N=2\) (a typical case being for example the partial occupancy of the considered crystallographic site by one type of atom, the rest being vacancies).
In the (frequent) case where more than one crystallographic site are disordered, the implemented algorithm handles them all at once (unless the user chooses to "freeze" some of them as shown in the program examples), assuming that the permutations within the different sites are independent. The total number of output structures will consequently be \(C=\prod _{i=1}^M P_i\), where \(P_i\) is number of possible permutations on site i, and M the total number of independent disordered sites.
The procedure performed by the supercell program consists of four main stages that are schematically depicted in Fig. 1.
Illustration on a hypothetical disordered 2D crystal of the main concepts and tasks of the supercell algorithm workflow. a Input structure consisting of crystallographic positions and occupation values for each atom type, as typically defined in a cif file. b Crystallographic sites are sorted into groups 1, 2, and 3. c All atom types and corresponding occupancies are then assigned to a group. d \(2\times 2\) supercell made from cell (b). e Atoms (in gray, red, green and blue) and vacancies (treated as special atoms, in white) used for permutations within the groups. f Two examples out of many possible resulting periodic structures with full occupancy (or vacancy) of all sites. Stages I–IV are described in detail in text. Numbers of permutations \(P_i\) were calculated with formula (1)
Stage I: Assigning crystallographic sites and atoms to groups
First of all, all sites and atoms in the initial structure (Fig. 1a) are sorted out to different groups. Those groups are constructed such that permutation theory can be applied independently to each group. Each object of this type consists of a set of positions and a set of atoms which can occupy all these positions with equal probability and without any restriction. The atom types and crystallographic sites are sorted to groups according to the following rules:
each atom type should be assigned strictly to one group,
each group should be associated with at least one atom type,
each crystallographic position belongs strictly to one group,
a group should be associated with at least one position,
all atom types within a group occupy all positions assigned to the group with the same probability, and
each position cannot be occupied by more than one atom.
In most cases it is enough to treat a group like a crystallographic site occupied by one or more element(s) and/or a vacancy with some population. Three different groups are presented in the example on Fig. 1. Group 1 consists of one crystallographic site, marked as "1" (Fig. 1b), and one atom type: "gray" (Fig. 1c) with 100 % occupancy. The same characteristics apply of course to all images of this site, whether by cell symmetry or periodicity. Group 2 consists of 1 crystallographic site, marked as "2", and one atom type "orange" occupying 50 % of the corresponding positions, while leaving 50 % vacancies on this site. Group 3 illustrates a more unusual but nevertheless plausible case. This group consists of two distinct but close crystallographic positions, each occupied by a distinct atom type: "green" and "blue", respectively with occupancies of 75 and 25 %. Because the two distinct crystallographic positions in group 3 are too close in space to be possibly occupied at the same time (a distance criterion that defaults to 0.75 Å, but can be modified by the user), the program reduces them to one single position, marked as "3" in Fig. 1b, and corresponding to the median point between them. This description in terms of groups serves as the basis for the application of the permutation theory.
The supercell program implementation assumes that atoms types (represented by different colors in Fig. 1) are separated into different atom labels obtained from the input cif file. Differently-labelled atoms will be treated like different permutation species even though they denote the same element and/or are located in positions that are related by symmetry (in the latter case the program will break the symmetry and assign the atoms to different groups).
Stage II: Supercell generation
A new (super)cell is produced by replication of the initial one (Fig. 1b) over cell vectors (\(\vec {a},\,\vec {b},\,\vec {c}\)).2 The new cell (Fig. 1d) will have a size of (\(l \vec {a},\,m \vec {b},\,n \vec {c}\)), where \(l,\,m\) and \(n\) are natural (positive integer) numbers. Each crystallographic site p in the initial cell with Cartesian coordinates \(\vec {q}_p\) will have a total of \(l \cdot m \cdot n\) images in the supercell with Cartesian coordinates \(\vec {q}_p^{i,j,k}= \vec {q}_p + i \vec {a} + j \vec {b} + k \vec {c}\), where \(i=0,1\ldots (l-1),\,j=0,1\ldots (m-1)\) and \(k=0,1\ldots (n-1)\). It is important to keep in mind that this supercell expansion approach is a special case: the simplest one. It does not allow for example transformations of a primitive cell into a conventional (super)cell, or the opposite. A more general approach exists [33], which creates supercell vectors on the basis of linear combinations of the initial cell vectors, a desirable improvement that will be considered for a future version of the program.
The choice of the \(l,\,m\) and n supercell-expansion factors are strongly dependent on the initial cell shape, on the targeted properties and (of course) on the computational cost of the calculations needed to predict them. For three-dimensional solids it is often desirable to describe the effects of local disorder to the longest-possible range in all directions of space (or parallel to the cleavage plane for 2D systems). This typically requires that \(l,\,m\) and \(n\) values are inversely proportional to their respective initial cell parameters to maximize the shortest distance between periodic images in all directions and hence minimize finite-size effects. On the contrary, in other cases where a long-range effect in one direction may be expected, it can be preferable to build instead largely anisotropic supercells [34]. Finally, while properties such as the density of states may often be satisfactorily computed with small supercells [35], other properties like Raman require very large ones.
Stage III: Occupancy correction
Within the supercell approximation, the composition of the supercell should be as consistent as possible with the composition determined experimentally for the system under study. The third stage of the procedure implemented in our program involves some algorithms that have been specifically designed to help the user obtain the desired composition. This step consists in transforming the occupancy values of all atoms (represented by their labels) in every group into integer numbers of atoms distributed among all allowed positions in the group (Fig. 1e). In the supercell, these positions include all images of the sites assigned to the group by symmetry and by translation of the original cell. This transformation is a classical task of integer programming. The implemented algorithm minimizes the \(\chi ^2\) defined below with some restrictions.
$$\begin{aligned} \chi ^2=\sum _{i \in G}\sum _{j \in L_i }{\left( R_i^j-\frac{N_i^j}{S_i}\right) ^2} \end{aligned}$$
In this equation the first sum (i indexes) runs over all groups (\(G\)), and the second (\(j\) indexes) over all atom labels (\(L_i\)) associated with group \(i\). \(N_i^j\) are variables (numbers of atoms) used to minimize (2), with \(R_i^j\) the occupancy value provided in the input file3 and \(S_i\) the number of allowed positions in group i. A first restriction is that the sum of all atom occupancies in a group should be less than or equal to the number of possible positions in the group: \(\sum _jN_i^j \le S_i\). The second restriction comes from charge balancing. If the corresponding option is switched on ("-c yes") the combinations of \(N_i^j\) which do not satisfy the system charge-balance condition \(\sum _{i, j} q_i^j N_i^j=0\) will be discarded. One more restriction is used for fully-occupied groups, i.e., groups i such that \(\sum _j{R_i^j} = 1\). In such cases the group should remain fully occupied after the procedure, which translates to imposing that \(\sum _j{N_i^j} = S_i\). Finally, it is possible to set manually some of the \(N_i^j\) values (with the "−p" option, see the examples for details), which will then remain fixed during the minimization procedure. The minimization algorithm implemented in the supercell program is quite simple. It goes through all possible values of \(N_i^j\), calculating the \(\chi ^2\) value and discarding combinations of \(N_i^j\) that do not fulfil the restriction conditions. The set of \(N_i^j\) values that minimizes \(\chi ^2\) will be kept for the next step.
Stage IV: Processing and storing result structures
Once the occupancy number of all sites has been determined, supercell structures with real atoms and vacancies (rather than partially-occupied or mixed compositions) may be constructed accordingly (Fig. 1f). As mentioned above, the permutation proceeds for all groups independently, which results in a set of structures representing all possible combinations of all groups. Some atom groups can be excluded from this procedure manually, in which case the generated structures will retain partial occupancies for all crystallographic sites within these groups.4
As mentioned above, the number of possible combinations may quickly become very large, to the point where storing and analyzing the results can be problematic. Different strategies have been implemented in the supercell program to overcome this problem. The program offers different ways to process the output structures before storing them. First of all (this is the default), the structures can be stored directly as is. Secondly, it is strongly advisable to merge structures that are equivalent by symmetry or by translation (using the "−m" option), which tends to considerably reduce the number of structures (depending on the symmetry of the system). Thirdly, Coulomb energy calculations can additionally be performed ("−q" option) to sort out structures on the basis of their (crudely-approximated) energies. This ranking may then be used to sample the resulting structures according to different criteria ("−n" option), which proves a particularly useful tool when the number of distinct structures is high. The number N of combinations to store may be picked in any of the following ways (possibly at the same time): randomly, the first and/or last N combinations generated, and/or the N combinations with lowest or highest Coulomb energy. The sampling algorithm runs after the Coulomb energy calculations and merging algorithm and requires extra memory to temporarily store the sampled combinations.
Merging equivalent structures
The purpose of this algorithm is to identify and offer the possibility to store only unique structures, i.e., that cannot be converted to each other by affine transformation. This is a truly central aspect to the treatment of atomic disorder in crystals because, as will be illustrated in various examples discussed below, the total number of combinations can be reduced by up to three orders of magnitude using this procedure, depending on the symmetry of the system. Efficient approaches to perform this task have been found as a result of intense research efforts by several groups in the past [28–30], which largely benefited to the supercell program. The method implemented here is based on the use of symmetries, similar in essence to the approach discussed by Hart and Forcade [29] and consists of two stages. During the first stage all possible symmetry operations that apply to the supercell are identified. The second stage is a part of the permutation iteration loop. During this stage, the symmetries calculated before are applied to the current atom configuration.
The symmetry-search algorithm (stage one) generates symmetries on-the-fly and does not use any symmetry information from the input structure5 or symmetry databases (no information about space group or Bravais lattice). The symmetry-search algorithm can be split in two parts. First, the algorithm searches for all possible crystallographic point groups for the lattice, to identify all linear transformations (rotations, inversions...etc), which transform the cell to itself. This step is performed on a supercell from which all disordered sites whose distribution is under investigation (i.e., those not fixed by the user) have been removed. This leads to a maximum number of potential symmetry operations which may or may not apply to each individual atomic configuration, once atoms and vacancies are distributed onto these disordered sites. In a second step, the symmetries identified are applied to all atomic positions in the supercell, to then search for translation symmetries (i.e., shift vectors).
Although this approach is not compatible with the "standard space groups notation" it has clear advantages. The first and most important advantage is that the program can be used sequentially: the output cif files of one supercell run in which some of the groups have been fixed (hence retaining partial occupancies) can be used as inputs to another run. This possibility greatly extends the applicability as well as the potentialities of the program for large, highly-symmetric and/or complicated structures containing several disordered sites (a principle that is exploited in some of the examples discussed in Additional file 1). Another advantage of this approach is that non-standard cells can be used (including for example structures that have been edited manually).
During the permutation loop the obtained symmetry operations are applied to each permutation. Lexicographic order of permutations allows to combine symmetry-equivalent structures with a run time that scales linearly with the total number of permutations [36], as also used in the implementation by Hart and Forcade [29] (readers are encouraged to look therein for a clear illustration of this principle). A weight parameter reflecting the number of non-unique structures merged together is finally attributed to each unique structure.
Coulomb energy calculations
The Coulomb energy calculations implemented in the supercell program uses the Ewald summation algorithm, which achieves considerably increased accuracy as compared to a simple truncation for long-range potentials with \(1/r\) asymptotic behavior. This algorithm has been described in many sources, and our implementation is based on equations published in [37], with a relative precision of the energy calculations set to \(10^{-7}\). The result of our implementation of the Ewald sum was tested by comparing the calculated energies with the Ewald energies obtained with the GULP code [38], which they match perfectly.
The supercell program is written in standard C++ language and should be compatible with all popular compilers (MSVC, GNU, Intel Compiler, MinGW, Clang), operating systems (Windows, Mac OS X, Linux, BSD, Windows/Cygwin) and platforms. The version of the program, presented in the paper successfully compiled on Linux platforms with GNU C++ 4.x, Clang 3.4 and Intel® 14.0 compilers. We expect fewer risks of facing technical problems during installation procedures performed with GNU compilers. The CMake configure system is used for dependency check and cross-platform build. The code uses the Boost library for command-line argument processing, advanced file operations and REgular EXpression (regex) parsing. All operations on the structure of the compound (read/write CIF file, structure exploration and modification) are done with a customized version of the OpenBabel [39] library, which was modified by KO to handle partial occupancy in CIF file reading and writing operations. This version of OpenBabel is available from the official Website [40], and the corresponding changes will be merged to the official OpenBabel repository [41]. The supercell program also uses the libarchive library to archive output structures on-the-fly.
A manual of the program is available in the distribution as a LaTeX source. The manual can be compiled to md (Unix man) and pdf file. Compiling to Unix man is carried out with a modified version of the latex2man script [42] (distributed with the program). This script requires Perl to be installed. To compile the manual to PDF, the pdflatex program should be available with latex2man style files.
Some limitations of the code should be taken into account. The program currently only has a serial implementation and consequently does not exploit the advantages of multiprocessor machines. The implementation of the different algorithms is nevertheless very efficient in terms of CPU and memory load. It can work satisfactorily even on netbooks for all examples discussed in this article. The maximum number of permutations should not exceed a limit of 800 million, above which the program will return an error. Importantly, systems with a total number of combinations well above this limit can still be treated provided that the number of permutations within each group is below the limit. As mentioned above, the program can indeed be executed step-by-step with some groups excluded from permutation. Large supercells can cause the program to crash with arithmetic overflow when the rapidly-increasing number of combinations exceeds \(2^{63} \approx 10^{18}\), due to the finite size of integer variables.
At the moment the supercell program distributed as a source code only. It is freely available from https://github.com/orex/supercell. Users are very welcome to submit bugs, features requests, code improvements, examples etc. Snapshot of supercell program can also be found in Additional file 2 (supercell.tar).
The supercell code is available for everybody without restrictions, which gives a possibility for all users to check the code, improve it and customize the program for their needs, respecting the license. We kindly ask users to keep in touch with authors to help them improve the program.
As already mentioned above, several solutions exist to systematically explore all possible atomic configurations from the long-range average structure of disordered crystals, which are listed in Table 1. Although all of them share the same general purposes, their implementations, sets of related algorithms and possible applications are quite different.
Comparisons of programs performing combinatorial treatments of disorder in crystals
enumlib
eumlib + pymathgen
License and availability
GPL, open access
Commercial, free demo version available
GPL, on demand
MIT, open access
Fortran/Python
Custom configuration file
Python script
Input from standard structure files
Preprocessing algorithms
Grouping, occupancy correction
(Any)a
Non-diagonal supercell expansion matrix
Multinominal distribution
Disorder on several independent sites supported
Random sampling
Coulomb energy sampling
Interface to calculations programs
Externalc
Internal, CRYSTAL only
Internal, VASP, GULP
Internal VASP
Internal VASP, GULP
Performance \(\hbox {Sn}_{0.5}\hbox {Pb}_{0.5}\hbox {Te}\) \(2\times 2\times 2\))d
15 se
N/Af
\(^\mathrm{a}\) Pymathgen supports a wide range of structure manipulation procedures [43]
\(^\mathrm{b}\) Coulomb energy sampling and merge algorithm are mutually exclusive within this framework
\(^\mathrm{c}\) Input for most calculation programs can be prepared with shell scripts and cif2cell or OpenBabel
\(^\mathrm{d}\) The reported time is a dry-run time on Intel® Xeon® X5550 processor. All time-consuming I/O operations were disabled. The example is particularly challenging because the number of symmetry operations (1536) is really high (the same number of permutations on systems of lower symmetry should be processed faster)
\(^\mathrm{e}\) The reported duration corresponds to the calculation of the total number of unique structures calculation. The sampling algorithm crashed
\(^\mathrm{f}\) The program crashed with memory error. The expected run time is more than a year
The Site-Occupancy Disorder (SOD) program was historically first to implement such functionalities [28]. Since 2007, the code has been used in studies of several classes of disordered solid-state compounds. The enumlib program [29] was introduced in 2008 and primarily used thereafter for the treatment of compositional disorder in metallic alloys. It was later (2013) wrapped in the pymatgen library [43], which considerably extended the possibilities of the program, allowing to customize the input and output of the program, and offering a variety of tools for data analysis. The commercial CRYSTAL program [31], a well-known solution for DFT calculations, was very recently extended with a part dedicated to the treatment of disorder in crystals [30] (in version CRYSTAL14).
The command line interface (CLI) was chosen for supercell to create a user-friendly and ready-to-use, but also flexible program, rather than the custom input-file interface used in SOD, CRYSTAL14 and enumlib codes. The CLI is suitable both for manual user input and scripting automation and, combined with a verbose output, allows to set output data properties easily and step-by-step. The pymatgen library offers a very powerful Python-script approach to use enumlib, which is extremely flexible, but requires at least a basic level of programming skills as well as knowledge of the pymatgen library structure and common templates. We note that the functionalities offered by pymatgen can also easily be used with our program. As most other solutions the supercell program is distributed with a manual (CLI parameters manual: Unix man) and a set of application examples illustrating all features of the program, which are described in detail below (and in Additional file 1).
The functionalities offered by the different programs also present several significant differences. Much attention has been paid in supercell to facilitate the initial structure processing steps: direct input from cif files, the grouping functionality, and a flexible occupancy correction procedure. Other programs require that users perform these operations manually (CRYSTAL14, SOD, enumlib) or create a Python script for this (pymatgen). Contrarily to other programs supercell is not distributed with extra tools for output structures analyses and conversion, and authors recommend to use external tools like OpenBabel [39] and cif2cell [44] for the preparation of DFT calculation inputs and fpNMR [45] for structure analyses.6 Finally, in contrast with SOD and CRYSTAL14 codes, which are limited to compounds having only one disordered crystallographic site involving two atom types (or one atom type and vacancies), Supercell, like enumlib, can handle multiple sites with complex compositions.
Some features of the supercell code have been developed specifically to facilitate the treatment of the (very frequent) cases where large numbers of permutations are encountered (typically \(10^{5}\)–\(10^{9}\)). The code was tested for \(\hbox {Sn}_{0.5}\hbox {Pb}_{0.5}\hbox {Te}\) with a supercell consisting of \(2\times 2\times 2\) conventional cells. Both SOD and CRYSTAL code crashed during the run, although the CRYSTAL14 program showed excellent performance as far as the calculation of the total number of unique structures is concerned. Supercell was 60 times faster than enumlib on this particular example. These results suggest that, at least to the best of our knowledge, the supercell program is currently the best solution for cases with a large number of permutations.
Distinct configurations generated with a given supercell size are expected to give rise to different calculated properties, but their numbers will often be too large to make systematic calculations practical. The crucial question for many research problems is therefore: how to select the configuration(s) that are most representative of the real system or/and how to average calculated properties from the set of calculated configurations? This question stretches far beyond the choice of the combinatorial approach and the answer will strongly depend on the type disorder present in the materials of interest. The next section illustrates the possibilities and limitations, advantages and disadvantages of the of supercell approach through examples based on data published in the literature, and representative of important types of compositional disorder in crystals. More examples can be found in Additional file 1.
Solid solutions: atomic substitutions in semiconductors
The atomic impurities and substitutions in semiconductors are crucial for building semiconductor electronic devices, lasers, thermoelectric materials etc. A typical example is the narrow-band semiconductor PbTe, which, when doped with tin, is used as an infra-red detector [46, 47]. The crystal structure of this compound is shown on Fig. 2. This IV–VI rocksalt semiconductor alloy system has a NaCl structure type, with typically fully-random atomic substitutions taking place at the 4b position. Some recent studies used an ab initio approach to investigate the band-structure properties of this system as a function of the amount and type of doping atoms, including Ga, In, Tl [48] or Sn [35].
Cubic crystal structure of \(\hbox {Sn}_\mathrm{x} \hbox {Pb}_{1-\mathrm{x}}\hbox {Te}\) [60]. a original unit cell with space group \(\hbox {Fm}\bar{3}\hbox {m}\) (225), and \(a = b = c = 6.32\text {-}6.46\) Å. The structure consists of mixed lead and tin (in dark gray and dark red, respectively) sites and tellurium sites (brass-colour) occupying 4a (0, 0, 0) and 4b \(\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\) positions with the same point symmetry \(\hbox {m}\bar{3}\hbox {m}\). b, c Two out of 8 possible unique structures for \(x=\frac{1}{2}\) and cell \(1\times 2\times 1\) (see Table 2). b The highest-symmetry structure corresponds to tetragonal \({\text{P}}4/{\text{mmm}}\) space group (123), whose corresponding primitive cell is highlighted with black lines. c Lowest-symmetry structure with orthorhombic \({\text{Pmm}}2\) space group (25)
During the alloy formation process, the minimization of the Gibbs energy (\(G=H-TS\)) of the system involves a competition between the enthalpy \(H\), which promotes ordering in the system, and entropy S, which drives it towards a fully disordered state. The result of this process depends on the alloy composition. It is important in this context to be able to generate both ordered and disordered configurations, and the supercell program can be a valuable tool for this purpose. In the particular case of \(\hbox {Sn}_\mathrm{x}\hbox {Pb}_{1-\mathrm{x}}\hbox {Te}\) compounds, the program was used with supercell sizes up to \(2\times 2\times 2\) (based on the conventional cell containing 8 atoms) and various concentrations of dopants (Table 2). These concentrations and supercell sizes were taken to reflect the structures calculated in Refs. [35, 48]. The symmetry-merging algorithm significantly reduces in all cases the number of configurations to treat. For most concentrations and supercell sizes presented in the table, it is feasible to conduct electronic-level calculations (i.e., DFT) of the desired properties for all atomic configurations. But in some cases such as, for instance, the \(2\times 2\times 2\) supercell with \(x=\frac{1}{2}\), the number of configurations is too high (larger than \(4\times 10^5\)) for this, and sampling strategies must consequently be employed.
Total number of possible atom combinations for different substitution levels x and different supercell sizes of the \(\hbox {A}_\mathrm{x} \hbox {Pb}_\mathrm{1-x}\hbox {Te}\) system
Cell, \(a\times b \times c\)
\(1\times 1 \times 1\)
Symmetry operations
\(x=\frac{1}{16}\)
\(x=\frac{1}{8}\)
35,960 (71)
12,870 (153)
601,080,390 (404,582)
N is the total number of atoms in the supercell. The number of unique (non-symmetric) combinations are given in parenthesis. The total number of combinations depends only on N, whereas the number of unique combinations can depend on the supercell formula \(a\times b \times c\). The number of combinations for substitutions \(1-x\) is equal to the number of combinations for \(x\) and are consequently not shown
Ordered structures
Order-disorder transitions are quite common in solid solutions [49]. A recent study showed in particular that the enthalpy of many type IV–VI solid-solution compounds was lower in the ordered state than in disordered states [50]. Such ordered configurations can easily be obtained with the supercell program. This is illustrated here for the supercell \(1\times 2\times 1\) of the \(\hbox {Sn}_{0.5}\hbox {Pb}_{0.5}\hbox {Te}\) system. As can be seen from Table 2, the total number of symmetry-unique structures for this case is 8. All structures can be easily be generated and processed. Although the supercell program does not produce information about spacegroup and/or primitive cell of the structures, the configurations of high and low symmetry can be easily separated on the basis of the weight of the structure (i.e., the number of configurations equivalent by symmetry and/or translation to the considered structure). The number of symmetry operations for configuration i can be calculated with the formula \(N{/}w_i\), where N is the total number of symmetry operations and \(w_i\) the weight of structure with index i. The two configurations of highest and lowest symmetry obtained with the supercell \(1\times 2\times 1\) of the \(\hbox {Sn}_{0.5}\hbox {Pb}_{0.5}\hbox {Te}\) system are shown on Fig. 2b, c, respectively. They are each characterized by 64 and 4 symmetry operations in the supercell representation, respectively. An analysis of the high-symmetry structure with a crystallographic visualization software indicates that it reduces to tetragonal \({\text{P}}4/{\text{mmm}}\) space group (123) with a 4-atom primitive cell shown in black in Fig. 2b whereas the lowest-symmetry structure corresponds to space group \({\text{Pmm}}2\).
Random disorder: special quasirandom structures
The most intuitive approach to model fully-random atom substitution is to randomly pick up a certain number of structures from the full set. This approach, however, is not efficient in the sense that a large number of randomly-selected structures are necessary to reliably predict average properties, which implies a large set of computationally-demanding electronic calculations. Zunger et al. showed that the properties of fully-random alloys can be obtained in a considerably more efficient manner by constructing non-random configurations called "special quasirandom structures" (SQS) [27]. SQS are special configurations that, for a certain supercell size, reproduce as closely as possible a set of close-range radial correlation functions (a simplified definition is given below) of the wholly-random system.
The combination of the Supercell program with a structure-analysis tool can be used to calculate the k-atom correlation functions (where k is typically limited to 2 or 3-atom interactions), to then identify the configurations that satisfy the SQS criteria. An implementation of this method for the \(\hbox {Sn}_{0.5}\hbox {Pb}_{0.5}\hbox {Te}\) system is provided as an example in the supercell program distribution. In this simple example, which has a single disordered site with mixed 50 %/50 % Sn/Pb composition, these correlation functions may be described with the following formulas, which are a simplification of the general theory for binary alloys considering two-atom correlations only. We first introduce the parameter \(S_i\) (with i the atom index), which takes the value +1 when i is an atom of type A and −1 for an atom of type B. In this case, the atom-pair correlation function can be calculated as:
$$\begin{aligned} \bar{\Pi }_{2,m} = \sum _{i,j \in r_{ij}=R_m } S_iS_j \end{aligned}$$
where the first index in \(\bar{\Pi }\) means that it is a two-atom correlation, the second index enumerates the coordination spheres of atoms, \(R_m\) is the radius of mth correlation sphere and \(r_{ij}\) is the distance between atoms i and j. The summation goes over all pairs i, j with distance between atoms equal to \(R_m\) (which takes discrete values in the average crystal system). Generally, the correlation functions for fully-random binary alloys can be calculated analytically with the formula \(\bar{\Pi }_{k,m}=(2x-1)^k\), where \(x=\frac{N_A}{N_A+N_B}\) is the substitution rate. For the special case of a fully-random alloy where \(x=0.5\) (equal numbers of A and B atoms), these correlation functions all reduce to zero. So the configurations matching the SQS criteria in this case should have null correlation functions in their first n coordination spheres, where the number n should be as high as possible. Out of the 153 distinct configurations identified for supercell size \(1\times 2\times 2\), our analysis of the \(\hbox {Sn}_{0.5}\hbox {Pb}_{0.5}\hbox {Te}\) system found 6 structures with null correlation coefficients for the first 3 shells (ignoring Te atoms), which means that their local Pb/Sn arrangements perfectly mimic those of a randomly-disordered system for cation-cation distances up to \(\approx\)8 Å.
The proposed approach calculates correlation functions for all possible configurations and is therefore technically limited to small structures.7 Traditionally, researchers use the alloy-theoretic automated toolkit (ATAT) package [51] to work with random alloys. The package is highly oriented towards this type of disordered systems and offers many useful functionalities such as the cluster expansion [52], SQS structure generation (both "brute-force" and stochastic algorithms) and interface to ab initio codes. The supercell program targets a broader range of systems, which potentially includes but is not limited to the randomly-disordered systems targeted by the SQS approach, and any type of target correlation function could be considered to favor or penalize instead some chosen interactions (e.g., A-A first-neighbor interactions in the simple two-atom example discussed above). The exhaustive searches performed by supercell and the set tools that is contains are particularly adapted to the treatment of non-random types of disorder, which are the focus of the next two sections.
Correlated disorder: ice \(\hbox {I}_\mathrm{h}\)
In some disordered systems the sitings of atoms onto the different disordered crystallographic sites are not independent from each other, but follow instead some restrictions. Correlated disorder appears when the restrictions impose a long-range correlation in atom locations, i.e., when the placement of one atom or a vacancy imposes or prevents the placement of another at a nearby position, which in turns affects the next position, etc. A well-known example of such type of disorder is the common form of ice (\(\hbox {I}_\mathrm{h}\)). Many specific properties of ice, including its residual entropy, large static dielectric permittivity, electrical polarizability and conductivity can be attributed to proton disorder [53]. The crystal structure of ice \(\hbox {I}_\mathrm{h}\) (as given in the cif file) is shown on Fig. 3. The system has a hexagonal lattice with a well-defined oxygen position in site 4f. Hydrogen atoms, on the other hand, occupy two distinct positions, each with 50 % probability: H1 (4f) and H2 (12k). The placement of hydrogen atoms should obey two rules. First, ice being a crystal made of water molecules, the number of H atoms in the first coordination sphere of every O atom should always be two. Second, the placement of a H atom depends on the presence (or absence) of another one on the close-by H site, whereby H–H close contact (dashed line in Fig. 3) is avoided. The disorder in this system is therefore clearly correlated: the placement of one H atom directly affects the placement of all other H atoms around it, which in turn affects the placement of others, etc. The comprehensive exploration of atomic configurations implemented in supercell can be used to generate structures that are then selected on the basis of the two criteria described above.
Crystal structure of ice \(\hbox {I}_\mathrm{h}\) [61] with correlated disorder. a original unit cell with space group \({\hbox {P}{6}_3/mmc}\) (194), \(a\, = \,b\, =\, 4.497479\) Å and \(c = 7.322382\) Å \((\alpha = \beta = 90^{\circ }, \gamma = 120^{\circ }\)). All O atoms (red) are on fully-occupied 4f site. H atoms have two positions: H1 4f (green) and H2 12k (gray), both with 50 % occupancy (see text). Dashed line shows unrealistic H–H distance of \(\approx\)0.8 Å. In b and c are two configurations satisfying the restrictions on H atoms positions that result in correlated disorder in this system, with space groups \(\hbox {Cmc2}_1\) (36) and c \({\hbox{Cc}}\) (9), respectively
Running the supercell program for the initial cell of ice (\(\hbox {I}_\mathrm{h}\)) yields a total of 5544 combinations, which reduces to 288 distinct configurations upon application of the symmetry-merging algorithm. The configurations were analyzed with the GULP program [38], which revealed that 30 out of the 288 structures have all their O atoms 2-coordinated, 23 have no H–H close contact, and only 2 structures satisfy both conditions simultaneously. Coulomb energy calculations (using the "−q" option with O and H charges set to −2 and +1, respectively) shows that these 2 structures have (not surprisingly) the lowest Coulomb energy, which illustrates the potential of this criterion for a pre-selection of structures in larger systems with different oxidation states on the substitution sites.
The configuration-generation step becomes more complex for cells larger than the initial one. For cell \(1\times 2\times 1\), for example, the total number of unique configurations will be 11 million, which is too large to process in the same way as the initial cell (this also applies of course to \(2\times 1\times 1\) and \(1\times 1\times 2\) supercells, but because the initial c parameter is larger than parameters a and b, the latter would be less interesting because of its strong anisotropy). The systematic structure-analysis approach (using GULP) described above reaches its limit considerably more quickly than the supercell program, which is able to generate ice \(\hbox {I}_\mathrm{h}\) configurations for supercells up to size \(2\times 2\times 1\) (\(\approx\) 4.2 \(\times 10^{17}\) combinations). This was nevertheless achieved, using a quite specific procedure that requires advanced understanding of combinatorics and of the algorithm implementation within the supercell program, and is consequently presented in Additional file 1. The results reveal that only 9 out of a total number of 11 million distinct \(1\times 2\times 1\)- (or \(2\times 1\times 1\)-) supercell configurations satisfy both the H–H avoidance and the presence of H\(_{2}\)O molecules only (see Additional file 1 for details).
The necessity to keep only configurations containing H\(_{2}\)O molecules is furthermore representative of another type of disorder that may be found in many molecular crystals or in systems consisting of guest molecules confined within a crystalline host matrix. The program should ideally be able to retain (or even better to explore) only the configurations where certain specified molecular fragments would be present. This is done as an external post-processing step in the \(\hbox {I}_\mathrm{h}\) example, with a procedure that may easily be extended to other systems. While this is an appealing functionality to possibly include in future developments of the supercell program, it should be done without loss of generality or ease-to-use, which makes it very challenging (and far beyond the scope of the present work).
Short-range order: the Loewenstein's rule in aluminosilicates
The disorder in many materials is neither random nor correlated. This intermediate case is characterized by large amounts of possible stable configurations, as in randomly-disordered crystalline systems. From another side, however, the difference in energy between different configurations is significant, as in correlated disordered materials, but to a smaller extent. The main difference between this type of compounds and systems with correlated disorder is that many local configurations of higher energy are stable enough to be present in the real system. In ice \(\hbox {I}_\mathrm{h}\), in contrast, structures with short H–H distances, which obviously have high energies, will be transformed to one of the low-energy configurations during an energy minimization procedure. Another scenario is the case where a propensity to local ordering (possibly a strict one) applies at the local level (typically two ions strongly repelling each other), but where the concentration of defects is such that this local ordering does not propagate further than a few atomic shells.
A typical example of such "intermediate" disorder is the Al–O–Al avoidance rule (also called the Loewenstein's rule [54]) between 4-coordinated Al atoms in alumino-silicate materials. The rule dictates that the Si/Al atom substitution in these systems tends to minimize the number of such Al–O–Al bonds, thus minimizing the total energy of the system. In contrast with the correlated disorder of the ice \(\hbox {I}_\mathrm{h}\) system discussed above, where violations of the structural restrictions yield unstable and hence forbidden configurations, structures that contain Al–O–Al bonds may very well be stable, albeit energetically unfavorable. And as a matter of fact the Loewenstein's rule is not strict and is violated in many systems [55]. The amount of violations typically increases with the synthesis temperature, which maximizes the effect of the entropy contribution to the Gibbs free energy, and with the cooling rate, which should be fast enough to limit local atomic rearrangements in the favor of Si–O–Al bonds.
The Si/Al disorder in the gehlenite system Ca\(_{2}\)Al\(_{2}\)SiO\(_{7}\), for example, was investigated in depth by Florian et al. with a combination of experimental NMR and DFT calculations [21]. Their work beautifully illustrates how a very simple type of disorder (50/50 mixed Al–Si composition on one crystallographic site) results in great extents of complexity at the local level, as probed with local spectroscopic techniques (NMR in this case). Such a complexity requires the combination of many advanced experimental and computational tools to decipher, and is furthermore expected to quickly increase as the number of disordered sites increases. This highlights the essential role of modelling to unravel the molecular-structure-property relationships in such materials. Figure 4a shows the layered structure of Ca\(_{2}\)Al\(_{2}\)SiO\(_{7}\), which consists of aluminosilicate sheets separated by Ca layers. The Si–Al substitution takes place at tetrahedral positions (T2 sites, shown as blue polyhedrons) forming pairs of two connected tetrahedra that are only connected to pure-Al T1 tetrahedral sites (shown in yellow). Due to this particular arrangement of T1 and T2 sites, the number of Al(T1)–O–Al(T2) connectivities in the system is constant (4 per cell) and the total number of Al–O–Al bonds is entirely determined by the number of Al(T2)–O(2c)–Al(T2) connections, i.e., the numbers of pairs of T2 sites occupied by two Al atoms.
Crystal structure of Ca2Al2SiO7 viewed a from the side and b from above layers and Al–O–Al bonds energy plot (b). a, b Tetragonal cell with spacegroup \(\hbox {P}\bar{4}{2}_\mathrm{1m}\) (113) and a = b = 7.716 Å and c = 5.089 Å. Wyckoff site 2a labelled T1 (yellow tetrahedra) is fully occupied by Al atoms and Wyckoff site 4e, labelled T2 (blue tetrahedra) is filled with a mix of Al (50 %) and Si (50 %) atoms. c Energy of configurations for cell \(1\times 1\times 2\) vs the number of Al–O–Al bonds in isolated groups. The slope of linear regression (0.48 eV) agrees well with previously reported value of 0.52 eV [21]
The supercell program was applied to a Ca\(_{2}\)Al\(_{2}\)SiO\(_{7}\) supercell of size \(1\times 1\times 2\), which contains 8 positions with mixed Al/Si composition, over four T2–T2 tetrahedron pairs. The total number of possible permutations is \(C_8^4=70\), which reduces to 10 distinct configurations. The total number of Al–O–Al bonds within each group varies from 8, when all T2–T2 pairs have one Al and one Si atom, to 10, when two of the pairs are occupied with only Al atoms and the other two with only Si atoms. A geometry optimization procedure (atomic positions only, see "Appendix" for details) was applied to each structure to obtain the total energy as a function of the number of Al–O–Al bonds that the structure contains. The results are presented on Fig. 4c, which shows a strong correlation between the total energy of the structures and the number of Al–O–Al bonds, which provides an estimation of the energetic cost of such bonds, in good quantitative agreement with previously-reported results and with the Loewenstein's rule [21]. This good agreement was rather unexpected given the exaggeratedly small system chosen here to illustrate how the supercell program may be used in such a context. The models used in Ref. [21] were based on much bigger supercells of size \(2\times 2\times 3\) (leading to over \(3\times 10^{13}\) configurations before symmetry merging), which were constructed "manually". It is important to realize that even such a big system could be treated by the supercell program, using a procedure that is described in Additional file 1.
In the present paper we describe the theory and methodology of the supercell approximation for the modelling of crystalline structures with compositional (i.e., chemical) disorder. The new supercell program contains an all-in-one implementation of a full set of algorithms that are crucial to conduct this type of analyses, all of which have been described here. It offers a good alternative to other software solutions dedicated to the treatment of disorder in crystals, which provided opportunities to fully test and validate our algorithms' implementations. The efficient exhaustive search over all possible configurations performed by supercell for small- to medium-size systems makes it possible to describe in depth the distributions of local compositions and resulting local geometrical distortions in systems with random as well as non-random types of disorder. The free open-source license of supercell as well as its compatibility with any ab initio code via the OpenBabel and Cif2cell package and use of CIF as input and output format, makes it a valuable alternative to the implementation embedded in the commercial CRYSTAL package [31]. We show that it offers significantly improved performance and versatility over the enumlib [29] and SOD program [28] (other comparable free open-source solutions) as well as additional tools for the crucial step of structure selection.
The supercell program should be treated as a completed software, even though we encourage users to suggest and/or implement improvements that would keep and possibly improve the broadness of its applicability. The released version has well-tested algorithms, a powerful command-line user interface that makes it easily embedded into shell scripts, and support material, including a manual and examples. It can be applied to a wide range of important materials whose properties are crucially impacted by atomic substitutions characterized by different types and extents of local to long-range (dis)order, some of which have been illustrated here (in the main text and/or in Additional file 1). This includes semiconductors and various oxides (including piezoelectric ceramics, see Additional file 1) and chalcogenides, but the list is far longer than this and may include for example ion-conducting materials and many others. Future developments will focus on improved ways to solve the problem of structure selection from large sets of configurations. For the moment this is done solely on the basis of fast and universal Coulomb energy calculations, which for many compounds can give good results [57]. Significant improvement can however be done by total energy calculations of each configuration using classical force fields, which by definition are not universal. This task is therefore very challenging, but would broadly increase the ability to solve complex Materials Science problems based on the supercell program. Other ways to improve the program would be to embed the SQS search functionality (which was done with an external program in the \(\hbox {Sn}_{0.5}\hbox {Pb}_{0.5}\hbox {Te}\) example discussed here), and to support non-diagonal supercell expansion matrices to explore different supercell shapes.
Project name: supercell
Project home page: https://github.com/orex/supercell
Operating system(s): Linux, can be compiled for other systems.
Programming language: C++
Other requirements: Boost 1.46 or higher, CMAKE 2.6 or higher, OpenBabel [40, 41], Eigen 3.x, LibArchive (optionally). Perl and LaTeX are needed to compile manual.
License: GNU GPL
Any restrictions to use by non-academics: according GNU GPL terms
Strictly speaking, permutations and combinations are a special case of multinomial distributions. But further below we will use these terms as synonyms of multinomial distribution sampling.
For simplification purposes, the figure illustrates a 2D cell, whereas the text describes the procedure in three dimensions.
In this procedure, vacancies are not treated like "special atoms", so \(N_i^j\) are numbers of real atoms in the groups. The number of vacant positions in group i can be calculated as \(V_i=S_i-\sum _j{N_i^j}\).
All sites in each group that the user wants to freeze should be specified as fixed, otherwise the program will return an error.
Regular cif files with space group information may of course be provided as the input, but this structure will be converted by OpenBabel library to a structure with P1 space group, which our program will use.
Although the fpNMR program is strongly oriented towards DFT calculations of NMR parameters, it offers a broad range of general functionalities for structure analyses.
For bigger systems is always possible to randomly pick a reasonable number of configurations to calculate the corresponding correlation coefficients, but the probability of finding good candidate SQSs (and even more so to find the best ones) would then be small. In this case it is better to directly construct the SQSs or use stochastic generation approach [56].
KO is the co-designer and developer of the program. SC co-designed, tested, and contributed to improving the program. All DFT calculations and structure analyses were performed by KO and TC. All authors read and approved the final manuscript.
We are grateful to Ricardo Grau-Crespo for the provided SOD code and maghemite input files. KO and SC are grateful to Franck Fayon (CEMHTI-CNRS, Univ. Orléans) for testing the program and for discussions that contributed to its improvement, and in particular the addition of Coulomb-energy calculations. We thank the French Agence Nationale de la Recherche for funding by Grants ANR-11-MONU-003-ExaviZ (KO and SC) and ANR-13-BS08-0012-DyStrAS (KO and TC), and to the "Centre de Calcul Scientifique en Région Centre" (CCSC, Orléans, France) for access to computational resources. We are also grateful to anonymous reviewer, who suggested applications on piezoelectric \(\hbox {PbZr}_{\mathrm{x}}\hbox {Ti}_{\mathrm{1-x}}\hbox{O}_3\) (PZT) ceramics to which our program can be successfully applied (see Additional file 1). Figures 2, 3 and 4 a, b were done in VESTA program [59].
Appendix: Ca\(_{2}\)Al\(_{2}\)SiO\(_{2}\) energy calculation details
The geometry optimizations were performed with fixed unit cell parameters using the plane-wave based DFT with pseudopotential implemented in the VASP code [58]. The input parameters were optimized to achieve a maximum performance with the required energy precision. The energy cut-off was 500 eV; k-points mesh was \(4\times 4\times 3\). Symmetrization was switched off. Ionic positions were optimized using the conjugate gradient method. Convergence threshold criterion was that all forces should be <0.005 eV/Å. We note that the energy of geometry-optimized structures is only an approximation to the total energy, which consists of both configurational (potential) and kinetic energies. More precise results could therefore in principle be obtained by replacing the energy optimizations by molecular dynamic runs (at the DFT level) at the target temperature, but the CPU-time requirements of such calculations make it irrelevant in the context of this work.
13321_2016_129_MOESM1_ESM.pdf Additional file 1. tutorial.pdf --- Short introduction to supercell program.
13321_2016_129_MOESM2_ESM.tar Additional file 2. supercell.zip --- The snapshot of supercell code from github.
NIMBE, CEA, CNRS, Université Paris-Saclay, CEA Saclay, 91191 Gif-sur-Yvette, France
CEMHTI - UPR3079 CNRS, Site Haute Température, 1D avenue de la Recherche Scientifique, 45071 Orléans Cedex 2, France
Present address: Institut des Matériaux Jean Rouxel (IMN), Université de Nantes, CNRS, 2 rue de la Houssinière, BP32229, 44322 Nantes Cedex 3, France
Kitaigorodskiy AI (1967) Order and disorder in the world of atoms. The Heidelberg Science Library. Springer, New York, NY. doi:10.1007/978-1-4615-7559-7 View ArticleGoogle Scholar
Schock RN (ed) (1985) Point defects in minerals. Geophysical Monograph Series, vol 31. American Geophysical Union, Washington, DC. doi:10.1029/GM031 Google Scholar
Kelly A, Knowles KM (2012) Crystallography and crystal defects. Wiley, Chichester. doi:10.1002/9781119961468 Google Scholar
Martineau C, Senker J, Taulelle F (2014) NMR crystallography, pp 1–57. doi:10.1016/B978-0-12-800184-4.00001-1. http://linkinghub.elsevier.com/retrieve/pii/B9780128001844000011
Tuschel D (2012) Raman crystallography in theory and in practice. Spectroscopy 27(3):22–27Google Scholar
Ashbrook SE, Dawson DM (2013) Exploiting periodic first-principles calculations in NMR spectroscopy of disordered solids. Acc Chem Res 46(9):1964–1974. doi:10.1021/ar300303w View ArticleGoogle Scholar
Massiot D, Messinger RJ, Cadars S, Deschamps M, Montouillout V, Pellerin N, Veron E, Allix M, Florian P, Fayon F (2013) Topological, geometric, and chemical order in materials: insights from solid-state NMR. Acc Chem Res 46(9):1975–1984. doi:10.1021/ar3003255 View ArticleGoogle Scholar
Charpentier T (2011) The PAW/GIPAW approach for computing NMR parameters: a new dimension added to NMR study of solids. Solid State Nucl Magn Reson 40(1):1–20. doi:10.1016/j.ssnmr.2011.04.006 View ArticleGoogle Scholar
Bonhomme C, Gervais C, Babonneau F, Coelho C, Pourpoint F, Azaïs T, Ashbrook SE, Griffin JM, Yates JR, Mauri F, Pickard CJ (2012) First-principles calculation of NMR parameters using the Gauge including projector augmented wave method: a chemists point of view. Chem Rev 112(11):5733–5779. doi:10.1021/cr300108a View ArticleGoogle Scholar
Sideris PJ, Nielsen UG, Gan Z, Grey CP (2008) Mg/Al ordering in layered double hydroxides revealed by multinuclear NMR spectroscopy. Science 321(5885):113–117. doi:10.1126/science.1157581 View ArticleGoogle Scholar
Cadars S, Layrac G, Gérardin C, Deschamps M, Yates JR, Tichit D, Massiot D (2011) Identification and quantification of defects in the cation ordering in Mg/Al layered double hydroxides. Chem Mater 23(11):2821–2831. doi:10.1021/cm200029q View ArticleGoogle Scholar
Cadars S, Guégan R, Garaga MN, Bourrat X, Le Forestier L, Fayon F, Huynh TV, Allier T, Nour Z, Massiot D (2012) New insights into the molecular structures, compositions, and cation distributions in synthetic and natural montmorillonite clays. Chem Mater 24(22):4376–4389. doi:10.1021/cm302549k View ArticleGoogle Scholar
Zhao L, Qi Z, Blanc F, Yu G, Wang M, Xue N, Ke X, Guo X, Ding W, Grey CP, Peng L (2014) Investigating local structure in layered double hydroxides with \(^{17}\text{O}\) NMR spectroscopy. Adv Funct Mater 24(12):1696–1702. doi:10.1002/adfm.201301157 View ArticleGoogle Scholar
Neuhoff PS, Kroeker S, Du LS, Fridriksson T, Stebbins JF, Tebbins JOFS (2002) Order/disorder in natrolite group zeolites: a \(^{29}\text{Si}\) and \(^{27}\text{Al}\) MAS NMR study. Am Mineral 87(10):1307–1320View ArticleGoogle Scholar
Shin J, Bhange DS, Camblor MA, Lee Y, Kim WJ, Nam I-S, Hong SB (2011) Tetrahedral atom ordering in a zeolite framework: a key factor affecting its physicochemical properties. J Am Chem Soc 133(27):10587–10598. doi:10.1021/ja202739j View ArticleGoogle Scholar
Garaga MN, Hsieh M-F, Nour Z, Deschamps M, Massiot D, Chmelka BF, Cadars S (2015) Local environments of boron heteroatoms in non-crystalline layered borosilicates. Phys Chem Chem Phys. doi:10.1039/C5CP03448E Google Scholar
Allix M, Chenu S, Véron E, Poumeyrol T, Kouadri-Boudjelthia EA, Alahraché S, Porcher F, Massiot D, Fayon F (2013) Considerable improvement of long-persistent luminescence in germanium and tin substituted ZnGa\(_2\)O\(_4\). Chem Mater 25(9):1600–1606. doi:10.1021/cm304101n View ArticleGoogle Scholar
Ashbrook SE, Mitchell MR, Sneddon S, Moran RF, de los Reyes M, Lumpkin GR, Whittle KR (2015) New insights into phase distribution, phase composition and disorder in Y\(_2\)(Zr, Sn)\(_2\)O\(_7\) ceramics from NMR spectroscopy. Phys Chem Chem Phys 17(14):9049–9059. doi:10.1039/C4CP05827E View ArticleGoogle Scholar
Clément RJ, Pell AJ, Middlemiss DS, Strobridge FC, Miller JK, Whittingham MS, Emsley L, Grey CP, Pintacuda G (2012) Spin-transfer pathways in paramagnetic lithium transition-metal phosphates from combined broadband isotropic solid-state MAS NMR spectroscopy and DFT calculations. J Am Chem Soc 134(41):17178–17185. doi:10.1021/ja306876u View ArticleGoogle Scholar
Strobridge F, Middlemiss D (2014) Characterising local environments in high energy density Li-ion battery cathodes: a combined NMR and first principles study of \(\text{LiFe}_{\rm x}\text{Co}_{1-{\rm x}}\text{PO}_4\). J Math. doi:10.1039/c4ta00934g
Florian P, Veron E, Green TFG, Yates JR, Massiot D (2012) Elucidation of the Al/Si ordering in gehlenite Ca\(_2\)Al\(_2\)SiO\(_7\) by combined \({^{29}\text{Si}}\) and \({^{27}\text{Al}}\) NMR spectroscopy/quantum chemical calculations. Chem Mater 24(21):4068–4079. doi:10.1021/cm3016935 View ArticleGoogle Scholar
Veron E, Garaga MN, Pelloquin D, Cadars S, Suchomel M, Suard E, Massiot D, Matzen G, Allix M (2013) Synthesis and structure determination of \(\text{CaSi}_{1/3}\text{B}_{2/3}\text{O}_{8/3}\): a new calcium borosilicate. Inorg Chem 52(08):4250–4258. doi:10.1021/ic302114t View ArticleGoogle Scholar
Choubrac L, Paris M, Lafond A, Guillot-Deudon C, Rocquefelte X, Jobic S (2013) Multinuclear (\({^{67}\text{Zn}}, {^{119}\text{Sn}}\) and \({^{65}\text{Cu}}\)) NMR spectroscopy-an ideal technique to probe the cationic ordering in Cu\(_2\)ZnSnS\(_2\) photovoltaic materials. Phys Chem Chem Phys 15(26):10722–5. doi:10.1039/c3cp51320c View ArticleGoogle Scholar
Paris M, Choubrac L, Lafond A, Guillot-Deudon C, Jobic S (2014) Solid-State NMR and Raman spectroscopy to address the local structure of defects and the tricky issue of the Cu/Zn disorder in Cu-poor, Zn-rich CZTS materials. Inorg Chem. doi:10.1021/ic5012346
Park S, Hu Y, Hwang JO, Lee E-S, Casabianca LB, Cai W, Potts JR, Ha H-W, Chen S, Oh J, Kim SO, Kim Y-H, Ishii Y, Ruoff RS (2012) Chemical structures of hydrazine-treated graphene oxide and generation of aromatic nitrogen doping. Nat Commun 3:638. doi:10.1038/ncomms1643 View ArticleGoogle Scholar
Freysoldt C, Grabowski B, Hickel T, Neugebauer J, Kresse G, Janotti A, Van De Walle CG (2014) First-principles calculations for point defects in solids. Rev Mod Phys 86(1):253–305. doi:10.1103/RevModPhys.86.253 View ArticleGoogle Scholar
Zunger A, Wei SH, Ferreira LG, Bernard JE (1990) Special quasirandom structures. Phys Rev Lett 65(3):353–356. doi:10.1103/PhysRevLett.65.353 View ArticleGoogle Scholar
Grau-Crespo R, Hamad S, Catlow CRA, Leeuw NHD (2007) Symmetry-adapted configurational modelling of fractional site occupancy in solids. J Phys Condens Matter 19(25):256201. doi:10.1088/0953-8984/19/25/256201 View ArticleGoogle Scholar
Hart GLW, Forcade RW (2008) Algorithm for generating derivative structures. Phys Rev B Condens Matter Mater Phys 77(22):1–12. doi:10.1103/PhysRevB.77.224115 View ArticleGoogle Scholar
Mustapha S, D'Arco P, De La Pierre M, Noël Y, Ferrabone M, Dovesi R (2013) On the use of symmetry in configurational analysis for the simulation of disordered solids. J Phys Condens Matter Inst Phys J 25(10):105401. doi:10.1088/0953-8984/25/10/105401 View ArticleGoogle Scholar
Dovesi R, Orlando R, Erba A, Zicovich-Wilson CM, Civalleri B, Casassa S, Maschio L, Ferrabone M, De La Pierre M, D'Arco P, Noël Y, Causà M, Rérat M, Kirtman B (2014) CRYSTAL14: a program for the ab initio investigation of crystalline solids. Int J Quantum Chem. doi:10.1002/qua.24658
Hart GLW, Nelson LJ, Forcade RW (2012) Generating derivative structures at a fixed concentration. Comput Mater Sci 59:101–107. doi:10.1016/j.commatsci.2012.02.015 View ArticleGoogle Scholar
Arnold H (2006) Transformations of the coordinate system (unit-cell transformations). In: International tables for crystallography. International Union of Crystallography, Chester, England, pp 78–85. doi:10.1107/97809553602060000510. http://xrpp.iucr.org/cgi-bin/itr?url_ver=Z39.88-2003&rft_dat=what%3Dchapter%26volid3DAb%26chnumo%3D5o1%26chvers%3Dv0001
Paglia G, Rohl AL, Buckley CE, Gale JD (2005) Determination of the structure of \(\gamma\)-alumina from interatomic potential and first-principles calculations: the requirement of significant numbers of nonspinel positions to achieve an accurate structural model. Phys Rev B Condens Matter Mater Phys 71(22):1–16. doi:10.1103/PhysRevB.71.224115 View ArticleGoogle Scholar
Gao X, Daw M (2008) Investigation of band inversion in (Pb, Sn)Te alloys using ab initio calculations. Phys Rev B 77(3):033103. doi:10.1103/PhysRevB.77.033103 View ArticleGoogle Scholar
Read RC (1978) Every one a winner or how to avoid isomorphism search when cataloguing combinatorial configurations. Ann Discrete Math 2:107–120. doi:10.1016/S0167-5060(08)70325-X View ArticleGoogle Scholar
Frenkel D, Smit B (eds) (1996) Understanding molecular simulation: from algorithms to applications, 1st edn. Academic Press Inc, Orlando, FLGoogle Scholar
Gale JD, Rohl AL (2003) The general utility lattice program (GULP). Mol Simul 29(5):291–341. doi:10.1080/0892702031000104887 View ArticleGoogle Scholar
O'Boyle NM, Banck M, Ca James, Morley C, Vandermeersch T, Hutchison GR (2011) Open Babel: an open chemical toolbox. J Cheminfor 3(1):33. doi:10.1186/1758-2946-3-33 View ArticleGoogle Scholar
Modified OpenBabel. https://github.com/orex/openbabel
OpenBabel Repository. https://github.com/openbabel/openbabel
Vollmer J (2010) Latex2Man. http://www.informatik-vollmer.de/software/latex2man.html
Ong SP, Richards WD, Jain A, Hautier G, Kocher M, Cholia S, Gunter D, Chevrier VL, Ka Persson, Ceder G (2013) Python Materials Genomics (pymatgen): a robust, open-source python library for materials analysis. Comput Mater Sci 68:314–319. doi:10.1016/j.commatsci.2012.10.028 View ArticleGoogle Scholar
Björkman T (2011) CIF2Cell: generating geometries for electronic structure programs. Comput Phys Commun 182(5):1183–1186. doi:10.1016/j.cpc.2011.01.013 View ArticleGoogle Scholar
Pedone A, Charpentier T, Menziani MC (2010) Multinuclear NMR of CaSiO\(_3\) glass: simulation from first-principles. Phys Chem Chem Phys 12(23):6054–66. doi:10.1039/b924489a View ArticleGoogle Scholar
Nimtz BG, Schlicht B (1983) Narrow-gap semiconductors. Springer Tracts in Modern Physics, vol 98. Springer, Berlin. doi:10.1007/BFb0044919 Google Scholar
Volkov BA, Ryabova LI, Khokhlov DR (2002) Mixed-valence impurities in lead telluride-based solid solutions. Phys Usp 45(8):819–846. doi:10.1070/PU2002v045n08ABEH001146 View ArticleGoogle Scholar
Hoang K, Mahanti SD (2008) Electronic structure of Ga-, In-, and Tl-doped PbTe: a supercell study of the impurity bands. Phys Rev B Condens Matter Mater Phys 78(8):085111. doi:10.1103/PhysRevB.78.085111 View ArticleGoogle Scholar
Faulkner JS, Jordan RG (eds) (1994) Metallic alloys: experimental and theoretical perspectives. Springer, Dordrecht. doi:10.1007/978-94-011-1092-1 Google Scholar
Doak JW, Wolverton C (2012) Coherent and incoherent phase stabilities of thermoelectric rocksalt IV–VI semiconductor alloys. Phys Rev B Condens Matter Mater Phys 86(14):1–16. doi:10.1103/PhysRevB.86.144202 View ArticleGoogle Scholar
van de Walle A, Asta M, Ceder G (2002) The alloy theoretic automated toolkit: a user guide. Calphad 26(4):539–553. doi:10.1016/S0364-5916(02)80006-2 View ArticleGoogle Scholar
Sanchez JM, Ducastelle F, Gratias D (1984) Generalized cluster description of multicomponent systems. Phys A Stat Mech Appl 128(1–2):334–350. doi:10.1016/0378-4371(84)90096-7 View ArticleGoogle Scholar
Petrenko VF, Whitworth RW (2002) Physics of ice. Oxford University Press, Oxford. doi:10.1093/acprof:oso/9780198518945.001.0001 View ArticleGoogle Scholar
Loewenstein W (1954) The distribution of aluminum in the tetrahedra of silicates and aluminates. Am Mineral 39(1–2):92–96Google Scholar
Pavón E, Osuna FJ, Alba MD, Delevoye L (2014) Direct evidence of Lowenstein's rule violation in swelling high-charge micas. Chem Commun 50(53):6984–6986. doi:10.1039/c4cc01632g View ArticleGoogle Scholar
Van De Walle A, Tiwary P, De Jong M, Olmsted DL, Asta M, Dick A, Shin D, Wang Y, Chen LQ, Liu ZK (2013) Efficient stochastic generation of special quasirandom structures. Calphad Comput Coupling Phase Diagr Thermochem 42:13–18. doi:10.1016/j.calphad.2013.06.006 View ArticleGoogle Scholar
Grau-Crespo R, Al-Baitai AY, Saadoune I, De Leeuw NH (2010) Vacancy ordering and electronic structure of \(\gamma\)-Fe\(_2\)O\(_3\) (maghemite): a theoretical investigation. J Phys Condens Matter Inst Phys J 22(25):255401. doi:10.1088/0953-8984/22/25/255401 View ArticleGoogle Scholar
Kresse G, Furthmüller J (1996) Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys Rev B 54(16):11169–11186. doi:10.1103/PhysRevB.54.11169 View ArticleGoogle Scholar
Momma K, Izumi F (2011) VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data. J Appl Crystallogr 44(6):1272–1276. doi:10.1107/S0021889811038970 View ArticleGoogle Scholar
Yakimova RT, Trifonova EP, Karagiozov L, Petrov S (1984) Structural and electrical characteristics of iodidely synthesized \(\text{Pb}_{1-{\rm x}}\text{Sn}_{{\rm x}}\text{Te}\) crystals. Cryst Res Technol 19(11):109–112. doi:10.1002/crat.2170191129 View ArticleGoogle Scholar
Fortes AD, Wood IG, Grigoriev D, Alfredsson M, Kipfstuhl S, Knight KS, Smith RI (2004) No evidence for large-scale proton ordering in Antarctic ice from powder neutron diffraction. J Chem Phys 120(24):11376–11379. doi:10.1063/1.1765099 View ArticleGoogle Scholar | CommonCrawl |
How time delay and network design shape response patterns in biochemical negative feedback systems
Anastasiya Börsch1,2 &
Jörg Schaber1
Negative feedback in combination with time delay can bring about both sustained oscillations and adaptive behaviour in cellular networks. Here, we study which design features of systems with delayed negative feedback shape characteristic response patterns with special emphasis on the role of time delay. To this end, we analyse generic two-dimensional delay differential equations describing the dynamics of biochemical signal-response networks.
We investigate the influence of several design features on the stability of the model equilibrium, i.e., presence of auto-inhibition and/or mass conservation and the kind and/or strength of the delayed negative feedback. We show that auto-inhibition and mass conservation have a stabilizing effect, whereas increasing abruptness and decreasing feedback threshold have a de-stabilizing effect on the model equilibrium. Moreover, applying our theoretical analysis to the mammalian p53 system we show that an auto-inhibitory feedback can decouple period and amplitude of an oscillatory response, whereas the delayed feedback can not.
Our theoretical framework provides insight into how time delay and design features of biochemical networks act together to elicit specific characteristic response patterns. Such insight is useful for constructing synthetic networks and controlling their behaviour in response to external stimulation.
Negative feedback is one of fundamental mechanisms in cellular networks [1–7], which fulfils a variety of functions such as mediating adaptation [8–10], stabilizing the abundance of biochemical components [1, 5, 11, 12], inducing oscillations [7, 13–15] and decoupling signal and response time [5]. Negative feedbacks are shown to be present in many biochemical systems including bacterial adaptation [9, 16], mammalian cell cycle [17, 18], stress response in mammalian cells [19] and yeast [20, 21].
Negative feedbacks may involve a time delay, which is needed for signal transduction and transcription, translation and formation of biochemical species [21–24]. The combination of negative feedback and time delay may result in oscillatory dynamics of components of the cellular network [25–27]. Oscillations induced by delayed negative feedbacks (DNFs) were experimentally observed in several biochemical systems as a response to external stimuli and stress, e.g., mammalian Hes1 [24, 28], p53 [29–31] and NF- κB [23, 32, 33] systems.
It is conceivable that oscillatory behaviour might be inappropriate in biological systems mediating adaptive responses. For example, in the hyperthermia treatment of cancer, large-amplitude temperature oscillation could result in tissue damage or patient discomfort [34]. We wondered, if there exist design features and mechanisms of systems containing DNF, which may suppress oscillatory behaviour caused by external stimulation. Recent studies [22, 35] demonstrated that nested negative feedbacks may suppress oscillations of biochemical species involved in DNF. However, these studies provided no insight into how time delay influences the dynamics of DNF systems and interacts with nested negative feedbacks.
In our previous study [36] we derived explicit thresholds and boundaries showing how time delay determines characteristic response patterns of biochemical networks containing DNF. In this manuscript, we continued our research and investigated how the combination of time delay and certain design features influences the dynamics of biochemical DNF systems. To this end, we constructed a range of generic two-dimensional DNF models using delay differential equations. Models differed in several properties:
presence of a nested negative (auto-inhibitory) feedback,
presence of mass conservation for biochemical compounds,
mechanism of DNF, i.e., input-inhibition or output-activation.
Further, we subjected these models to computational and analytical stability analyses with respect to time delay. Our computational analyses demonstrate that
the presence of auto-inhibition and mass conservation have a stabilizing influence on the model equilibrium independent of the DNF strength.
increasing abruptness and decreasing DNF threshold have a de-stabilizing effect on the model equilibrium.
Our theoretical analyses show that
nested auto-inhibitory feedbacks may increase the range of time delay, where the system is stable, through the abruptness of the feedback function.
Applying our theoretical framework to the oscillating p53 system in mammalian cells [37] indicates that
both period and amplitude of p53 oscillations increase with time delay, and
a nested auto-inhibitory feedback can decouple period and amplitude of oscillations, whereas the delayed feedback can not.
Our analysis provides insight into how time delay and specific design features act in concert to shape the systems dose-response relationship. This knowledge can be used for constructing synthetic networks with the fine-tuned behaviour.
The dataset used for the parametrization of the p53 model was digitized from the supplementary material of [30] from Additional file 1: Figure S6 as described in [22]. It represents an averaged oscillation pattern that was meant to resemble an idealized undamped oscillation with peak characteristics that correspond to the average peak characteristics of oscillating cells.
Model simulation and analysis
All simulations of the delay differential equations were carried out in Mathematica 9 (Wolfram Research, Champaign, Illinois) using the function NDSolve based on the method of steps.
We used DDE-BIFTOOL v. 2.00 [38] and MATLAB R2008b (The MathWorks, Natick, MA) to calculate dependencies between the value of time delay τ and amplitude and period of oscillations of the p53 model.
Monte-Carlo analysis was performed in Mathematica 9.
For the parameter estimation we used the least-squares method minimizing the sum of squared residuals (SSR):
$$ SSR(p)=\sum_{i=1}^{n} \left(x(t_{i},p)-x_{i}\right)^{2}, $$
where p=(p 1,p 2,…,p m ) denotes a set of parameters to be estimated, x(t,p) is a numerical solution depending on parameters p, x i is a measured data point at the time t i , n is a number of the data points.
For minimizing S S R(p) with respect to parameter values we utilized the numerical function NMinimize in Mathematica 9, which, by default, uses the "Nelder-Mead" method. In case "Nelder-Mead" performs poorly, it automatically switches to the "Differential Evolution" method. The parameter optimization process is assumed to have converged to a local minimum if the difference between the new best and the old best function value S S R(p), as well as the distance between the new best and the old best parameter values, are less than a tolerance of 10−8.
Robustness with respect to model parameters
We analysed the robustness of the parameter fit for the p53 model with respect to noise. Parameter values were randomly sampled within ±10 % of their respective fitted values using a uniform distribution for 100 times. Then the p53 model was simulated using perturbed parameter sets. The relative variation of the integral of the first transient response after the initial stimulus was calculated (Additional file 1: Figure S6). Namely, we calculated the integral of the simulated p53 model from the time point 0 until the time of the first minimum after stimulation. This way, the robustness of both initial activation amplitude and timing of the first transient response, two characteristic measures of system dynamics, have been estimated concomitantly.
Model formulation
We investigated four different two-dimensional models containing DNF that describe generic signal-response networks (Fig. 1). Models differ in design of the DNF and presence of mass conservation for a biochemical compound. All models can have a nested negative (auto-inhibitory) feedback.
Generic signal-response models with DNF. Squares indicate model variables, circles indicate model functions. Arrows between and to components indicate biochemical reactions, arrows on arrows indicate modifying influences and arrows to functions indicate the respective influence on the function. The models differ in design of the delayed negative feedback (DNF) as well as in presence of mass conservation for the component C and auto-inhibitory feedback. a Model with input-inhibition as DNF and without mass conservation. b Model with input-inhibition as DNF and with mass conservation. c Model with output-activation as DNF and without mass conservation. d Model with output-activation as DNF and with mass conservation. In all models the time delay τ is before activation of the response variable R. Dashed lines indicate an alternative auto-inhibitory feedback
We mathematically formulated models from Fig. 1 using deterministic delay differential equations: Model 1: DNF with input-inhibition and without mass conservation (Fig. 1 a):
$$ \begin{aligned} \frac{dC}{dt}=&I\,S_{1}(R)\,F(C)-\alpha\,C,\\ \frac{dR}{dt}=&\,C(t-\tau)-\beta\,R. \end{aligned} $$
Model 2: DNF with input-inhibition and with mass conservation (Fig. 1 b):
$$ \begin{aligned} \frac{dC}{dt}=&I\,S_{1}(R)\,F(C)\,(1-C)-\alpha\,C,\\ \frac{dR}{dt}=&\,C(t-\tau)-\beta\,R. \end{aligned} $$
Model 3: DNF with output-activation and without mass conservation (Fig. 1 c):
$$ \begin{aligned} \frac{dC}{dt}=&I\,F(C)-\alpha\,C-\delta\,C\,S_{2}(R),\\ \frac{dR}{dt}=&\,C(t-\tau)-\beta\,R. \end{aligned} $$
Model 4: DNF with output-activation and with mass conservation (Fig. 1 d):
$$ \begin{aligned} \frac{dC}{dt}=&I\,F(C)\,(1-C)-\alpha\,C-\delta\,C\,S_{2}(R),\\ \frac{dR}{dt}=&\,C(t-\tau)-\beta\,R. \end{aligned} $$
The function \(S_{1}\colon [0,\infty)\to \mathbb {R}_{+}\) is twice continuously differentiable and monotonically decreasing with R. The function \(S_{2}\colon [0,\infty)\to \mathbb {R}_{+}\) is twice continuously differentiable and monotonically increasing with R. The twice continuously differentiable monotonically decreasing function F:[0,∞)→(0,1] mimics an optional auto-inhibitory feedback.
Parameters of Models 1-4 have real positive constant values. For convenience, we combined them into the vector p:
$$ p=\left(I,\alpha,\beta,\delta\right). $$
Note that all model parameters represent lumped biological processes and therefore have only limited physical or biological meaning. For Models 1, 2 the parameter δ equals to 0.
In all models the input I mimics some constant stimulus (e.g., radiation, see below) that activates a gene transcription network (Fig. 1 a, c) or a signalling cascade (Fig. 1 b, d) starting with the component C. Further, the component C activates the response variable R with a certain time delay τ caused by processes like transport, transcription, translation, etc. Further, the response R negatively feeds back through S 1 or S 2 performing the DNF. Depending on the model design the response R mediates DNF using different mechanisms.
We refer to models from Fig. 1 a, c as a transcription network, because C is not reversibly converted into different states, but rather produced and degraded. In the model from Fig. 1 a the response variable R has the inhibiting influence on C by means of the function S 1(R). Therefore, we refer to this DNF mechanism as input-inhibition. In the model from Fig. 1 c the DNF mediated by the response R acts through activating the degradation of C by means of the function S 2(R). We refer to this DNF mechanism as output-activation. We used this model to simulate p53 protein dynamics [30, 37].
We considered both input-inhibition and output-activation architectures together with a so-called signalling component (see Fig. 1 b, d, respectively). The component C activating the response R originates from another component \(\breve {C}\) to which it is constitutively converted back. The sum of both components \(C_{t}=C+\breve {C}\) is a constant and assumed to be unity (C t =1). We refer to this model feature as mass conservation. This modelling technique mimics a fast and reversible post-translational protein modification, e.g., phosphorylation, leaving the total protein content unchanged, as it is often described in signalling cascades.
In the following sections we presented theoretical and computational analyses of Models 1-4 with application to p53 system in mammalian cells.
Stability analysis of systems with delayed negative feedback
Presented in this section stability analysis can be applied to all Models 1-4. Therefore, we do not make an explicit distinction between models, unless necessary.
As the equilibria of Models 1-4 we considered the vector E=(C s ,R s )T. The equilibria E always exist (see Additional file 1) and implicitly depend on the parameter vector p (5). Model equilibria E also depend on the input I and, therefore, Models 1-4 are not able to show a perfect adaptation.
We linearised Models 1-4 about their respective equilibria E:
$$ \begin{pmatrix} \frac{d C}{d t}\\[0.7em] \frac{d R}{d t} \end{pmatrix}= \begin{pmatrix} -x& -y\\[0.7em] 0&-\beta \end{pmatrix} \begin{pmatrix} C\\[0.7em] R \end{pmatrix}+ \begin{pmatrix} 0&0\\[0.7em] 1&0 \end{pmatrix} \begin{pmatrix} C(t-\tau)\\[0.7em] R(t-\tau) \end{pmatrix}, $$
where for Model 1, we have
$$ \begin{aligned} x&=I\,S_{1}(R_{s})|F'(C_{s})|+\alpha>0,\\ y&=I\,|S_{1}'(R_{s})|\,F\left(C_{s}\right)>0 \end{aligned} $$
for Model 2, we have
$$ \begin{aligned} x&=I\,S_{1}\left(R_{s}\right)\left[F(C_{s})+\left(1-C_{s}\right)\,|F'(C_{s})|\,\right]+\alpha>0,\\ y&=I\,|S_{1}'\left(R_{s}\right)|\left(1-C_{s}\right)\,F\left(C_{s}\right)>0 \end{aligned} $$
$$ \begin{aligned} x&=I\,\left|F'\left(C_{s}\right)\right|+\alpha+\delta\,S_{2}\left(R_{s}\right)>0,\\ y&=\delta\,C_{s}\,S_{2}'\left(R_{s}\right)>0. \end{aligned} $$
$$ \begin{aligned} x&=I\,\left[F(C_{s})+\left(1-C_{s}\right)\,|F'(C_{s})|\,\right]+\alpha+\delta S_{2}\left(R_{s}\right)>0,\\ y&=\delta\,C_{s}\,S_{2}'\left(R_{s}\right)>0. \end{aligned} $$
The analysis of the model (6) revealed the following stability properties of the equilibrium E with respect to x, y, β, τ (for details refer to Additional file 1):
If x β≥y holds, then the equilibrium E is absolutely stable, i.e., stable for any τ≥0.
If x β<y holds, then there exists a marginal time delay τ m (the Hopf bifurcation point) such that the equilibrium of the model (6) is stable for any 0<τ<τ m and unstable for any τ≥τ m . The marginal time delay τ m is calculated as a product of functions f and g that depend on β and x, y from (7)–(10), respectively:
$$ \tau_{m}(x,y,\beta)=f(x,y,\beta)\times g(x,y,\beta), $$
$$ \begin{aligned} &f(x,y,\beta)\\&= \frac{\sqrt{2}}{\sqrt{-x^{2}-\beta^{2}+\sqrt{(x^{2}+\beta^{2})^{2}+4(y^{2}-x^{2} \beta^{2})}}}>0,\\ &g(x,y,\beta)\\&=\arccos{\frac{-(x+\beta)^{2}+\sqrt{(x^{2}+\beta^{2})^{2}+4(y^{2}-x^{2} \beta^{2})}}{2 y}}>0. \end{aligned} $$
The derivation of functions f and g is represented in Additional file 1.
In the next section, we considered features and mechanisms of systems with DNF that may stabilize the system equilibrium after stimulation.
Design features stabilizing biochemical delayed negative feedback systems
Recently, two research articles indicated that nested auto-inhibitory feedbacks repress oscillatory dynamics of simple biochemical networks involving a non-linear DNF [21, 35]. We wondered how other design features of systems with DNF influence the stability of the model equilibrium. Thus, in addition to auto-inhibition we investigated the influence of the following designs on the stability of the model equilibrium:
Mechanism of DNF: input-inhibition or output-activation,
Strength of DNF: strong or weak,
Presence of mass conservation for a chemical compound.
We also considered how different combinations of delayed and auto-inhibitory negative feedbacks affect the stability of the equilibrium:
Weak DNF with and without auto-inhibition,
Strong DNF with and without auto-inhibition.
For analysing the influence of these design features on the stability of the equilibrium we performed a Monte-Carlo analysis of Models 1-4. First, we defined concrete DNF functions S 1 and S 2 and an auto-inhibitory feedback function F.
We defined a reverse Hill function as the input-inhibition DNF function S 1(R). As the output-activation function S 2(R) we defined a Hill function. Thus, functions S 1 and S 2 have the following form:
$$\begin{array}{@{}rcl@{}} S_{1}(R)&=&\frac{{K_{m}}^{n}}{{K_{m}}^{n}+R^{n}}, \end{array} $$
$$\begin{array}{@{}rcl@{}} S_{2}(R)&=&\frac{R^{n}}{{K_{m}}^{n}+R^{n}} \end{array} $$
with K m >0 being the half-saturation constant, characterizing the activation threshold beyond which the feedback takes effect, and n≥1 being the Hill coefficient, characterizing how abrupt the feedback takes effect after having passed the activation threshold. Thus, parameters K m and n specify the strength of the DNF: the lower the activation threshold K m and the higher the abruptness n, the stronger the DNF is. Note that applying the same parameters make functions S 1 and S 2 symmetric, allowing a fair comparison of the influence of input-inhibition and output-activation on the model stability.
As the auto-inhibitory feedback we employed a reverse Hill function F(C):
$$ F(C)=\frac{1}{1+\left(\kappa\,C\right)^{\nu}},\,\nu\ge 1,\,\kappa\ge 0. $$
Then, we randomly generated parameter values I=0.87, α=0.11, β=0.17, δ=58.2, n=12.77, K m =0.23 in the way that Models 1-4 without auto-inhibition, i.e., κ=0, have similar values of τ m . They correspond to τ m =1.23, τ m =1.27, τ m =1.22, τ m =1.24 for Models 1-4, respectively. Thus, applying this parameter set we guarantee that any value of the time delay τ has a similar distance to the Hopf bifurcation point τ m for all considered models. Simulations of Models 1-4 with these parameters and τ=2.5 are depicted in Fig. 2 a.
Results of Monte-Carlo analysis of Models 1-4. a Simulation of Models 1-4 with parameter values I=0.87, α=0.11, β=0.17, δ=58.2, n=12.77, K m =0.23, τ=2.5 without auto-inhibition (κ=0). b,c Stability analysis of Monte-Carlo simulations of Models 1-4. Model parameters were randomly sampled 10000 times in the certain range. The range was defined according to assumptions about model characteristics: strength of DNF (strong or weak) and presence of auto-inhibition. The percentage of parameter sets (see Fig. b), which induced absolute stability, and the mean value of marginal time delay τ m (see Fig. c) were quantified
Further, using these parameter values we performed a Monte-Carlo analysis for Models 1-4 with the constant input I=0.87. Namely, the rate constants α, β, δ have been sampled in the range from 0.1 to 10 times of their respective values for 10000 times. The parameter values defining the system design, i.e., n, K m , ν and κ, were sampled in the following way:
in the case of weak DNF without auto-inhibition (κ=0) we sampled n in the range from 0.1 to 1 time of its value and K m in the range from 10 to 20 times of its value 10000 times.
in the case of strong DNF without auto-inhibition (κ=0) we sampled n in the range from 1 to 2 times of its value and K m in the range from 0.1 to 10 times of its value 10000 times.
in the case of weak DNF with auto-inhibition we sampled n and K m as in (i), κ in the interval [0.1,10], ν in the interval [1,20] 10000 times.
in the case of strong DNF with auto-inhibition we sampled n and K m as in (ii), κ in the interval [0.1,10], ν in the interval [1,20] 10000 times.
Thus, for each Model 1-4 and each combination of DNF and auto-inhibition we obtained 10000 parameter sets. For each parameter set we calculated x and y according to (7)–(10) for Models 1-4, respectively. Then, we defined the percentage of parameter sets for which x β≥y holds indicating absolute stability of the model equilibrium (see Fig. 2 b). For all other parameter sets, we calculated the mean value of the marginal time delay τ m (11), i.e., the Hopf-bifurcation point (see Fig. 2 c).
Figure 2 b, c shows that considered models with auto-inhibition have a higher percentage of parameter sets leading to absolute stability and higher mean value of τ m than the same models without auto-inhibition. This observation confirms previous results [21, 35] showing that nested auto-inhibitory feedbacks repress oscillatory dynamics in networks containing DNF.
Additionally, for models with weak DNF there are more parameter sets, which induce absolute stability of the model equilibrium, than for models with the strong DNF. Accordingly, models with weak DNF have higher mean value of τ m , i.e., are less prone to oscillations, than models with the strong one. Thus, the nested auto-inhibitory feedback and the DNF have opposing effects on the system's stability. The stronger the auto-inhibitory feedback and the weaker the DNF, the less probable an oscillatory response of the system is.
Moreover, Models 2 and 4 with mass conservation have a higher percentage of parameter sets leading to absolute stability of the model equilibrium and higher mean value of τ m than respective Models 1 and 3 without mass conservation. In order to explain this effect, for each model we quantified the dependence between τ m and each parameter value in the range from 0.1 to 10 times of its default value leaving the other parameters fixed (see Additional file 1: Figure S11a–d). This sensitivity analysis shows that the presence of mass conservation influences the sensitivity of τ m only with respect to the half-saturation rate K m leaving all other parameter sensitivities unchanged (compare Additional file 1: Figure S11a to b and S11c to d for models without and with mass activation, respectively, and Additional file 1: Figure S11e). In the presence of mass conservation τ m increases much stronger with increasing feedback activation threshold K m and, therefore, stabilizes the equilibrium. Moreover, in the presence of mass conservation the value of K m beyond which τ m does not exist any more, i.e., the system's equilibrium becomes absolutely stable, also decreases (Additional file 1: Figure S11e).
Concerning the design of the DNF, Monte-Carlo analysis shows that Models 1, 2 with input-inhibition have a higher percentage of parameter sets leading to absolute stability and a higher mean value of τ m than Models 3, 4 with output-activation (see Fig. 2 b, c). However, we were not able to support these results with an alternative parameter set I=0.48, α=0.14, β=0.44, δ=83.71, n=10, K m =0.9. Refer to Additional file 1: Figure S12a for simulations of Models 1-4 using the alternative parameter set and τ=10. In comparison, for the alternative parameter set Models 1, 2 with input-inhibition have higher values of τ m , i.e., τ m =1.27 and τ m =1.86, than Models 3, 4 with output-activation, i.e., τ m =0.36 and τ m =0.52. Thus, the distance of τ to τ m is smaller for Models 1, 2, and one may expect a higher percentage of parameter sets inducing absolute stability. Nevertheless, for the alternative parameter set the Monte-Carlo analysis showed that models with input-inhibition have approximately the same percentage of parameter sets leading to absolute stability as corresponding models with output-activation. In comparison, all other conclusions presented above were confirmed for model simulations with the alternative parameter set (Additional file 1: Figure S12b).
Taken together, we conclude that auto-inhibition as well as mass conservation have a stabilizing influence on the model equilibrium independent of the strength of DNF and allow systems with DNF to adapt to an external stimulus without producing sustained oscillations. Moreover, the higher the activation threshold and the less abrupt the DNF, the less prone the system is to an oscillatory behaviour.
Auto-inhibition increases τ m
Computational analysis presented in the above section demonstrated the opposing behaviour of auto-inhibitory and delayed negative feedbacks with respect to stability (Fig. 2). In this section, we analytically investigated how the auto-inhibitory feedback stabilizes the equilibrium of the system.
We proved that τ m (12) increases with x and decreases with y (see Additional file 1):
$$ \partial\,\tau_{m}(x,y,\beta)/\partial\,x>0,\, \partial\,\tau_{m}(x,y,\beta)/\partial\,y<0 \quad\text{for \(x,\,y>0\).}\quad $$
Further, we derived upper and lower bounds for x and y from (7)–(10) for Models 1-4, respectively (see Additional file 1):
$$ \begin{aligned} 0<\varepsilon_{lb}(|F'\left(C_{s}\right)|)&<x<\varepsilon_{ub}(|F'\left(C_{s}\right)|),\\ 0\le\sigma_{lb}&<y<\sigma_{ub}. \end{aligned} $$
Both the lower and upper bound of x, i.e., ε lb and ε ub , are increasing with |F ′(C s )| for Models 1-4. Therefore, we can always increase a given x by choosing an appropriate value of |F ′(C s )|. The lower and upper bound of y, i.e., σ lb and σ ub , have non-negative constant values. Consequently, according to (16), we can always increase τ m by increasing |F ′(C s )|.
Taken together, we showed that auto-inhibitory feedback can increase the range of the interval [0,τ m ) and stabilise the model equilibrium.
Application to the p53 system
In this section we applied Model 3 to the p53 system to gain novel insights into the functioning of this system.
The tumour suppressor protein p53 is activated in response to many stress signals and activates various stress-response programs including cell-cycle arrest, senescence and apoptosis [39]. It is also well established that p53 acts within a negative feedback loop, including Mdm2 as the negative regulator of p53: p53 transcriptionally activates Mdm2, which in turn targets p53 for degradation [29, 40].
Several mathematical models of p53-Mdm2 feedback loop have been published [22, 28, 30, 37]. One of these models (model III from Table 1 in [30]) is a particular case of the model from Fig. 1 c corresponding to the Model 3 with F(C)≡1. Therefore, we wondered, whether our framework would also be able to explain measured p53 dynamics upon DNA damage. In our designations C and R correspond to p53 and Mdm2, respectively. The input I is defined here as a scaled DNA damage signal and is measured in arbitrary units. The negative feedback by output-activation is modelled by a non-linear Hill function S 2(R) (14).
We fitted parameters of the p53 model (3) to the experimental data of an averaged oscillation pattern of the p53-Mdm2 system after DNA damage from [30] (Additional file 1: Fig. S6 therein). The results of the fit are presented in Additional file 1: Table S1. Figure 3a shows the simulation of the p53 model (3) with fitted parameters. The model well recapitulates measured p53 dynamics after DNA damage. Moreover, the fitted optimal solution is also robust with respect to noise in the fitted parameters (see Additional file 1). Indeed, the integrated first transient response varies only by 8.7 % assuming a parameter noise of ±10 % (see Methods Section and Additional file 1).
Simulation and response analysis of the p53 model. a Simulation of the p53 model (3) with fitted parameters from Table S1 (see Additional file 1), dots – experimental data from [30], Fig. S6 therein. b Dependence between the stimulus value I and τ m for the p53 model (3) with fitted parameters from Table S1 (see Additional file 1) without and with synthetically activated auto-inhibitory feedback F(C) (with ν=2, κ=1.23 and ν=3, κ=1.73). Dots designate values of τ m calculated for the fitted value of I=0.23 for the p53 model (3) with and without auto-inhibitory feedback
The model analysis shows that the fitted time delay τ=1.37 h is almost two times larger than the corresponding τ m =0.76 h that was calculated for the fitted DNA damage signal I=0.23 (Fig. 3 b). Therefore, the p53 model (3) with fitted parameters from Table S1 (see Additional file 1) produces sustained oscillatory response.
It was earlier reported that distinct p53 dynamics such as oscillations or sustained activation may lead to different cell fate decisions [31, 39]. Recent study [41] showed that the system's response is modulated by DNA damage strength. Namely, after high DNA damage p53 level was monotonically increased and cells activated apoptosis, whereas after low DNA damage p53 level underwent periodic pulsing resulting in a cell-cycle arrest. We checked if our generic p53 model (3) is able to reproduce this transition with respect to the DNA damage level. Figure 3 b shows that τ m is decreasing with respect to the DNA damage signal I. Hence the fitted value of time delay τ is greater than τ m for any I>0.23 (fitted value). Therefore, the p53 model (3) produces sustained oscillations for any I greater than the fitted value and is not able to perform the transition from oscillatory to adaptive behaviour with respect to increased DNA damage signal I. This conclusion is also applicable to other model alternatives (1)–(4) used in our previous study presenting models of HOG pathway in yeast and NF- κB signalling in mammalian cells (see Figs. 3b and 6b in [36]). However, according to [41] the switch from sustained oscillations to monotonic increase of p53 level is regulated by a mechanism attenuating Mdm2 expression that is not present in the current p53 model. Studies [39, 41] considered DNA damage kinases ATM and ATR as negative regulators of Mdm2 expression. Using this knowledge, we extended the p53 model (3) by including the additional component ATM activating p53 and attenuating Mdm2 (see Additional file 1: Figure S7). As a result, the extended p53 model was able to qualitatively reproduce the switch from oscillations to monotonic increase of p53 level (see Additional file 1: Figure S8). Simulations of the extended p53 model suggest that this transition between response types originates from the competition between ATM and p53 for the inhibition and activation of Mdm2, respectively. In case of high DNA damage, ATM level is high and suppresses Mdm2 giving a monotone increase of p53 level. In case of low DNA damage, Mdm2 activity is not effectively suppressed by ATM resulting in sustained oscillations of both p53 and Mdm2.
Further, we applied our theoretical analysis to explore under what conditions sustained oscillations of p53 model (3) can be suppressed by the activation of a nested auto-inhibitory feedback to the model component C preserving all values of fitted parameters. Our theoretical analysis suggested that the marginal time delay τ m , beyond which any time delay leads to sustained oscillations, can be increased by increasing the slope of the auto-inhibitory feedback function at the equilibrium |F ′(C s )|. As in the previous section, as the auto-inhibitory feedback function we utilized a reverse Hill-function F(C) (15). Further, we adjusted parameters ν and κ of F(C) and calculated τ m (see Additional file 1). For ν=3 the marginal value of time delay τ m is larger than the fitted time delay τ=1.37 h (Fig. 3 b). As a result, p53 model (3) with parameters from Table S1 in Additional file 1 produced damped oscillatory response (Additional file 1: Figure S4).
In a similar DNF system it was shown that the period of oscillations increases with the Hill coefficient n of the DNF function for a given time delay [24]. We wondered how parameters of the delayed negative and auto-inhibitory feedbacks influence the amplitude and period of oscillations in our system. Figure 4 demonstrates that the auto-inhibitory feedback (with parameters ν=3, κ=1.73) decreases and stabilizes the amplitude of oscillations, whereas the amplitude of oscillations increases with respect to the Hill coefficient n of the DNF function S 2. Moreover, increasing the abruptness of the DNF has no substantial influence on the increase of period with respect to time delay τ. The period of oscillations is a linear function of the time delay τ irrespective of values of ν, κ and n. Thus, opposed to the delayed feedback, the auto-inhibitory feedback has the potential to de-couple the increase of amplitude and period of oscillations with respect to τ. Moreover, auto-inhibitory and delayed negative feedbacks have an opposing influence on the amplitude of oscillations.
Amplitude/period curves of the p53 model under variation of τ. The analysis is performed for the p53 model (3) without and with synthetically activated (ν=2, κ=1.23; ν=3, κ=1.73) auto-inhibitory feedback using values of the Hill coefficient n=3 and n=5 (fitted value) of the DNF function S 2. Period and amplitude were quantified for the time delay τ varied in the range from 1 to 8 hours with the step 0.2 hour. Both amplitude and period of oscillations increase with τ
Thus, our analysis showed that for the p53 model (3) an auto-inhibitory feedback can be a potential mechanism increasing the marginal time delay τ m , decreasing the amplitude of oscillations and turning sustained oscillations into damped oscillations.
The experimental study of p53 oscillations [29] concluded that the mean number of p53 pulses in individual cells increased with DNA damage. Moreover, the authors suggest that the p53-Mdm2 feedback loop generates a "digital" clock making the number of p53 pulses relevant for the cell fate, and not their amplitude and duration. However, this hypothesis has not been proven yet. Therefore, we wondered which parameters of p53 model (3) play a prominent role in controlling the length of p53 pulses. Using p53 model (3) we split the p53 simulation curve on "On" and "Off" states (see Additional file 1: Figure S9). Then we checked how different model parameters control the duration of p53 pulses (see Additional file 1: Figure S10). The analysis showed that time delay τ is the only parameter that significantly changed the duration of "On" and "Off" states of the model response (see Additional file 1: Figure S10b). Namely, time delay τ increases the duration of "Off" states and decreases the duration of "On" states. In addition, τ increases the amplitude and period of pulses (see Fig. 4). Note that the same conclusion can be applied to the relation between time delay τ and marginal time delay τ m : the higher τ/τ m , the higher the amplitude and period of pulses are. It would be interesting to validate these predictions experimentally and check the physiological effect of changing time delay between p53 and Mdm2 activation after DNA damage.
Negative feedback in combination with time delay can induce oscillations in cellular networks [25–27]. However, oscillations might be inappropriate in biological systems with adaptive behaviour [34].
Here, we systematically study how design features in combination with time delay tune the response patterns of biochemical networks. To this end, we create a range of models containing an explicit time delay and a DNF differing in several aspects: presence of a nested negative (auto-inhibitory) feedback, presence of mass conservation for a system component and mechanism of DNF, i.e., input-inhibition or output-activation (Fig. 1). The obtained models (Models 1-4) are mathematically described by two-dimensional delay differential Eqs. 1–(4) and subjected to computational and theoretical stability analyses with respect to time delay. The general idea to specifically address the interaction of time delay and design features was that all these design features act on the system stability and response pattern by modifying the time delay threshold, i.e., the bifurcation point, beyond which the system's stability properties change.
We show that
nested auto-inhibitory feedbacks and overall delayed negative feedbacks have opposing roles with respect to the characteristic response pattern. Indeed, nested auto-inhibitory feedbacks have the potential to suppress oscillatory behaviour, whereas the increasing strength of the DNF promotes oscillations. Moreover, in oscillatory systems auto-inhibitory feedbacks de-couple amplitude and period of oscillations.
mass conservation has a stabilizing effect on the system's equilibrium.
depending on the parameter set, the type of DNF can also influence the response pattern. We found that input-inhibition can be more stabilizing compared to output-activation.
Thus, biochemical networks have a range of design possibilities shaping both their dynamic as well as their equilibrium properties. Our systematic analysis of different design features allows predicting what kind of biochemical network underlies a certain characteristic response. For example, in oscillatory systems with a long time delay, it is reasonable to assume a limited number of post-translational modifications (mass conservation), no nested feedbacks and a strong overall negative feedback. Whereas adaptive systems with long time delay are likely to harbour nested negative feedbacks and post-translational modifications. Systems with low number of components and short time delay that are meant to oscillate, will need an abrupt negative feedback with low activation threshold, whereas short time delay and a weak negative feedback are good designs principles for adaptive systems.
Our framework of delayed and non-delayed feedbacks can serve to support a design process for novel synthetic gene-regulatory networks. Indeed, our study allows to approximate the value of time delay and the structure of the DNF system for obtaining a certain type of the system dynamics. As an example, we considered p53 system in mammalian cells that contains DNF and is able to produce both oscillatory and adaptive responses depending on the source and strength of DNA damage [37, 41]. Although many studies are dedicated to studying DNA damage response in cells, the purpose of oscillations in p53 system remains unclear [29, 37, 39, 41]. The earlier study [37] suggested that oscillatory behaviour can be advantageous for the p53 system to achieve a trade-off between irreversible biological outcome, e.g., irreversible cell cycle arrest or apoptosis, and insufficiently low levels of p53. Thus, oscillations have been viewed as repetitive repair efforts allowing the system to check after every p53 pulse whether the damage has been properly repaired. Our analysis showed that time delay increases the duration of "Off" states and decreases the duration of "On" states. Additionally, time delay may increase the amplitude and period of oscillations. According to our analysis the auto-inhibitory feedback is able to decouple the amplitude and period of oscillations with respect to time delay. Thus, our study suggests that auto-inhibition and time delay may control oscillations in p53 system. The experimental validation of these predictions may help to better understand the role of p53 oscillations and indicate more efficient treatment of diseases caused by violation of p53 regulation.
DNF:
delayed negative feedback
SSR:
sum of squared residuals
Alon U, Vol. 10. An Introduction to Systems Biology: Design Principles of Biological Circuits. Boca Raton: Chapman & Hall/CRC; 2006.
Alon U. Network motifs: theory and experimental approaches. Nat Rev Genet. 2007; 8(6):450–61.
Legewie S, Herzel H, Westerhoff HV, Blüthgen N. Recurrent design patterns in the feedback regulation of the mammalian signalling network.Mol Syst Biol. 2008; 4:190.
Kholodenko BN, Hancock JF, Kolch W. Signalling ballet in space and time. Nat Rev Mol Cell Biol. 2010; 11(6):414–26.
Tyson JJ, Chen KC, Novak B. Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell. Curr Opin Cell Biol. 2003; 15(2):221–31.
Sauro HHM, Kholodenko BBN. Quantitative analysis of signaling networks. Prog Biophys Mol Biol. 2004; 86(1):5–43.
Tsai TY-C, Choi YS, Ma W, Pomerening JR, Tang C, Ferrell JE. Robust, tunable biological oscillations from interlinked positive and negative feedback loops. Science. 2008; 321(5885):126–9.
Ma W, Trusina A, El-Samad H, Lim WA, Tang C. Defining network topologies that can achieve biochemical adaptation. Cell. 2009; 138(4):760–73.
Yi TM, Huang Y, Simon MI, Doyle J. Robust perfect adaptation in bacterial chemotaxis through integral feedback control. Proc Natl Acad Sci. 2000; 97(9):4649–653.
Ni XY, Drengstig T, Ruoff P. The control of the controller: molecular mechanisms for robust perfect adaptation and temperature compensation. Biophys J. 2009; 97(5):1244–53.
Hasty J, Dolnik M, Rottschäfer V, Collins JJ. Synthetic gene network for entraining and amplifying cellular oscillations. Phys Rev Lett. 2002; 88(14):148101.
Sturm OE, Orton R, Grindlay J, Birtwistle M, Vyshemirsky V, Gilbert D, Calder M, Pitt A, Kholodenko B, Kolch W. The mammalian MAPK/ERK pathway exhibits properties of a negative feedback amplifier. Sci Signal. 2010; 3(153):90.
Novak B, Tyson JJ, Gyorffy B, Csikasz-Nagy A. Irreversible cell-cycle transitions are due to systems-level feedback. Nat Cell Biol. 2007; 9(7):724–8.
Elowitz MB, Leibler S. A synthetic oscillatory network of transcriptional regulators. Nature. 2000; 403(6767):335–8.
Kholodenko BN. Negative feedback and ultrasensitivity can bring about oscillations in the mitogen-activated protein kinase cascades. Eur J Biochem. 2000; 267(6):1583–1588.
Kollmann M, Løvdok L, Bartholomé K, Timmer J, Sourjik V. Design principles of a bacterial signalling network. Nature. 2005; 438(7067):504–7.
Novak B, Kapuy O, Domingo-Sananes MR, Tyson JJ. Regulated protein kinases and phosphatases in cell cycle decisions. Curr Opin Cell Biol. 2010; 22(6):801–8.
Ferrell JE, Tsai TY-C, Yang Q. Modeling the cell cycle: why do certain circuits oscillate?. Cell. 2011; 144(6):874–85.
Blüthgen N. Transcriptional feedbacks in mammalian signal transduction pathways facilitate rapid and reliable protein induction. Mol Biosyst. 2010; 6(7):1277–1284.
Klipp E, Nordlander B, Krüger R, Gennemark P, Hohmann S. Integrative model of the response of yeast to osmotic shock. Nat Biotechnol. 2005; 23(8):975–82.
Schaber J, Baltanas R, Bush A, Klipp E, Colman-Lerner A. Modelling reveals novel roles of two parallel signalling pathways and homeostatic feedbacks in yeast. Mol Syst Biol. 2012; 8:622.
Schaber J, Lapytsko A, Flockerzi D. Nested autoinhibitory feedbacks alter the resistance of homeostatic adaptive biochemical networks. J R Soc Interface. 2014; 11(91):20130971.
Hoffmann A, Levchenko A, Scott ML, Baltimore D. The I κB-NF- κB signaling module: Temporal control and selective gene activation. Science. 2002; 298(5596):1241–1245.
Bernard S, Cajavec B, Pujo-Menjouet L, Mackey MC, Herzel H. Modelling transcriptional feedback loops: the role of Gro/TLE1 in Hes1 oscillations. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2006; 364(1842):1155–1170.
Goodwin BC. Temporal Organization in Cells: a Dynamic Theory of Cellular Control Processes. London: Academic Press; 1963.
Griffith JS. Mathematics of cellular control processes i. negative feedback to one gene. J Theor Biol. 1968; 20(2):202–8.
Mahaffy J, Pao C. Models of genetic control by repression with time delays and spatial effects. J Math Biol. 1984; 20(1):39–57.
Monk NAM. Oscillatory expression of hes1, p53, and NF- κB driven by transcriptional time delays. Curr Biol. 2003; 13(16):1409–1413.
Lahav G, Rosenfeld N, Sigal A, Geva-Zatorsky N, Levine AJ, Elowitz MB, Alon U. Dynamics of the p53-Mdm2 feedback loop in individual cells. Nat Genet. 2004; 36(2):147–50.
Geva-Zatorsky N, Rosenfeld N, Itzkovitz S, Milo R, Sigal A, Dekel E, Yarnitzky T, Liron Y, Polak P, Lahav G. Oscillations and variability in the p53 system. Mol Syst Biol. 2006; 2(1):2006–0033.
Purvis JE, Karhohs KW, Mock C, Batchelor E, Loewer A, Lahav G. p53 dynamics control cell fate. Sci Signal. 2012; 336(6087):1440.
Nelson D, Ihekwaba A, Elliott M, Johnson J, Gibney C, Foreman B, Nelson G, See V, Horton C, Spiller D, et al. Oscillations in NF- κB signaling control the dynamics of gene expression. Sci Signal. 2004; 306(5696):704.
Ashall L, Horton CA, Nelson DE, Paszek P, Harper CV, Sillitoe K, Ryan S, Spiller DG, Unitt JF, Broomhead DS, et al. Pulsatile stimulation determines timing and specificity of NF- κB-dependent transcription. Sci Signal. 2009; 324(5924):242.
Tharp HS, Zhang WW. Robust control of a nonlinear time-delay system. Dyn Control. 1994; 4(1):21–38.
Nguyen LK. Regulation of oscillation dynamics in biochemical systems with dual negative feedback loops. J R Soc Interface. 2012; 9(73):1998–2010.
Lapytsko A, Schaber J. The role of time delay in adaptive cellular negative feedback systems. J Theor Biol. 2016; 398:64–73.
Lev Bar-Or R, Maya R, Segel LA, Alon U, Levine AJ, Oren M. Generation of oscillations by the p53-Mdm2 feedback loop: A theoretical and experimental study. Proc Natl Acad Sci. 2000; 97(21):11250–11255.
Engelborghs K, Luzyanina T, Samaey G. Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Transactions on Mathematical Software, 28. 2002;:1–21.
Batchelor E, Loewer A, Mock C, Lahav G. Stimulus-dependent dynamics of p53 in single cells. Mol Syst Biol. 2011; 7:488.
Kubbutat MH, Ludwig RL, Ashcroft M, Vousden KH. Regulation of Mdm2-directed degradation by the C terminus of p53. Mol Cell Biol. 1998; 18(10):5690–698.
Chen X, Chen J, Gan S, Guan H, Zhou Y, Ouyang Q, Shi J. DNA damage strength modulates a bimodal switch of p53 dynamics for cell-fate control. BMC Biol. 2013; 11(1):73.
The authors thank Prof. Dietrich Flockerzi for fruitful discussions and Prof. Steffen Waldherr for critical comments on the manuscript. The authors are also affiliated to the "International Max Planck Research School (IMPRS) for Advanced Methods in Process and System Engineering (Magdeburg)".
This work was supported by the German Ministry of Science and Education [Federal Ministry of Education and Research project 0135779 to JS].
The data sets supporting the result of the article are included within the article and its additional files.
AB and JS designed research, AB did the mathematical and data analysis, JS and AB wrote the paper. Both authors read and approved the final manuscript.
Institute for Experimental Internal Medicine, Medical Faculty, Otto-von-Guericke University, Pfälzer Platz 2, Magdeburg, 39106, Germany
Anastasiya Börsch & Jörg Schaber
Biozentrum, University of Basel and Swiss Institute of Bioinformatics, Klingelbergstrasse 50–70, Basel, 4056, Switzerland
Anastasiya Börsch
Jörg Schaber
Correspondence to Jörg Schaber.
Additional file 1
Supporting material. The file contains detailed stability analysis of Models 1-4; theoretical analysis showing how auto-inhibition increases τ m ; demonstration how τ m can be increased by auto-inhibition for the p53 model; details of robustness analysis of the optimal solution for the p53 model; modelling the switch from oscillatory to adaptive response of the p53 system; calculating the duration of "On" and "Off" states of p53 pulses; figure demonstrating the dependence between parameter values of Models 1-4 and τ m ; figure with results of Monte-Carlo analysis of Models 1-4 applied to the alternative parameter set; table with the best-fit parameters for the p53 model. (PDF 1044 kb)
Börsch, A., Schaber, J. How time delay and network design shape response patterns in biochemical negative feedback systems. BMC Syst Biol 10, 82 (2016). https://doi.org/10.1186/s12918-016-0325-9
Auto-inhibiton
Mass conservation
Networks and information flow | CommonCrawl |
\begin{definition}[Definition:Trace of Element of Algebra over Ring]
Let $A$ be a commutative ring with unity.
Let $B$ be an algebra over $A$ such that $B$ is a finite-dimensional free module over $A$.
Let $b \in B$.
The '''trace''' $\operatorname{Tr}_{B/A}(b)$ of $b$ is the trace of the regular representation $\lambda_b : B \to B$ over $A$.
\end{definition} | ProofWiki |
\begin{definition}[Definition:Neighborhood (Complex Analysis)]
Let $z_0 \in \C$ be a complex number.
Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number.
The '''$\epsilon$-neighborhood''' of $z_0$ is defined as:
:$\map {N_\epsilon} {z_0} := \set {z \in \C: \cmod {z - z_0} < \epsilon}$
\end{definition} | ProofWiki |
Solenoid (mathematics)
In mathematics, a solenoid is a compact connected topological space (i.e. a continuum) that may be obtained as the inverse limit of an inverse system of topological groups and continuous homomorphisms
$f_{i}:S_{i+1}\to S_{i}\quad \forall i\geq 0$
This page discusses a class of topological groups. For the wrapped loop of wire, see Solenoid.
Algebraic structure → Group theory
Group theory
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• Cyclic group Zn
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• Alternating group An
• Dihedral group Dn
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• Cauchy's theorem
• Lagrange's theorem
• Sylow theorems
• Hall's theorem
• p-group
• Elementary abelian group
• Frobenius group
• Schur multiplier
Classification of finite simple groups
• cyclic
• alternating
• Lie type
• sporadic
• Discrete groups
• Lattices
• Integers ($\mathbb {Z} $)
• Free group
Modular groups
• PSL(2, $\mathbb {Z} $)
• SL(2, $\mathbb {Z} $)
• Arithmetic group
• Lattice
• Hyperbolic group
Topological and Lie groups
• Solenoid
• Circle
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Euclidean E(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
• G2
• F4
• E6
• E7
• E8
• Lorentz
• Poincaré
• Conformal
• Diffeomorphism
• Loop
Infinite dimensional Lie group
• O(∞)
• SU(∞)
• Sp(∞)
Algebraic groups
• Linear algebraic group
• Reductive group
• Abelian variety
• Elliptic curve
where each $S_{i}$ is a circle and fi is the map that uniformly wraps the circle $S_{i+1}$ for $n_{i+1}$ times ($n_{i+1}\geq 2$) around the circle $S_{i}$. This construction can be carried out geometrically in the three-dimensional Euclidean space R3. A solenoid is a one-dimensional homogeneous indecomposable continuum that has the structure of a compact topological group.
Solenoids were first introduced by Vietoris for the $n_{i}=2$ case,[1] and by van Dantzig the $n_{i}=n$ case, where $n\geq 2$ is fixed.[2] Such a solenoid arises as a one-dimensional expanding attractor, or Smale–Williams attractor, and forms an important example in the theory of hyperbolic dynamical systems.
Construction
Geometric construction and the Smale–Williams attractor
Each solenoid may be constructed as the intersection of a nested system of embedded solid tori in R3.
Fix a sequence of natural numbers {ni}, ni ≥ 2. Let T0 = S1 × D be a solid torus. For each i ≥ 0, choose a solid torus Ti+1 that is wrapped longitudinally ni times inside the solid torus Ti. Then their intersection
$\Lambda =\bigcap _{i\geq 0}T_{i}$
is homeomorphic to the solenoid constructed as the inverse limit of the system of circles with the maps determined by the sequence {ni}.
Here is a variant of this construction isolated by Stephen Smale as an example of an expanding attractor in the theory of smooth dynamical systems. Denote the angular coordinate on the circle S1 by t (it is defined mod 2π) and consider the complex coordinate z on the two-dimensional unit disk D. Let f be the map of the solid torus T = S1 × D into itself given by the explicit formula
$f(t,z)=\left(2t,{\tfrac {1}{4}}z+{\tfrac {1}{2}}e^{it}\right).$
This map is a smooth embedding of T into itself that preserves the foliation by meridional disks (the constants 1/2 and 1/4 are somewhat arbitrary, but it is essential that 1/4 < 1/2 and 1/4 + 1/2 < 1). If T is imagined as a rubber tube, the map f stretches it in the longitudinal direction, contracts each meridional disk, and wraps the deformed tube twice inside T with twisting, but without self-intersections. The hyperbolic set Λ of the discrete dynamical system (T, f) is the intersection of the sequence of nested solid tori described above, where Ti is the image of T under the ith iteration of the map f. This set is a one-dimensional (in the sense of topological dimension) attractor, and the dynamics of f on Λ has the following interesting properties:
• meridional disks are the stable manifolds, each of which intersects Λ over a Cantor set
• periodic points of f are dense in Λ
• the map f is topologically transitive on Λ
General theory of solenoids and expanding attractors, not necessarily one-dimensional, was developed by R. F. Williams and involves a projective system of infinitely many copies of a compact branched manifold in place of the circle, together with an expanding self-immersion.
Construction in toroidal coordinates
In the toroidal coordinates with radius $R$, the solenoid can be parametrized by $t\in \mathbb {R} $ as
$\zeta =2\pi t,\quad re^{i\theta }=\sum _{k=1}^{\infty }r_{k}e^{2\pi i\omega _{k}t}$
where
$\omega _{k}={\frac {1}{n_{1}\cdots n_{k}}},\quad r_{k}=R\delta _{1}\cdots \delta _{k}$
Here, $\delta _{k}$ are adjustable shape-parameters, with constraint $0<\delta <1-{\frac {1}{1+\sin {\frac {\pi }{n_{k}}}}}$. In particular, $\delta ={\frac {1}{2n_{k}}}$ works.
Let $S\subset \mathbb {R} ^{3}$ be the solenoid constructed this way, then the topology of the solenoid is just the subset topology induced by the Euclidean topology on $\mathbb {R} ^{3}$.
Since the parametrization is bijective, we can pullback the topology on $S$ to $\mathbb {R} $, which makes $\mathbb {R} $ itself the solenoid. This allows us to construct the inverse limit maps explicitly:
$g_{k}:\mathbb {R} \to S_{k},\quad g_{k}(t)=(r,\theta ,\zeta ){\text{ in toroidal coordinates, where }}\zeta =2\pi t,\quad re^{i\theta }=\sum _{k=1}^{k}r_{k}e^{2\pi i\omega _{k}t}$
Construction by symbolic dynamics
Viewed as a set, the solenoid is just a Cantor-continuum of circles, wired together in a particular way. This suggests to us the construction by symbolic dynamics, where we start with a circle as a "racetrack", and append an "odometer" to keep track of which circle we are on.
Define $S=S^{1}\times \prod _{k=1}^{\infty }\mathbb {Z} _{n_{k}}$ as the solenoid. Next, define addition on the odometer $\mathbb {Z} \times \prod _{k=1}^{\infty }\mathbb {Z} _{n_{k}}\to \prod _{k=1}^{\infty }\mathbb {Z} _{n_{k}}$, in the same way as p-adic numbers.
Next, define addition on the solenoid $+:\mathbb {R} \times S\to S$ by
$r+(\theta ,n)=((r+\theta \mod 1),\lfloor r+\theta \rfloor +n)$
The topology on the solenoid is generated by the basis containing the subsets $S'\times Z'_{(m_{1},...,m_{k})}$, where $S'$ is any open interval in $S^{1}$, and $Z'_{(m_{1},...,m_{k})}$ is the set of all elements of $\prod _{k=1}^{\infty }\mathbb {Z} _{n_{k}}$ starting with the initial segment $(m_{1},...,m_{k})$.
Pathological properties
Solenoids are compact metrizable spaces that are connected, but not locally connected or path connected. This is reflected in their pathological behavior with respect to various homology theories, in contrast with the standard properties of homology for simplicial complexes. In Čech homology, one can construct a non-exact long homology sequence using a solenoid. In Steenrod-style homology theories,[3] the 0th homology group of a solenoid may have a fairly complicated structure, even though a solenoid is a connected space.
See also
• Protorus, a class of topological groups that includes the solenoids
• Pontryagin duality
• Inverse limit
• p-adic number
• Profinite integer
References
1. Vietoris, L. (December 1927). "Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen". Mathematische Annalen. 97 (1): 454–472. doi:10.1007/bf01447877. ISSN 0025-5831. S2CID 121172198.
2. van Dantzig, D. (1930). "Ueber topologisch homogene Kontinua". Fundamenta Mathematicae. 15: 102–125. doi:10.4064/fm-15-1-102-125. ISSN 0016-2736.
3. "Steenrod-Sitnikov homology - Encyclopedia of Mathematics".
• D. van Dantzig, Ueber topologisch homogene Kontinua, Fund. Math. 15 (1930), pp. 102–125
• "Solenoid", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Clark Robinson, Dynamical systems: Stability, Symbolic Dynamics and Chaos, 2nd edition, CRC Press, 1998 ISBN 978-0-8493-8495-0
• S. Smale, Differentiable dynamical systems, Bull. of the AMS, 73 (1967), 747 – 817.
• L. Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann. 97 (1927), pp. 454–472
• Robert F. Williams, Expanding attractors, Publ. Math. IHÉS, t. 43 (1974), p. 169–203
Further reading
• Semmes, Stephen (12 January 2012), Some remarks about solenoids, arXiv:1201.2647, Bibcode:2012arXiv1201.2647S
| Wikipedia |
Pedal equation
For a plane curve C and a given fixed point O, the pedal equation of the curve is a relation between r and p where r is the distance from O to a point on C and p is the perpendicular distance from O to the tangent line to C at the point. The point O is called the pedal point and the values r and p are sometimes called the pedal coordinates of a point relative to the curve and the pedal point. It is also useful to measure the distance of O to the normal $p_{c}$ (the contrapedal coordinate) even though it is not an independent quantity and it relates to $(r,p)$ as $p_{c}:={\sqrt {r^{2}-p^{2}}}$.
Some curves have particularly simple pedal equations and knowing the pedal equation of a curve may simplify the calculation of certain of its properties such as curvature. These coordinates are also well suited for solving certain type of force problems in classical mechanics and celestial mechanics.
Equations
Cartesian coordinates
For C given in rectangular coordinates by f(x, y) = 0, and with O taken to be the origin, the pedal coordinates of the point (x, y) are given by:[1]
$r={\sqrt {x^{2}+y^{2}}}$
$p={\frac {x{\frac {\partial f}{\partial x}}+y{\frac {\partial f}{\partial y}}}{\sqrt {\left({\frac {\partial f}{\partial x}}\right)^{2}+\left({\frac {\partial f}{\partial y}}\right)^{2}}}}.$
The pedal equation can be found by eliminating x and y from these equations and the equation of the curve.
The expression for p may be simplified if the equation of the curve is written in homogeneous coordinates by introducing a variable z, so that the equation of the curve is g(x, y, z) = 0. The value of p is then given by[2]
$p={\frac {\frac {\partial g}{\partial z}}{\sqrt {\left({\frac {\partial g}{\partial x}}\right)^{2}+\left({\frac {\partial g}{\partial y}}\right)^{2}}}}$
where the result is evaluated at z=1
Polar coordinates
For C given in polar coordinates by r = f(θ), then
$p=r\sin \phi $
where $\phi $ is the polar tangential angle given by
$r={\frac {dr}{d\theta }}\tan \phi .$
The pedal equation can be found by eliminating θ from these equations.[3]
Alternatively, from the above we can find that
$\left|{\frac {dr}{d\theta }}\right|={\frac {rp_{c}}{p}},$
where $p_{c}:={\sqrt {r^{2}-p^{2}}}$ is the "contrapedal" coordinate, i.e. distance to the normal. This implies that if a curve satisfies an autonomous differential equation in polar coordinates of the form:
$f\left(r,\left|{\frac {dr}{d\theta }}\right|\right)=0,$
its pedal equation becomes
$f\left(r,{\frac {rp_{c}}{p}}\right)=0.$
Example
As an example take the logarithmic spiral with the spiral angle α:
$r=ae^{{\frac {\cos \alpha }{\sin \alpha }}\theta }.$
Differentiating with respect to $\theta $ we obtain
${\frac {dr}{d\theta }}={\frac {\cos \alpha }{\sin \alpha }}ae^{{\frac {\cos \alpha }{\sin \alpha }}\theta }={\frac {\cos \alpha }{\sin \alpha }}r,$
hence
$\left|{\frac {dr}{d\theta }}\right|=\left|{\frac {\cos \alpha }{\sin \alpha }}\right|r,$
and thus in pedal coordinates we get
${\frac {r}{p}}p_{c}=\left|{\frac {\cos \alpha }{\sin \alpha }}\right|r,\qquad \Rightarrow \qquad |\sin \alpha |p_{c}=|\cos \alpha |p,$
or using the fact that $p_{c}^{2}=r^{2}-p^{2}$ we obtain
$p=|\sin \alpha |r.$
This approach can be generalized to include autonomous differential equations of any order as follows:[4] A curve C which a solution of an n-th order autonomous differential equation ($n\geq 1$) in polar coordinates
$f\left(r,|r'_{\theta }|,r''_{\theta },|r'''_{\theta }|\dots ,r_{\theta }^{(2j)},|r_{\theta }^{(2j+1)}|,\dots ,r_{\theta }^{(n)}\right)=0,$
is the pedal curve of a curve given in pedal coordinates by
$f(p,p_{c},p_{c}p_{c}',p_{c}(p_{c}p_{c}')',\dots ,(p_{c}\partial _{p})^{n}p)=0,$
where the differentiation is done with respect to $p$.
Force problems
Solutions to some force problems of classical mechanics can be surprisingly easily obtained in pedal coordinates.
Consider a dynamical system:
${\ddot {x}}=F^{\prime }(|x|^{2})x+2G^{\prime }(|x|^{2}){\dot {x}}^{\perp },$
describing an evolution of a test particle (with position $x$ and velocity ${\dot {x}}$) in the plane in the presence of central $F$ and Lorentz like $G$ potential. The quantities:
$L=x\cdot {\dot {x}}^{\perp }+G(|x|^{2}),\qquad c=|{\dot {x}}|^{2}-F(|x|^{2}),$
are conserved in this system.
Then the curve traced by $x$ is given in pedal coordinates by
${\frac {\left(L-G(r^{2})\right)^{2}}{p^{2}}}=F(r^{2})+c,$
with the pedal point at the origin. This fact was discovered by P. Blaschke in 2017.[5]
Example
As an example consider the so-called Kepler problem, i.e. central force problem, where the force varies inversely as a square of the distance:
${\ddot {x}}=-{\frac {M}{|x|^{3}}}x,$
we can arrive at the solution immediately in pedal coordinates
${\frac {L^{2}}{p^{2}}}={\frac {2M}{r}}+c,$,
where $L$ corresponds to the particle's angular momentum and $c$ to its energy. Thus we have obtained the equation of a conic section in pedal coordinates.
Inversely, for a given curve C, we can easily deduce what forces do we have to impose on a test particle to move along it.
Pedal equations for specific curves
Sinusoidal spirals
For a sinusoidal spiral written in the form
$r^{n}=a^{n}\sin(n\theta )$
the polar tangential angle is
$\psi =n\theta $
which produces the pedal equation
$pa^{n}=r^{n+1}.$
The pedal equation for a number of familiar curves can be obtained setting n to specific values:[6]
n Curve Pedal point Pedal eq.
1 Circle with radius a Point on circumference pa = r2
−1 Line Point distance a from line p = a
1⁄2 Cardioid Cusp p2a = r3
−1⁄2 Parabola Focus p2 = ar
2 Lemniscate of Bernoulli Center pa2 = r3
−2 Rectangular hyperbola Center rp = a2
Spirals
A spiral shaped curve of the form
$r=c\theta ^{\alpha },$
satisfies the equation
${\frac {dr}{d\theta }}=\alpha r^{\frac {\alpha -1}{\alpha }},$
and thus can be easily converted into pedal coordinates as
${\frac {1}{p^{2}}}={\frac {\alpha ^{2}c^{\frac {2}{\alpha }}}{r^{2+{\frac {2}{\alpha }}}}}+{\frac {1}{r^{2}}}.$
Special cases include:
$\alpha $ Curve Pedal point Pedal eq.
1 Spiral of Archimedes Origin ${\frac {1}{p^{2}}}={\frac {1}{r^{2}}}+{\frac {c^{2}}{r^{4}}}$
−1 Hyperbolic spiral Origin ${\frac {1}{p^{2}}}={\frac {1}{r^{2}}}+{\frac {1}{c^{2}}}$
1⁄2 Fermat's spiral Origin ${\frac {1}{p^{2}}}={\frac {1}{r^{2}}}+{\frac {c^{4}}{4r^{6}}}$
−1⁄2 Lituus Origin ${\frac {1}{p^{2}}}={\frac {1}{r^{2}}}+{\frac {r^{2}}{4c^{4}}}$
Epi- and hypocycloids
For an epi- or hypocycloid given by parametric equations
$x(\theta )=(a+b)\cos \theta -b\cos \left({\frac {a+b}{b}}\theta \right)$
$y(\theta )=(a+b)\sin \theta -b\sin \left({\frac {a+b}{b}}\theta \right),$
the pedal equation with respect to the origin is[7]
$r^{2}=a^{2}+{\frac {4(a+b)b}{(a+2b)^{2}}}p^{2}$
or[8]
$p^{2}=A(r^{2}-a^{2})$
with
$A={\frac {(a+2b)^{2}}{4(a+b)b}}.$
Special cases obtained by setting b=a⁄n for specific values of n include:
n Curve Pedal eq.
1, −1⁄2 Cardioid $p^{2}={\frac {9}{8}}(r^{2}-a^{2})$
2, −2⁄3 Nephroid $p^{2}={\frac {4}{3}}(r^{2}-a^{2})$
−3, −3⁄2 Deltoid $p^{2}=-{\frac {1}{8}}(r^{2}-a^{2})$
−4, −4⁄3 Astroid $p^{2}=-{\frac {1}{3}}(r^{2}-a^{2})$
Other curves
Other pedal equations are:,[9]
Curve Equation Pedal point Pedal eq.
Line $ax+by+c=0$ Origin $p={\frac {|c|}{\sqrt {a^{2}+b^{2}}}}$
Point $(x_{0},y_{0})$ Origin $r={\sqrt {x_{0}^{2}+y_{0}^{2}}}$
Circle $|x-a|=R$ Origin $2pR=r^{2}+R^{2}-|a|^{2}$
Involute of a circle $r={\frac {a}{\cos \alpha }},\ \theta =\tan \alpha -\alpha $ Origin $p_{c}=|a|$
Ellipse ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$ Center ${\frac {a^{2}b^{2}}{p^{2}}}+r^{2}=a^{2}+b^{2}$
Hyperbola ${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1$ Center $-{\frac {a^{2}b^{2}}{p^{2}}}+r^{2}=a^{2}-b^{2}$
Ellipse ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$ Focus ${\frac {b^{2}}{p^{2}}}={\frac {2a}{r}}-1$
Hyperbola ${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1$ Focus ${\frac {b^{2}}{p^{2}}}={\frac {2a}{r}}+1$
Logarithmic spiral $r=ae^{\theta \cot \alpha }$ Pole $p=r\sin \alpha $
Cartesian oval $|x|+\alpha |x-a|=C,$ Focus ${\frac {(b-(1-\alpha ^{2})r^{2})^{2}}{4p^{2}}}={\frac {Cb}{r}}+(1-\alpha ^{2})Cr-((1-\alpha ^{2})C^{2}+b),\ b:=C^{2}-\alpha ^{2}|a|^{2}$
Cassini oval $|x||x-a|=C,$ Focus ${\frac {(3C^{2}+r^{4}-|a|^{2}r^{2})^{2}}{p^{2}}}=4C^{2}\left({\frac {2C^{2}}{r^{2}}}+2r^{2}-|a|^{2}\right).$
Cassini oval $|x-a||x+a|=C,$ Center $2Rpr=r^{4}+R^{2}-|a|^{2}.$
See also
• Pedal curve
References
1. Yates §1
2. Edwards p. 161
3. Yates p. 166, Edwards p. 162
4. Blaschke Proposition 1
5. Blaschke Theorem 2
6. Yates p. 168, Edwards p. 162
7. Edwards p. 163
8. Yates p. 163
9. Yates p. 169, Edwards p. 163, Blaschke sec. 2.1
• R.C. Yates (1952). "Pedal Equations". A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards. pp. 166 ff.
• J. Edwards (1892). Differential Calculus. London: MacMillan and Co. pp. 161 ff.
• P. Blaschke (2017). "Pedal coordinates, dark Kepler and other force problems" (PDF). Journal of Mathematical Physics. 58/6. arXiv:1704.00897. doi:10.1063/1.4984905.
External links
• Weisstein, Eric W. "Pedal coordinates". MathWorld.
| Wikipedia |
Tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve.[1] More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c if the line passes through the point (c, f(c)) on the curve and has slope f'(c), where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
For the tangent function, see Tangent (trigonometry). For other uses, see Tangent (disambiguation).
As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.
The tangent line to a point on a differentiable curve can also be thought of as a tangent line approximation, the graph of the affine function that best approximates the original function at the given point.[2]
Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space.
The word "tangent" comes from the Latin tangere, "to touch".
History
Euclid makes several references to the tangent (ἐφαπτομένη ephaptoménē) to a circle in book III of the Elements (c. 300 BC).[3] In Apollonius' work Conics (c. 225 BC) he defines a tangent as being a line such that no other straight line could fall between it and the curve.[4]
Archimedes (c. 287 – c. 212 BC) found the tangent to an Archimedean spiral by considering the path of a point moving along the curve.[4]
In the 1630s Fermat developed the technique of adequality to calculate tangents and other problems in analysis and used this to calculate tangents to the parabola. The technique of adequality is similar to taking the difference between $f(x+h)$ and $f(x)$ and dividing by a power of $h$. Independently Descartes used his method of normals based on the observation that the radius of a circle is always normal to the circle itself.[5]
These methods led to the development of differential calculus in the 17th century. Many people contributed. Roberval discovered a general method of drawing tangents, by considering a curve as described by a moving point whose motion is the resultant of several simpler motions.[6] René-François de Sluse and Johannes Hudde found algebraic algorithms for finding tangents.[7] Further developments included those of John Wallis and Isaac Barrow, leading to the theory of Isaac Newton and Gottfried Leibniz.
An 1828 definition of a tangent was "a right line which touches a curve, but which when produced, does not cut it".[8] This old definition prevents inflection points from having any tangent. It has been dismissed and the modern definitions are equivalent to those of Leibniz, who defined the tangent line as the line through a pair of infinitely close points on the curve.
Tangent line to a plane curve
Further information: Differentiable curve § Tangent vector, and Frenet–Serret formulas
The intuitive notion that a tangent line "touches" a curve can be made more explicit by considering the sequence of straight lines (secant lines) passing through two points, A and B, those that lie on the function curve. The tangent at A is the limit when point B approximates or tends to A. The existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as "differentiability." For example, if two circular arcs meet at a sharp point (a vertex) then there is no uniquely defined tangent at the vertex because the limit of the progression of secant lines depends on the direction in which "point B" approaches the vertex.
At most points, the tangent touches the curve without crossing it (though it may, when continued, cross the curve at other places away from the point of tangent). A point where the tangent (at this point) crosses the curve is called an inflection point. Circles, parabolas, hyperbolas and ellipses do not have any inflection point, but more complicated curves do have, like the graph of a cubic function, which has exactly one inflection point, or a sinusoid, which has two inflection points per each period of the sine.
Conversely, it may happen that the curve lies entirely on one side of a straight line passing through a point on it, and yet this straight line is not a tangent line. This is the case, for example, for a line passing through the vertex of a triangle and not intersecting it otherwise—where the tangent line does not exist for the reasons explained above. In convex geometry, such lines are called supporting lines.
Analytical approach
The geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th century. In the second book of his Geometry, René Descartes[9] said of the problem of constructing the tangent to a curve, "And I dare say that this is not only the most useful and most general problem in geometry that I know, but even that I have ever desired to know".[10]
Intuitive description
Suppose that a curve is given as the graph of a function, y = f(x). To find the tangent line at the point p = (a, f(a)), consider another nearby point q = (a + h, f(a + h)) on the curve. The slope of the secant line passing through p and q is equal to the difference quotient
${\frac {f(a+h)-f(a)}{h}}.$
As the point q approaches p, which corresponds to making h smaller and smaller, the difference quotient should approach a certain limiting value k, which is the slope of the tangent line at the point p. If k is known, the equation of the tangent line can be found in the point-slope form:
$y-f(a)=k(x-a).\,$
More rigorous description
To make the preceding reasoning rigorous, one has to explain what is meant by the difference quotient approaching a certain limiting value k. The precise mathematical formulation was given by Cauchy in the 19th century and is based on the notion of limit. Suppose that the graph does not have a break or a sharp edge at p and it is neither plumb nor too wiggly near p. Then there is a unique value of k such that, as h approaches 0, the difference quotient gets closer and closer to k, and the distance between them becomes negligible compared with the size of h, if h is small enough. This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function f. This limit is the derivative of the function f at x = a, denoted f ′(a). Using derivatives, the equation of the tangent line can be stated as follows:
$y=f(a)+f'(a)(x-a).\,$
Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the power function, trigonometric functions, exponential function, logarithm, and their various combinations. Thus, equations of the tangents to graphs of all these functions, as well as many others, can be found by the methods of calculus.
How the method can fail
Calculus also demonstrates that there are functions and points on their graphs for which the limit determining the slope of the tangent line does not exist. For these points the function f is non-differentiable. There are two possible reasons for the method of finding the tangents based on the limits and derivatives to fail: either the geometric tangent exists, but it is a vertical line, which cannot be given in the point-slope form since it does not have a slope, or the graph exhibits one of three behaviors that precludes a geometric tangent.
The graph y = x1/3 illustrates the first possibility: here the difference quotient at a = 0 is equal to h1/3/h = h−2/3, which becomes very large as h approaches 0. This curve has a tangent line at the origin that is vertical.
The graph y = x2/3 illustrates another possibility: this graph has a cusp at the origin. This means that, when h approaches 0, the difference quotient at a = 0 approaches plus or minus infinity depending on the sign of x. Thus both branches of the curve are near to the half vertical line for which y=0, but none is near to the negative part of this line. Basically, there is no tangent at the origin in this case, but in some context one may consider this line as a tangent, and even, in algebraic geometry, as a double tangent.
The graph y = |x| of the absolute value function consists of two straight lines with different slopes joined at the origin. As a point q approaches the origin from the right, the secant line always has slope 1. As a point q approaches the origin from the left, the secant line always has slope −1. Therefore, there is no unique tangent to the graph at the origin. Having two different (but finite) slopes is called a corner.
Finally, since differentiability implies continuity, the contrapositive states discontinuity implies non-differentiability. Any such jump or point discontinuity will have no tangent line. This includes cases where one slope approaches positive infinity while the other approaches negative infinity, leading to an infinite jump discontinuity
Equations
When the curve is given by y = f(x) then the slope of the tangent is $dy/dx,$ so by the point–slope formula the equation of the tangent line at (X, Y) is
$y-Y={\frac {dy}{dx}}(X)\cdot (x-X)$
where (x, y) are the coordinates of any point on the tangent line, and where the derivative is evaluated at $x=X$.[11]
When the curve is given by y = f(x), the tangent line's equation can also be found[12] by using polynomial division to divide $f\,(x)$ by $(x-X)^{2}$; if the remainder is denoted by $g(x)$, then the equation of the tangent line is given by
$y=g(x).$
When the equation of the curve is given in the form f(x, y) = 0 then the value of the slope can be found by implicit differentiation, giving
${\frac {dy}{dx}}=-{\frac {\partial f}{\partial x}}{\bigg /}{\frac {\partial f}{\partial y}}.$
The equation of the tangent line at a point (X,Y) such that f(X,Y) = 0 is then[11]
${\frac {\partial f}{\partial x}}(X,Y)\cdot (x-X)+{\frac {\partial f}{\partial y}}(X,Y)\cdot (y-Y)=0.$
This equation remains true if
${\frac {\partial f}{\partial y}}(X,Y)=0,\quad {\frac {\partial f}{\partial x}}(X,Y)\neq 0,$
in which case the slope of the tangent is infinite. If, however,
${\frac {\partial f}{\partial y}}(X,Y)={\frac {\partial f}{\partial x}}(X,Y)=0,$
the tangent line is not defined and the point (X,Y) is said to be singular.
For algebraic curves, computations may be simplified somewhat by converting to homogeneous coordinates. Specifically, let the homogeneous equation of the curve be g(x, y, z) = 0 where g is a homogeneous function of degree n. Then, if (X, Y, Z) lies on the curve, Euler's theorem implies
${\frac {\partial g}{\partial x}}\cdot X+{\frac {\partial g}{\partial y}}\cdot Y+{\frac {\partial g}{\partial z}}\cdot Z=ng(X,Y,Z)=0.$
It follows that the homogeneous equation of the tangent line is
${\frac {\partial g}{\partial x}}(X,Y,Z)\cdot x+{\frac {\partial g}{\partial y}}(X,Y,Z)\cdot y+{\frac {\partial g}{\partial z}}(X,Y,Z)\cdot z=0.$
The equation of the tangent line in Cartesian coordinates can be found by setting z=1 in this equation.[13]
To apply this to algebraic curves, write f(x, y) as
$f=u_{n}+u_{n-1}+\dots +u_{1}+u_{0}\,$
where each ur is the sum of all terms of degree r. The homogeneous equation of the curve is then
$g=u_{n}+u_{n-1}z+\dots +u_{1}z^{n-1}+u_{0}z^{n}=0.\,$
Applying the equation above and setting z=1 produces
${\frac {\partial f}{\partial x}}(X,Y)\cdot x+{\frac {\partial f}{\partial y}}(X,Y)\cdot y+{\frac {\partial g}{\partial z}}(X,Y,1)=0$
as the equation of the tangent line.[14] The equation in this form is often simpler to use in practice since no further simplification is needed after it is applied.[13]
If the curve is given parametrically by
$x=x(t),\quad y=y(t)$
then the slope of the tangent is
${\frac {dy}{dx}}={\frac {dy}{dt}}{\bigg /}{\frac {dx}{dt}}$
giving the equation for the tangent line at $\,t=T,\,X=x(T),\,Y=y(T)$ as[15]
${\frac {dx}{dt}}(T)\cdot (y-Y)={\frac {dy}{dt}}(T)\cdot (x-X).$
If
${\frac {dx}{dt}}(T)={\frac {dy}{dt}}(T)=0,$
the tangent line is not defined. However, it may occur that the tangent line exists and may be computed from an implicit equation of the curve.
Normal line to a curve
Further information: Normal (geometry)
The line perpendicular to the tangent line to a curve at the point of tangency is called the normal line to the curve at that point. The slopes of perpendicular lines have product −1, so if the equation of the curve is y = f(x) then slope of the normal line is
$-{\frac {1}{\frac {dy}{dx}}}$
and it follows that the equation of the normal line at (X, Y) is
$(x-X)+{\frac {dy}{dx}}(y-Y)=0.$
Similarly, if the equation of the curve has the form f(x, y) = 0 then the equation of the normal line is given by[16]
${\frac {\partial f}{\partial y}}(x-X)-{\frac {\partial f}{\partial x}}(y-Y)=0.$
If the curve is given parametrically by
$x=x(t),\quad y=y(t)$
then the equation of the normal line is[15]
${\frac {dx}{dt}}(x-X)+{\frac {dy}{dt}}(y-Y)=0.$
Angle between curves
See also: Angle § Angles between curves
The angle between two curves at a point where they intersect is defined as the angle between their tangent lines at that point. More specifically, two curves are said to be tangent at a point if they have the same tangent at a point, and orthogonal if their tangent lines are orthogonal.[17]
Multiple tangents at a point
The formulas above fail when the point is a singular point. In this case there may be two or more branches of the curve that pass through the point, each branch having its own tangent line. When the point is the origin, the equations of these lines can be found for algebraic curves by factoring the equation formed by eliminating all but the lowest degree terms from the original equation. Since any point can be made the origin by a change of variables (or by translating the curve) this gives a method for finding the tangent lines at any singular point.
For example, the equation of the limaçon trisectrix shown to the right is
$(x^{2}+y^{2}-2ax)^{2}=a^{2}(x^{2}+y^{2}).\,$
Expanding this and eliminating all but terms of degree 2 gives
$a^{2}(3x^{2}-y^{2})=0\,$
which, when factored, becomes
$y=\pm {\sqrt {3}}x.$
So these are the equations of the two tangent lines through the origin.[18]
When the curve is not self-crossing, the tangent at a reference point may still not be uniquely defined because the curve is not differentiable at that point although it is differentiable elsewhere. In this case the left and right derivatives are defined as the limits of the derivative as the point at which it is evaluated approaches the reference point from respectively the left (lower values) or the right (higher values). For example, the curve y = |x | is not differentiable at x = 0: its left and right derivatives have respective slopes −1 and 1; the tangents at that point with those slopes are called the left and right tangents.[19]
Sometimes the slopes of the left and right tangent lines are equal, so the tangent lines coincide. This is true, for example, for the curve y = x 2/3, for which both the left and right derivatives at x = 0 are infinite; both the left and right tangent lines have equation x = 0.
Tangent line to a space curve
This section is an excerpt from Tangent vector.[edit]
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in terms of germs. Formally, a tangent vector at the point $x$ is a linear derivation of the algebra defined by the set of germs at $x$.
Tangent circles
Main article: Tangent circles
Two circles of non-equal radius, both in the same plane, are said to be tangent to each other if they meet at only one point. Equivalently, two circles, with radii of ri and centers at (xi, yi), for i = 1, 2 are said to be tangent to each other if
$\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}\pm r_{2}\right)^{2}.\,$
• Two circles are externally tangent if the distance between their centres is equal to the sum of their radii.
$\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}+r_{2}\right)^{2}.\,$
• Two circles are internally tangent if the distance between their centres is equal to the difference between their radii.[20]
$\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}-r_{2}\right)^{2}.\,$
Tangent plane to a surface
Further information: Differential geometry of surfaces § Tangent plane, and Parametric surface § Tangent plane
See also: Normal plane (geometry)
The tangent plane to a surface at a given point p is defined in an analogous way to the tangent line in the case of curves. It is the best approximation of the surface by a plane at p, and can be obtained as the limiting position of the planes passing through 3 distinct points on the surface close to p as these points converge to p.
Higher-dimensional manifolds
Main article: Tangent space
More generally, there is a k-dimensional tangent space at each point of a k-dimensional manifold in the n-dimensional Euclidean space.
See also
• Newton's method
• Normal (geometry)
• Osculating circle
• Osculating curve
• Perpendicular
• Subtangent
• Supporting line
• Tangent cone
• Tangential angle
• Tangential component
• Tangent lines to circles
• Tangent vector
• Multiplicity (mathematics)#Behavior of a polynomial function near a multiple root
• Algebraic curve#Tangent at a point
References
1. Leibniz, G., "Nova Methodus pro Maximis et Minimis", Acta Eruditorum, Oct. 1684.
2. Dan Sloughter (2000) . "Best Affine Approximations"
3. Euclid. "Euclid's Elements". Retrieved 1 June 2015.
4. Shenk, Al. "e-CALCULUS Section 2.8" (PDF). p. 2.8. Retrieved 1 June 2015.
5. Katz, Victor J. (2008). A History of Mathematics (3rd ed.). Addison Wesley. p. 510. ISBN 978-0321387004.
6. Wolfson, Paul R. (2001). "The Crooked Made Straight: Roberval and Newton on Tangents". The American Mathematical Monthly. 108 (3): 206–216. doi:10.2307/2695381. JSTOR 2695381.
7. Katz, Victor J. (2008). A History of Mathematics (3rd ed.). Addison Wesley. pp. 512–514. ISBN 978-0321387004.
8. Noah Webster, American Dictionary of the English Language (New York: S. Converse, 1828), vol. 2, p. 733,
9. Descartes, René (1954) [1637]. The Geometry of René Descartes. Translated by Smith, David Eugene; Latham, Marcia L. Open Court. p. 95.
10. R. E. Langer (October 1937). "Rene Descartes". American Mathematical Monthly. Mathematical Association of America. 44 (8): 495–512. doi:10.2307/2301226. JSTOR 2301226.
11. Edwards Art. 191
12. Strickland-Constable, Charles, "A simple method for finding tangents to polynomial graphs", Mathematical Gazette, November 2005, 466–467.
13. Edwards Art. 192
14. Edwards Art. 193
15. Edwards Art. 196
16. Edwards Art. 194
17. Edwards Art. 195
18. Edwards Art. 197
19. Thomas, George B. Jr., and Finney, Ross L. (1979), Calculus and Analytic Geometry, Addison Wesley Publ. Co.: p. 140.
20. "Circles For Leaving Certificate Honours Mathematics by Thomas O'Sullivan 1997".
Sources
• J. Edwards (1892). Differential Calculus. London: MacMillan and Co. pp. 143 ff.
External links
Wikimedia Commons has media related to Tangency.
Wikisource has the text of the 1921 Collier's Encyclopedia article Tangent.
• "Tangent line", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Tangent Line". MathWorld.
• Tangent to a circle With interactive animation
• Tangent and first derivative — An interactive simulation
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| Wikipedia |
Kleene's O
In set theory and computability theory, Kleene's ${\mathcal {O}}$ is a canonical subset of the natural numbers when regarded as ordinal notations. It contains ordinal notations for every computable ordinal, that is, ordinals below Church–Kleene ordinal, $\omega _{1}^{\text{CK}}$. Since $\omega _{1}^{\text{CK}}$ is the first ordinal not representable in a computable system of ordinal notations the elements of ${\mathcal {O}}$ can be regarded as the canonical ordinal notations.
Kleene (1938) described a system of notation for all computable ordinals (those less than the Church–Kleene ordinal). It uses a subset of the natural numbers instead of finite strings of symbols. Unfortunately, there is in general no effective way to tell whether some natural number represents an ordinal, or whether two numbers represent the same ordinal. However, one can effectively find notations which represent the ordinal sum, product, and power (see ordinal arithmetic) of any two given notations in Kleene's ${\mathcal {O}}$; and given any notation for an ordinal, there is a computably enumerable set of notations which contains one element for each smaller ordinal and is effectively ordered.
Definition
The basic idea of Kleene's system of ordinal notations is to build up ordinals in an effective manner. For members $p$ of ${\mathcal {O}}$, the ordinal for which $p$ is a notation is $|p|$. ${\mathcal {O}}$ and $<_{\mathcal {O}}$ (a partial ordering of Kleene's ${\mathcal {O}}$) are the smallest sets such that the following holds.
• $0\in {\mathcal {O}}\land |0|=0$.
• $i\in {\mathcal {O}}\land |i|=\alpha \rightarrow 2^{i}\in {\mathcal {O}}\land |2^{i}|=\alpha +1\land i<_{\mathcal {O}}2^{i}$
• Suppose $\{e\}$ is the $e$-th partial computable function. If $e$ is total and ${\textrm {range}}(\{e\})\subset {\mathcal {O}}\land \forall n(\{e\}(n)<_{\mathcal {O}}\{e\}(n+1))$, then $3\cdot 5^{e}\in {\mathcal {O}}\land \forall n,\{e\}(n)<_{\mathcal {O}}3\cdot 5^{e}\land |3\cdot 5^{e}|=\lim _{k}|\{e\}(k)|$
• $p<_{\mathcal {O}}q\land q<_{\mathcal {O}}r\rightarrow p<_{\mathcal {O}}r$
This definition has the advantages that one can computably enumerate the predecessors of a given ordinal (though not in the $<_{\mathcal {O}}$ ordering) and that the notations are downward closed, i.e., if there is a notation for $\gamma $ and $\alpha <\gamma $ then there is a notation for $\alpha $. There are alternate definitions, such as the set of indices of (partial) well-orderings of the natural numbers.[1]
Basic properties of <O
• If $|i|=\alpha $ and $|j|=\beta $ and $i<_{\mathcal {O}}j\,,$ then $\alpha <\beta $; but the converse may fail to hold.
• $<_{\mathcal {O}}$ induces a tree structure on ${\mathcal {O}}$, so ${\mathcal {O}}$ is well-founded.
• ${\mathcal {O}}$ only branches at limit ordinals; and at each notation of a limit ordinal, ${\mathcal {O}}$ is infinitely branching.
• Since every computable function has countably many indices, each infinite ordinal receives countably many notations; the finite ordinals have unique notations, $n$ usually denoted $n_{\mathcal {O}}$.
• The first ordinal that doesn't receive a notation is called the Church–Kleene ordinal and is denoted by $\omega _{1}^{\text{CK}}$. Since there are only countably many computable functions, the ordinal $\omega _{1}^{\text{CK}}$ is evidently countable.
• The ordinals with a notation in Kleene's ${\mathcal {O}}$ are exactly the computable ordinals. (The fact that every computable ordinal has a notation follows from the closure of this system of ordinal notations under successor and effective limits.)
• $<_{\mathcal {O}}$ is not computably enumerable, but there is a computably enumerable relation which agrees with $<_{\mathcal {O}}$ precisely on members of ${\mathcal {O}}$.
• For any notation $p$, the set $\lbrace q\mid q<_{\mathcal {O}}p\rbrace $ of notations below $p$ is computably enumerable. However, Kleene's ${\mathcal {O}}$, when taken as a whole, is $\Pi _{1}^{1}$ (see analytical hierarchy) and not arithmetical because of the following:
• ${\mathcal {O}}$ is $\Pi _{1}^{1}$-complete (i.e. ${\mathcal {O}}$ is $\Pi _{1}^{1}$ and every $\Pi _{1}^{1}$ set is Turing reducible to it)[2] and every $\Sigma _{1}^{1}$ subset of ${\mathcal {O}}$ is effectively bounded in ${\mathcal {O}}$ (a result of Spector).
• In fact, any $\Pi _{1}^{1}$ set is many-one reducible to ${\mathcal {O}}$.[2]
• ${\mathcal {O}}$ is the universal system of ordinal notations in the sense that any specific set of ordinal notations can be mapped into it in a straightforward way. More precisely, there is a computable function $f$ such that if $e$ is an index for a computable well-ordering, then $f(e)$ is a member of ${\mathcal {O}}$ and $<_{e}$ is order-isomorphic to an initial segment of the set $\{p\mid p<_{\mathcal {O}}f(e)\}$.
• There is a computable function $+_{\mathcal {O}}$, which, for members of ${\mathcal {O}}$, mimics ordinal addition and has the property that $\max\{p,q\}\leq p+_{\mathcal {O}}q$. (Jockusch)
Properties of paths in <O
A path in ${\mathcal {O}}$ is a subset ${\mathcal {P}}$ of ${\mathcal {O}}$ which is totally ordered by $<_{\mathcal {O}}$ and is closed under predecessors, i.e. if $p$ is a member of a path ${\mathcal {P}}$ and $q<_{\mathcal {O}}p$ then $q$ is also a member of ${\mathcal {P}}$. A path ${\mathcal {P}}$ is maximal if there is no element of ${\mathcal {O}}$ which is above (in the sense of $<_{\mathcal {O}}$) every member of ${\mathcal {P}}$, otherwise ${\mathcal {P}}$ is non-maximal.
• A path ${\mathcal {P}}$ is non-maximal if and only if ${\mathcal {P}}$ is computably enumerable (c.e.). It follows by remarks above that every element $p$ of ${\mathcal {O}}$ determines a non-maximal path ${\mathcal {P}}$; and every non-maximal path is so determined.
• There are $2^{\aleph _{0}}$ maximal paths through ${\mathcal {O}}$; since they are maximal, they are non-c.e.
• In fact, there are $2^{\aleph _{0}}$ maximal paths within ${\mathcal {O}}$ of length $\omega ^{2}$. (Crossley, Schütte)
• For every non-zero ordinal $\lambda <\omega _{1}^{CK}$, there are $2^{\aleph _{0}}$ maximal paths within ${\mathcal {O}}$ of length $\omega ^{2}\cdot \lambda $. (Aczel)
• Further, if ${\mathcal {P}}$ is a path whose length is not a multiple of $\omega ^{2}$ then ${\mathcal {P}}$ is not maximal. (Aczel)
• For each c.e. degree $d$, there is a member $e_{d}$ of ${\mathcal {O}}$ such that the path ${\mathcal {P}}=\lbrace p\mid p<_{\mathcal {O}}e_{d}\rbrace $ has many-one degree $d$. In fact, for each computable ordinal $\alpha \geq \omega ^{2}$, a notation $e_{d}$ exists with $|e_{d}|=\alpha $. (Jockusch)
• There exist $\aleph _{0}$ paths through ${\mathcal {O}}$ which are $\Pi _{1}^{1}$. Given a progression of computably enumerable theories based on iterating Uniform Reflection, each such path is incomplete with respect to the set of true $\Pi _{1}^{0}$ sentences. (Feferman & Spector)
• There exist $\Pi _{1}^{1}$ paths through ${\mathcal {O}}$ each initial segment of which is not merely c.e., but computable. (Jockusch)
• Various other paths in ${\mathcal {O}}$ have been shown to exist, each with specific kinds of reducibility properties. (See references below)
See also
• Computable ordinal
• Large countable ordinal
• Ordinal notation
References
1. S. G. Simpson, The Hierarchy Based on the Jump Operator, p.269. The Kleene Symposium (North-Holland, 1980)
2. Ash, Knight, *Computable Structures and the Hyperarithmetical Hierarchy* p.83. Studies in Logic and the Foundations of Mathematics vol. 144 (2000), ISBN 0-444-50072-3.
• Church, Alonzo (1938), "The constructive second number class", Bull. Amer. Math. Soc., 44 (4): 224–232, doi:10.1090/S0002-9904-1938-06720-1
• Kleene, S. C. (1938), "On Notation for Ordinal Numbers", The Journal of Symbolic Logic, Association for Symbolic Logic, 3 (4): 150–155, doi:10.2307/2267778, JSTOR 2267778, S2CID 34314018
• Rogers, Hartley (1987) [1967], The Theory of Recursive Functions and Effective Computability, First MIT press paperback edition, ISBN 978-0-262-68052-3
• Feferman, Solomon; Spector, Clifford (1962), "Incompleteness along paths in progressions of theories", Journal of Symbolic Logic, 27 (4): 383–390, doi:10.2307/2964544, JSTOR 2964544, S2CID 33892829
| Wikipedia |
High spatio-temporal-resolution detection of chlorophyll fluorescence dynamics from a single chloroplast with confocal imaging fluorometer
Yi-Chin Tseng1 &
Shi-Wei Chu ORCID: orcid.org/0000-0001-7728-43291,2
Chlorophyll fluorescence (CF) is a key indicator to study plant physiology or photosynthesis efficiency. Conventionally, CF is characterized by fluorometers, which only allows ensemble measurement through wide-field detection. For imaging fluorometers, the typical spatial and temporal resolutions are on the order of millimeter and second, far from enough to study cellular/sub-cellular CF dynamics. In addition, due to the lack of optical sectioning capability, conventional imaging fluorometers cannot identify CF from a single cell or even a single chloroplast.
Here we demonstrated a fluorometer based on confocal imaging, that not only provides high contrast images, but also allows CF measurement with spatiotemporal resolution as high as micrometer and millisecond. CF transient (the Kautsky curve) from a single chloroplast is successfully obtained, with both the temporal dynamics and the intensity dependences corresponding well to the ensemble measurement from conventional studies. The significance of confocal imaging fluorometer is to identify the variation among individual chloroplasts, e.g. the temporal position of the P–S–M phases, and the half-life period of P–T decay in the Kautsky curve, that are not possible to analyze with wide-field techniques. A linear relationship is found between excitation intensity and the temporal positions of P–S–M peaks/valleys in the Kautsky curve. Based on the CF transients, the photosynthetic quantum efficiency is derived with spatial resolution down to a single chloroplast. In addition, an interesting 6-order increase in excitation intensity is found between wide-field and confocal fluorometers, whose pixel integration time and optical sectioning may account for this substantial difference.
Confocal imaging fluorometers provide micrometer and millisecond CF characterization, opening up unprecedented possibilities toward detailed spatiotemporal analysis of CF transients and its propagation dynamics, as well as photosynthesis efficiency analysis, on the scale of organelles, in a living plant.
Chlorophyll fluorescence (CF) has been proven to be one of the most powerful and widely used techniques for plant physiologists [1,2,3,4,5,6,7]. Despite of its low quantum efficiency (2–10% of absorbed light [8]), CF detections are meaningful due to its intricate connection with numerous internal processes during photosynthesis, such as reduction of photosystem reaction centers, non-photochemical quenching, etc. [9, 10]. It is well known that the efficiency of photosynthesis can be derived from CF dynamics, thus providing noninvasive, fast and accurate characterization for photosynthesis. CF characterization has been widely adopted to study plant physiology, including stress tolerance, nitrogen balance, carbon fixation efficiency, etc. [11]. It is not too exaggerated to say that nowadays, no investigation about photosynthetic process would be complete without CF analysis.
Conventionally, the tool of choice to study CF is a fluorometer. There are many different fluorometry techniques, such as plant efficiency analyzer (PEA) [12], pulse amplitude modulation (PAM) [13], the pump and probe (P&P) [14, 15] and the fast repetition rate (FRR) [16]. It is interesting to note that these various detection approaches are all based on the same principle, i.e. the Kautsky effect [7], or equivalent, CF transient when moving photosynthetic material from dark adaption to light environment.
Conventional imaging fluorometers (e.g. PAM and P&P fluorometers) are based on wide-field detection, and are routinely adopted to study ensemble of CF transients from a large area of a leaf, significantly limiting its spatiotemporal resolution. For example, to study stress propagation in a plant leaf [17], current imaging fluorometers only provide spatial resolution on the order of millimeter, with temporal resolution on the order of second. To unravel the more detailed propagation dynamics, the required spatial resolution should be at least on single cell or sub-cellular level, while the temporal resolution should be enhanced to millisecond scale.
The concept of introducing fluorescent microscope to study high-resolution CF dynamics has been realized two decades ago [18], but the drawback of the early microscopic fluorometer version is the lack of optical section capability due to its wide-field nature, and thus prevents study of CF transient on a truly single cell or even a single chloroplast level. Confocal microscopy, which is known to provide optical sectioning with exceptionally high axial contrast, has been extensively used for CF imaging with sub-micrometer resolution [19,20,21,22,23]. However, the high-speed time-lapsed imaging capability is less explored in earlier works.
Here we introduce a concept of confocal imaging fluorometer, which is the combination of confocal microscopy and CF transient detection. The technique not only detects CF signals with millisecond temporal resolution, but also attains micrometer spatial resolution in all three dimensions. The CF transient (Kautsky curve) within a single chloroplast is successfully retrieved. With statistical comparison, the CF transients of a group of palisade cells and the ensemble of single chloroplasts are found to be similar to each other, and both correspond well to the result of conventional imaging fluorometers, showing the reliability of our result. Nevertheless, the CF transient of individual chloroplast can be substantially different, manifesting the value of the unusual capability to study plant cell organelles. Furthermore, we found that the shape of transients is highly intensity-dependent, which is also shown in an earlier study [24]. We also found that the short integration time and optical section characteristic of confocal image fluorometer make a significant difference of illumination intensity comparing to that of conventional fluorometers. Given CF transient from a single chloroplast, it is possible to investigate degree of influence from external or internal plant-stress with scale of organelle, and confocal imaging fluorometer has paved the way for this high spatiotemporal resolution CF detection.
Basic concept of confocal imaging fluorometer
The optical principle of confocal imaging fluorometer is basically the same as confocal laser-scanning microscopy [25], which is an optical imaging technique for increasing contrast and resolution. The essential components of a confocal imaging fluorometer is shown in Fig. 1, including a laser system, a dichroic mirror, a scanning mirror system, an objective lens, a pinhole and a photomultiplier tube (PMT).
Principle and basic components of a confocal imaging fluorometer. Laser beam is reflected by a dichroic mirror and goes through a set of scanning mirrors, then focused by an objective lens onto the specimen. Fluorescence signals is epi-collected in the same path, and filtered out by the dichroic mirror. A confocal pinhole is used to allow only fluorescence emitted from the focal plane being detected by the PMT
The laser system in a confocal imaging fluorometer provides strong and monochromatic illumination, whose wavelength can be selected to meet sample request. The laser beam is sent to the objective after the scanning mirror system to achieve two-dimensional raster scanning at the focal plane. The backward fluorescence signal is collected by the same objective, de-scanned through the scanning mirrors, and separated from residual laser by the dichroic mirror. The fluorescence signal then is focused onto the pinhole, which is placed at the conjugate plane of objective focus, to achieve optical sectioning by excluding out-of-focus signals. One or more PMTs are placed behind the pinhole to collect the in-focus fluorescence signals, which are reconstructed into images by synchronization with the scanning mirrors [25].
In general, a confocal imaging system is capable of collecting signal with a well-defined optical section on the order of 1 µm [26]. This high axial resolution makes confocal system an invaluable tool to observe single cell or sub-cellular organelles [27,28,29].
The objective lens is characterized by magnification and numerical aperture (NA). To enable large field-of-view observation, low magnification objectives are typically required. However, please note that resolution is determined by NA, which can be independent from magnification. NA describes the light acceptance cone of an objective lens and hence light gathering ability and resolution. The definition of NA is:
$$NA \equiv n \times \sin \theta ,$$
where n is the index of refraction of the immersion medium, and θ is the half-angle of the maximum light acceptance cone. Both lateral (xy-direction) and axial (z-direction) resolutions for fluorescence imaging mode are defined by NA and the wavelength (λ) [30].
$$r_{lateral} = \frac{0.43 \times \lambda }{NA}$$
$$r_{axial} = \frac{0.67 \times \lambda }{{n - \sqrt {n^{2} - NA^{2} } }}$$
To compare the actinic light illumination in a conventional fluorometer, e.g. PAM, and in a laser scanning confocal fluorometer, there are several aspects. First, in PAM the actinic light is provided by a lamp or an LED, which is an incoherent light source; while in a confocal system, the laser excitation is coherent. Second, the spectral bandwidth of a laser is in general much narrower than that of a lamp, which is typically tens of nanometers even after adding bandpass filters. Third, wide-field illumination is adopted in PAM, while point-scan is used in confocal.
Although there are many differences between the illumination method of the conventional fluorometer and the confocal one, in an early work [31], it has been shown that the actinic effect of using a Xe lamp or a laser is equivalent. In a more recent work [23], they have shown that frequency of scanning (~300 s−1) does not seem to affect the response, even when compared to wide-field illumination. In our current work, the scanning frequency on each chloroplast is about 10,000 s−1. However, as we will show in the results, clear OPSMT transitions and similar intensity-dependent CF dynamics are all observable. Therefore, it seems that the high-frequency laser beam movement does not cause significant effects on CF dynamics.
Kautsky effect
Kautsky effect, discovered in 1931, describes the dynamics of CF when dark-adapted photosynthetic chlorophyll suddenly exposes to continuous light illumination [32]. After initial light absorption, chlorophyll becomes excited and soon releases its energy into one of the three internal decay pathways, including photosynthesis (photochemical quenching, qP), heat (non-photochemical quenching, NPQ) and light emission (CF). Owing to energy conservation, the sum of quantum efficiencies for these three pathways should be unity. Therefore, the yield of CF is strongly related to the efficiency of both qP and NPQ [33].
To be more specific, when transferring a photosynthetic material from dark adaption into light illumination, CF yield typically exhibits a fast rising phase (within 1 s) and a slow decay phase (few minute duration), as shown by the green curve in Fig. 2. The fast rising phase is labeled as O–P, where O is for origin, and P is the peak [24]. It is mainly caused by the reduction of qP; that is, depletion of electron acceptors, quinine (Qa) in the electron transport chain [34]. The slow decay phase is labeled as P–S–M–T, where S stands for semisteady state, M for a local maximum, and T for a terminal steady state level [24]. One very interesting phenomenon is the shape of this decay phase depends strongly on illumination intensity. At low intensity (32 μmol/m2/s), the Kautsky curve is the green one. When the intensity grows one order larger, the amplitude of S–M rise in the transient is smaller, as shown by the red curve. At one more order higher intensity, the blue curve shows that the S–M section disappears completely, leaving an exponential decay in the P–T section. This is known as saturation state, which is critical to derive the quantum efficiency of photosynthesis. Such intensity-dependent curve transition is the result of photosynthetic state transition, and more detailed discussion can be found in the references [1, 10, 13, 35,36,37].
(Modified figure from [1], with copyright permission)
The Kautsky effect, showing the CF transient as well as its intensity dependence. Wavelength of excitation: 650 nm. Excitation light intensity for curves labeled 1, 2 and 3 was 32, 320 and 3200 μmol/m2/s, respectively. For definition of OPSMT, O is the origin, P is the peak, S stands for semi-steady state, M for a local maximum, and T for a terminal steady state level
Plant sample
Brugmansia suaveolens (solanaceae), also known as Angel's Trumpet, was a woody plant usually 3–4 m in height with pendulous flowers and furry leaves distributed widely in Taiwan, especially in wet areas. Being interested in spatiotemporal dynamics of CF, we selected B. suaveolens as our target material since the CF of its cousin Datura wrightii, also known as Devil's Trumpet, had been studied in depth [17]. B. suaveolens leaves were collected from the Botanical Garden of National Taiwan University, Taipei, Taiwan (25°1′N, 121°31′E, 9 m a.s.l.). All sample leaves were picked as fully expanded leaves that had neither experienced detectable physical damage nor herbivory. In order to minimize the sampling error, three leaves were chosen within plants that grew in similar micro-climate. Furthermore, all the measurements were completed no longer than two hours after disleaving. Fresh leaves were sealed in slide glass (76 × 26 mm), and slide samples were dark-adapted under constant temperature and constant humidity dark environment (20 °C, 70%RH) for 20 min.
A confocal microscope (Leica TCS SP5) in the Molecular Imaging Center of National Taiwan University was adopted. CF was excited by a HeNe laser, whose wavelength (633 nm) was the same as that used in popular conventional fluorometers, such as LI-6400 from LI-COR. A relatively low-NA objective (HC PL Apo 10×/0.4 CS) was selected to allow not only large field of view over a few millimeters, but also spatial resolution better than a single chloroplast. From Eqs. (2) and (3), the lateral and axial resolutions were 1 and 5 µm, respectively. Although this was not particularly high compared to common confocal imaging system, due to the low-NA objective here, the three-dimensional spatial resolution was much better than conventional imaging fluorometers.
To operate the confocal fluorometer, the initial step was to bring the sample to focus by weak excitation (~1 kW/cm2, or equivalently 5.56 × 107 μmol/m2/s for intensity conversion, please see "Discussion"), and then the leaf was left in dark again for 5 min. To observe the Kautsky effect, the 633-nm laser was focused on the sample, and the fluorescence emission was recorded in the spectral range of 670–690 nm. The intensity-dependent CF transient curves were obtained by taking time-lapsed images while varying the 633-nm excitation intensity from 1 to 55 kW/cm2, at different sample regions. For experimental details, the scanning speed was 1400 Hz (1400 rows per second), the pinhole size was 52 μm (one Airy diameter), the built-in PMT voltage was set at 600 V, and a dichroic filter TD 488/543/633 was included in the optical path. With different number of total pixels, the temporal resolution of the CF transient varies from 10 ms (16 × 16 pixels) to about 200 ms (256 × 256 pixels). No significant photobleaching of CF was expected at this intensity range [38].
Fluorescence dynamics from a single chloroplast
Conventional fluorometers observe CF dynamics over a large area on a leaf, and here we demonstrate that the confocal imaging fluorometer allows us to obtain CF transients from a precisely chosen cell or even a single chloroplast. Figure 3a shows the confocal images of a leaf sample. (a1) is the large-area view, showing the distribution of vascular bundles, while (a2) gives a zoom-in view of a group of palisade cells, showing clear distribution of chloroplasts in each cell. By further zooming in, the field of view is focused onto a single chloroplast, as given in (a3), showing the distribution of chlorophyll density inside the organelle [39].
Confocal images and CF transients on different spatial scales inside a living leaf. A 633-nm laser, with 3 kW/cm2 intensity, is adopted for 50 s continuous confocal imaging. The sample leaf was kept in darkness for ~20 min before imaging. a1 The image over a large area of the leaf, a2 zoomed into show a group of palisade cells, and a3 further zoomed into focus onto a single chloroplast. b1, b2 CF transient from a group of palisade cells and a single chloroplast, respectively. b2 Noisier since less pixels are involved
Figure 3b presents the CF transients at low intensity illumination (3 kW/cm2) from a group of palisade cells (b1) and a single chloroplast (b2). The latter is noisier due to less pixels involved. The characteristic P–S decay and S–M rise of Kautsky curve are obvious in both (b1) and (b2). In Fig. 3b1, based on the statistics of 30 chloroplasts, the averaged timing points for P–S–M states are 1.8, 5.9 and 10.4 s, respectively, corresponding well with the reported values in the literature (Fig. 2). On the other hand, box plots are embedded in Fig. 3b1 to show the variations of time and intensity in P–S–M states between the 30 individual chloroplasts. The bottom and top of the box are the first and third quartiles, while the ends of the whiskers represent the maximum and minimum values. This result not only confirms that the averaged Kautsky curves acquired by the confocal fluorometer are similar to the curves taken with conventional fluorometers, but also shows that the variations between individual chloroplasts are indeed significant.
Intensity dependent fluorescence transient
As we have mentioned in Fig. 2, it is well known that the Kautsky curve changes with intensity. Figure 4 shows the intensity-dependent Kautsky curves from ~780 chloroplasts (colored lines) along with their standard error (gray lines), obtained by the confocal fluorometer. Note that to make the standard error visible on the same scale, it is multiplied by 16. Figure 4a is acquired with low laser intensity (3 kW/cm2), and a temporal variation similar to curve 1 of Fig. 2 is found, i.e. a complete O–P–S–M–T curve. The CF intensity rises to its first peak within 1 s (O–P rise), quickly decreasing to a local minimum (P–S fall), rising again to a second peak (S–M rise) then slowly falling as exponential decay (M–T decay). At slightly higher intensity (10 kW/cm2), a temporal variation similar to curve 2 of Fig. 2 is observed. The P–S fall and S–M rise still exist, but become much smaller, while the positions of P, S, and M appear earlier in the curve. At high intensity (55 kW/cm2), the S–M part disappears completely, leaving a single exponential P–T decay, similar to curve 3 of Fig. 2, i.e. saturation state. This result matches very well to the conventional wide-field fluorometer [1, 13, 36], but with much higher spatiotemporal resolution, manifesting again the reliability and usefulness of the confocal technique.
Averaged CF transients from ~780 chloroplasts (colored) with standard error (×16, gray) under excitation intensity at a 3, b 10, and c 55 kW/cm2, respectively, showing clearly the intensity-dependent Kautsky curves
From Fig. 4, not only the curve shape is intensity-dependent, but the positions of local maxima and minimum (P, S, M points) are strongly dependent on excitation intensities. Figure 5a shows the detailed curve variation relative to intensity, in the range of 3–55 kW/cm2, and the corresponding temporal position of local maximum of induced transients, i.e. point M, is given in Fig. 5b. Surprisingly, an almost perfect linear trend is observed. Similar linear results are found for the semi-steady state point S in Fig. 5c, and for the peak point P in Fig. 5d. Due to the limitation of temporal resolution (200 ms for 256 × 256 pixels), S and P points are analyzed with intensity range 3–40 kW/cm2 and 3–20 kW/cm2, respectively. The linear trends indicate that the state transition rate increases with higher excitation intensity. The underlying mechanism relies more investigation in the future.
a Detailed Kautsky curve variation in the intensity range of 3–55 kW/cm2. The temporal positions of b the local maximum (M), c semisteady state (S), and d peak (P), all change linearly with excitation intensity. The grey area represents 95% confidence region
Fluorescence dynamics under saturation intensity
In the last section, we have shown that at high excitation intensity, the CF is driven into saturation, which is very important for the quantum efficiency calculation. Thus, here we provide further characterization of the saturation states across individual chloroplasts. In Fig. 6a, the green color provides the spatial distribution of CF intensity over many living cells, and the red color shows the distribution of P–T phase decay time constant. For better identification, the two colors are shown separately in Fig. 6b, c. The statistical analysis for the P–T decay time constant of transients from individual chloroplasts is derived from Fig. 6c. The averaged decay time constant of a large area of leaf is 34.6 s, again matching well to the reported values in Fig. 2. Nevertheless, the standard variation of the decay time constant is 10.6 s, which reaches one-third of the average value, so significant divergence exists between each chloroplast. This decay time divergence is manifested by explicitly showing four Kautsky curves from individual chloroplasts in Fig. 6c1–c4.
High-resolution spatial distribution of CF intensity (green in a, b) and of the PT-phase time constant (red in a, c). The fluorescence transients of four selected chloroplasts within a living leaf are shown in the bottom panels, manifesting the significant difference in the time constants. The dataset is acquired at 40 kW/cm2 with a HeNe laser (633 nm)
Please note that error values in Fig. 6c1–c4 are the least square errors when fitting the curves with an exponential decay, different from the statistical standard deviation above. When analyzing data from a single chloroplast, the signal-to-noise ratio is relatively low, resulting in about 10% error in the time constant determination. Fortunately, the variation of time constants among chloroplasts is much larger than 10%, so this error is still tolerable. In the case where reduced error is necessary, the confocal system provides the flexibility to increase the integration time (reducing temporal resolution), so that higher signal-to-noise ratio can be achieved.
Since the laser intensity is relatively strong, it is necessary to confirm the reproducibility of the Kautsky curve in the same region of chloroplasts. Figure 7a1 shows the confocal CF image of a group of cells, and the corresponding averaged Kautsky curve is given in Fig. 7b1. The excitation intensity is 55 kW/cm2, which is adequate to saturate the photosystem, so a curve similar to 3 in Fig. 2 is observed. The sample was then kept in dark for 5 min, before the same intensity was applied again. The results of second excitation is given in Fig. 7a2, b2. Apparently, the Kautsky curve is fully recoverable, even under relatively high illumination intensity.
The reproducibility of Kautsky curve under strong illumination intensity (55 kW/cm2). a1, b1 are confocal CF image and Kautsky curve for the first set of excitation. a2, b2 are the corresponding results with the second set of excitation after 5 min in dark
Deriving quantum efficiency of photosystem II
We have shown that intensity-dependent CF transient is found on the scale of cells and chloroplasts, it is then straightforward to derive the physiologically important factors, such as the maximal quantum efficiency of photosystem II (ΦPSII). To derive maximal ΦPSII, the first step is to quantify the fluorescence yield, which is the ratio between CF intensity and excitation intensity. In our work, the relative quantum yield values are obtained by normalizing the CF intensities to the fluorescence intensity of a commercial fluorescent slide (92001, Chroma Tech., VT) under the same excitation intensity. The values of relative quantum yield at low excitation intensity (ΦF0, at 3 kW/cm2) and at saturation intensity (ΦFm, at 55 kW/cm2) are given in Table 1. Then the spatial distribution of ΦPSII is obtained with pixel-by-pixel calculation of (ΦFm − ΦF0)/ΦFm, as shown in Fig. 8. Apparently, the effect of the fluorescent slide is removed when calculating the quantum efficiency with the above equation. Numerical values of quantum efficiencies on different scales are also listed in Table 1. Similar to the results of Kautsky curves, the mean values of quantum efficiency are similar throughout a large area of leaf to a single chloroplast. On the other hand, from Table 1 and Fig. 8, the value of ΦPSII can be very different among individual chloroplasts, once again manifesting the significance of high-resolution mapping of the CF dynamics inside a living plant.
Table 1 Relative quantum yields and quantum efficiencies at different spatial scales
High-resolution spatial distribution of quantum efficiency of photosystem II inside a living leaf
We have successfully obtained the Kautsky curve, as well as its intensity dependence, with the confocal imaging fluorometer. Comparing to conventional wide-field imaging fluorometers, the confocal technique allows much better spatial confinement due to optical sectioning capability, and thus observation from a single chloroplast becomes possible. With the statistical analyses for P, S, M, T states of the Kautsky curves, at low and high intensities in Figs. 3 and 6 respectively, it can be concluded that the behavior of individual chloroplasts under our confocal imaging fluorometer is indeed similar to a large area of leaf under a conventional wide-field fluorometer. However, the value of the confocal technique lies in the capability to unravel the significant difference between individual chloroplasts, as highlighted by the box plot in Fig. 3 and the clear variation of P–T decay time constants in Fig. 6c.
In terms of the temporal resolution performance, the confocal and wide-field fluorometers should be similar in terms of a single pixel detection, which takes about 1–10 μs in both cases. As mentioned in [17], the wide-field fluorometer takes about 1 s to record one image. Nevertheless, the advantage of the confocal scheme is the freedom to select number of pixels, as well as the position of these pixels, significantly enhancing the temporal responses. By using more advanced scanning approaches, such as random-access microscopy [40], high-speed CF detection among distant chloroplasts is possible. In addition, by adopting a multi-focus scanning approach, such as being demonstrated by spinning disk confocal microscopy in 2009 [23], the frame rate of confocal fluorometer can be significantly improved.
Although spinning disk technique may potentially provide higher frame rate, there are several limitations that prevent it to be an ideal choice for fluorometry application [41]. First of all, due to the size limitation of camera, spinning disk confocal microscopy typically exhibits a small field of view, often only the size of a few cells, which is problematic when studying tissues. A good comparison is given in Fig. 8 of [41], where laser scanning confocal microscope provides much larger field of view.
Second, due to the existence of multiple pinholes on a pinhole array, the optical sectioning capability of spinning disk microscopy is in general less ideal than laser scanning confocal microscopy, especially when observing thick and scattering tissues. In addition, when using a low-magnification lens for large-area study, the spinning disk technique can significantly lose its optical sectioning ability. The reason is that most spinning disk system has a pinhole array comprising pinholes with a fixed size, which is designed for high-magnification and high-NA immersion objective lens, such as a 100×/NA 1.4 objective. However, for plant tissues, low magnification lens with moderate NA is preferred for large area observation. In our case, a 10×/NA 0.4 objective is employed, providing 1.5 mm × 1.5 mm field of view. If the 10× objective is used in a spinning disk system, whose pinhole diameter cannot be adjusted, both the axial sectioning capability and the lateral resolution shall be far less optimal than a confocal system. On the other hand, in a point-scanning confocal system, the pinhole size is easily adjustable, allowing observation for both high and low magnifications. Even with the low NA objective, as we mentioned in the main text, our confocal fluorometer still provides 1-micrometer lateral resolution and 5-micrometer axial resolution, adequate for single chloroplast imaging.
The third concern is image uniformity. In a spinning disk system, when using a Gaussian laser beam, the excitation intensity of the center region is larger than the edge, making it difficult to quantify the response from individual chloroplast. On the other hand, the image uniformity of laser scanning confocal fluorometer is much better than typical spinning disk one.
Last but not least, when comparing spinning disk and laser scanning confocal techniques, it is commonly accepted that the laser intensity at each focus is less for the former, so photobleaching is reduced. However, we would like to point out that the overall accumulated power/energy on the plant tissue is in fact higher, since the laser power is spread over hundreds of foci across the entire field of view. Therefore, more powerful lasers are required for the spinning disk system, and the issue of potential photothermal damage in the tissues has to be considered.
Another important aspect to notice is that the illumination intensity of the confocal fluorometer is much higher than that of the wide-field fluorometers. As shown in Figs. 5 and 6, to eliminate the semi-steady state S in the CF transient, about 55 kW/cm2 is required for the confocal fluorometer. However, in the case of wide-field fluorometer, as shown in the example of Fig. 2 [1], to eliminate S, 3200 μmol/m2/s is required. Considering the wavelength to be 650 nm in [1], the photon energy is 1240/650 = 1.9 eV = 3 × 10−19 J. Therefore, the intensity unit (μmol/m2/s) is equivalent to [10−6 × 6 × 1023 (# of photons)] × [3 × 10−19 (J/photon)]/104 cm2/s = 18 × 10−9 kW/cm2. As a result, in the wide-field fluorometer, the required illumination intensity is 3200 × 18 × 10−9 kW/cm2 = 5.76 × 10−5 kW/cm2, six orders of magnitude smaller than that in the confocal one.
To explain this 6-order intensity difference, optical sectioning and illumination time of the confocal imaging fluorometer have to be considered. In a conventional fluorometer (wide-field detection), CF signals are emitted throughout the whole leaf in the axial direction, so the depth of field (i.e. signal collection depth) is equivalent to the thickness of a leaf, which is usually 100–1000 µm. On the other hand, for a confocal fluorometer, a pinhole is inserted before the detector to reject most out-of-focus fluorescence, and thus the total signal strength is significantly reduced. The typical depth of field in a confocal fluorometer is about 1–10 µm, which is 2-orders less than that of the wide-field one. Hence, the signal strength of the confocal fluorometer is expected to be 2-orders weaker than the wide-field counterpart.
In terms of the illumination time, in a conventional wide-field imaging fluorometer, the whole leaf sample is illuminated continuously, so the illumination time for each pixel is the same as the frame acquisition time. On the other hand, a small laser focus scans across the sample in the confocal scheme, making the illumination time for each pixel much shorter than the frame time. For example, in the case of Fig. 4a2, 1 frame takes about 1 s, and the frame is composed of 256 × 256 pixels, so the illuminating time for each pixel (1 pixel is roughly 1 µm2 in this case) of the confocal imaging fluorometer is about 4-orders shorter than that of conventional wide-field imaging fluorometer.
Combining the above two reasons, it is reasonable that the illumination intensity in the confocal imaging fluorometer needs to be much higher than that in the wide-field fluorometer to achieve similar CF signal strength, as well as the Kautsky curves. The latter is somewhat surprising since it indicates that the physiological response of the chlorophyll remains the same with such high-intensity, yet short-period, illumination. One possible reason is that there is a slow reaction during photosynthesis and CF generation, so the chlorophyll only responses to the average intensity, not the instantaneous intensity. By looking into the electron transport chains in the photosystem, the bottleneck reaction might be the reduction of plastoquinone (PQ), which has a relatively slow reaction rate (100 molChl mmol−1 s−1) [42]. Further studies are necessary to identify the underlying photochemical mechanism.
In this work, we demonstrated a confocal imaging fluorometer that can provide high spatiotemporal characterization of CF inside a living leaf. The three-dimensional spatial resolution is on the order of micrometer, and the temporal resolution reaches tens of milliseconds, allowing us to study CF transient, i.e. the Kautsky effect, from even a single chloroplast. Although the ensemble behavior of CF transient, as well as the intensity-dependent Kautsky curves, agree well with the results of conventional wide-field fluorometers, confocal imaging fluorometer provides valuable information toward the difference of CF dynamics among individual chloroplasts. The features of optical sectioning and laser focus scanning in the confocal fluorometer result in much higher illumination intensity compared to conventional techniques, while maintaining normal cellular physiological responses. Our work not only opens up new possibilities to study CF dynamics on the level of organelles, but also is promising to unravel more spatial/temporal details in the associated photosynthetic processes.
Strasser RJ, Srivastava A, Govindjee. Polyphasic chlorophyll a fluorescence transient in plants and cyanobacteria. Photochem Photobiol. 1995;61:32–42.
Zarco-Tejada PJ, Miller JR, Mohammed GH, Noland TL. Chlorophyll fluorescence effects on vegetation apparent reflectance: I. Leaf-level measurements and model simulation. Remote Sens Environ. 2000;74:582–95.
Krause GH, Weis E. Chlorophyll fluorescence and photosynthesis: the basics. Annu Rev Plant Biol. 1991;42:313–49.
Horton P, Ruban AV, Walters RG. Regulation of light harvesting in green plants (indication by nonphotochemical quenching of chlorophyll fluorescence). Plant Physiol. 1994;106:415.
Dobrowski SZ, Pushnik JC, Zarco-Tejada PJ, Ustin SL. Simple reflectance indices track heat and water stress-induced changes in steady-state chlorophyll fluorescence at the canopy scale. Remote Sens Environ. 2005;97:403–14.
Csintalan Z, Proctor MCF, Tuba Z. Chlorophyll fluorescence during drying and rehydration in the mosses Rhytidiadelphus loreus (hedw.) warnst., Anomodon viticulosus (hedw.) hook. & tayl. and Grimmia pulvinata (hedw.) sm. Ann Bot. 1999;84:235–44.
Bolhar-Nordenkampf HR, Long SP, Baker NR, Oquist G, Schreiber U, Lechner EG. Chlorophyll fluorescence as a probe of the photosynthetic competence of leaves in the field: A review of current instrumentation. Funct Ecol. 1989;3:497–514.
Trissl HW, Gao Y, Wulf K. Theoretical fluorescence induction curves derived from coupled differential equations describing the primary photochemistry of photosystem ii by an exciton-radical pair equilibrium. Biophys J. 1993;64:974.
Schansker G, Tóth SZ, Holzwarth AR, Garab G. Chlorophyll a fluorescence: beyond the limits of the QA model. Photosynth Res. 2014;120:43–58.
Papageorgiou GC. Photosystem ii fluorescence: slow changes–scaling from the past. J Photochem Photobiol B Biol. 2011;104:258–70.
DeEll JR, Toivonen PMA. Practical applications of chlorophyll fluorescence in plant biology. Berlin: Springer; 2012.
Lazár D, Ilik P. High-temperature induced chlorophyll fluorescence changes in barley leaves comparison of the critical temperatures determined from fluorescence induction and from fluorescence temperature curve. Plant Sci. 1997;124:159–64.
Schreiber U, Schliwa U, Bilger W. Continuous recording of photochemical and non-photochemical chlorophyll fluorescence quenching with a new type of modulation fluorometer. Photosynth Res. 1986;10:51–62.
Mauzerall D. Light-induced fluorescence changes in chlorella, and the primary photoreactions for the production of oxygen. Proc Nat Acad Sci. 1972;69:1358–62.
Falkowski PG, Wyman K, Ley AC, Mauzerall DC. Relationship of steady-state photosynthesis to fluorescence in eucaryotic algae. BBA-Bioenergetics. 1986;849:183–92.
Kolber ZS, Prášil O, Falkowski PG. Measurements of variable chlorophyll fluorescence using fast repetition rate techniques: defining methodology and experimental protocols. BBA-Bioenergetics. 1998;1367:88–106.
Barron-Gafford GA, Rascher U, Bronstein JL, Davidowitz G, Chaszar B, Huxman TE. Herbivory of wild manduca sexta causes fast down-regulation of photosynthetic efficiency in datura wrightii: an early signaling cascade visualized by chlorophyll fluorescence. Photosynth Res. 2012;113:249–60.
Oxborough K, Baker N. An instrument capable of imaging chlorophyll a fluorescence from intact leaves at very low irradiance and at cellular and subcellular levels of organization. Plant, Cell Environ. 1997;20:1473–83.
Scholes JD, Rolfe SA. Photosynthesis in localised regions of oat leaves infected with crown rust (Puccinia coronata): quantitative imaging of chlorophyll fluorescence. Planta. 1996;199:573–82.
Osmond B, Schwartz O, Gunning B. Photoinhibitory printing on leaves, visualised by chlorophyll fluorescence imaging and confocal microscopy, is due to diminished fluorescence from grana. Funct Plant Biol. 1999;26:717–24.
Hibberd JM, Quick WP. Characteristics of c4 photosynthesis in stems and petioles of c3 flowering plants. Nature. 2002;415:451–4.
Kim S-J, Hahn E-J, Heo J-W, Paek K-Y. Effects of leds on net photosynthetic rate, growth and leaf stomata of chrysanthemum plantlets in vitro. Sci Hortic. 2004;101:143–51.
Omasa K, Konishi A, Tamura H, Hosoi F. 3d confocal laser scanning microscopy for the analysis of chlorophyll fluorescence parameters of chloroplasts in intact leaf tissues. Plant Cell Physiol. 2009;50:90–105.
Papageorgiou G. Light-induced changes in the fluorescence yield of chlorophyll a in vivo: I. Anacystis nidulans. Biophys J. 1968;8:1299–315.
Paddock S, Fellers TJ, Davidson MW. Confocal microscopy. Berlin: Springer; 2001.
Shotton DM. Confocal scanning optical microscopy and its applications for biological specimens. J Cell Sci. 1989;94:175–206.
Roselli L, Paparella F, Stanca E, Basset A. New data-driven method from 3d confocal microscopy for calculating phytoplankton cell biovolume. J Microsc. 2015;258:200–11.
Matsumoto B. Cell biological applications of confocal microscopy. Cambridge: Academic Press; 2003.
Hibbs AR. Confocal microscopy for biologists. Berlin: Springer; 2004.
Pawley JB, Masters BR. Handbook of biological confocal microscopy. Opt Eng. 1996;35:2765–6.
Omasa K. Image instrumentation of chlorophyll a fluorescence. In SPIE Proc. 1998;3382:91–9.
Kautsky H, Appel W. Chlorophyllfluorescenz und kohlensaureassimilation. Biochemistry. 1960;322:279–92.
Bradbury M, Baker NR. Analysis of the slow phases of the in vivo chlorophyll fluorescence induction curve. Changes in the redox state of photosystem ii electron acceptors and fluorescence emission from photosystems i and ii. BBA-Bioenergetics. 1981;635:542–51.
Maxwell K, Johnson GN. Chlorophyll fluorescence—a practical guide. J Exp Bot. 2000;51:659–68.
Stirbet A, Govindjee. The slow phase of chlorophyll a fluorescence induction in silico: origin of the s–m fluorescence rise. Photosynth Res. 2016;130:193–213.
Omasa K, Shimazaki KI, Aiga I, Larcher W, Onoe M. Image analysis of chlorophyll fluorescence transients for diagnosing the photosynthetic system of attached leaves. Plant Physiol. 1987;84:748–52.
Kodru S, Malavath T, Devadasu E, Nellaepalli S, Stirbet A, Subramanyam R. The slow s to m rise of chlorophyll a fluorescence reflects transition from state 2 to state 1 in the green alga chlamydomonas reinhardtii. Photosynth Res. 2015;125:219–31.
Vermaas WF, Timlin JA, Jones HD, Sinclair MB, Nieman LT, Hamad SW, Melgaard DK, Haaland DM. In vivo hyperspectral confocal fluorescence imaging to determine pigment localization and distribution in cyanobacterial cells. Proc Nat Acad Sci USA. 2008;105:4050–5.
Chen M-Y, Zhuo G-Y, Chen K-C, Wu P-C, Hsieh T-Y, Liu T-M, Chu S-W. Multiphoton imaging to identify grana, stroma thylakoid, and starch inside an intact leaf. BMC Plant Biol. 2014;14:175.
Reddy GD, Kelleher K, Fink R, Saggau P. Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity. Nat Neurosci. 2008;11:713–20.
Jonkman J, Brown CM. Any way you slice it—a comparison of confocal microscopy techniques. J Biomol Tech. 2015;26:54–65.
de Wijn R, van Gorkom HJ. Kinetics of electron transfer from QA to QB in photosystem ii. Biochemistry. 2001;40:11912–22.
YCT designed the experiment, carried out signal analysis, and wrote most of the manuscript. SWC envisioned the idea, provided the experimental hardware, and helped to polish the manuscript. Both authors read and approved the final manuscript.
The authors appreciate the inspirational discussion with Prof. Govindjee from University of Illinois at Urbana-Champaign. SWC acknowledge the generous support from the Foundation for the Advancement of Outstanding Scholarship.
The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.
This work is supported by the Molecular Imaging Center of NTU (105R8916, 105R7732), and by the Ministry of Science and Technology, Taiwan, under grant MOST-105-2628-M-002-010-MY4 and MOST-106-2321-B-002-020.
Department of Physics, National Taiwan University, No. 1, Section 4, Roosevelt Rd, Da'an District, Taipei City, 10617, Taiwan
Yi-Chin Tseng & Shi-Wei Chu
Molecular Imaging Center, National Taiwan University, No. 81, Changxing Street, Da'an District, Taipei, 10672, Taiwan
Shi-Wei Chu
Yi-Chin Tseng
Correspondence to Shi-Wei Chu.
Tseng, YC., Chu, SW. High spatio-temporal-resolution detection of chlorophyll fluorescence dynamics from a single chloroplast with confocal imaging fluorometer. Plant Methods 13, 43 (2017). https://doi.org/10.1186/s13007-017-0194-2
Optical section
3D microscopy
Kautsky curve
Chlorophyll fluorescence transient | CommonCrawl |
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{\Large\bf Definitizability of normal operators on Krein spaces and their functional calculus}
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\textsc{Michael Kaltenb\"ack\footnote{This work was supported by a joint project of the Austrian Science Fund
(FWF, I1536--N25) and the Russian Foundation for Basic Research (RFBR, 13-01-91002-ANF).}}
\\[6mm]
{\small
\textbf{Abstract: We discuss a new concept of definitizability of a normal
operator on Krein spaces. For this new concept we develop
a functional calculus $\phi \mapsto \phi(N)$ which is the
proper analogue of $\phi \mapsto \int \phi \, dE$ in the Hilbert space situation.}
} \end{flushleft}
\begin{flushleft}
{\small
{\bf Mathematics Subject Classification (2010):} 47A60, 47B50, 47B15.
} \end{flushleft} \begin{flushleft}
{\small
{\bf Keywords:} Krein space, definitizable operators, normal operators, spectral theorem
} \end{flushleft}
\section{Introduction}
A bounded linear operator $N$ on a Krein space $(\mc K,[.,.])$ is normal, if $N$ commutes with its Krein space adjoint $N^+$. If we write $N$ as $A+iB$ with the selfadjoint real part $A:=\RE N:=\frac{N+N^+}{2}$ and the selfadjoint imaginary part $B:=\IM N:=\frac{N-N^+}{2i}$, then $N$ is normal if and only if $AB=BA$. In \cite{Ka2015} we called a normal $N$ definitizable whenever $A$ and $B$ were both definitizable in the classical sense, i.e.\ there exist so-called definitizing polynomials $p(z),q(z)\in \bb R[z]\setminus\{0\}$ such that $[p(A)x,x] \geq 0$ and $[q(B)x,x] \geq 0$ for all $x\in \mc K$.
For such definitizable operators in \cite{Ka2015} we could build a functional calculus in analogy to the functional calculus $\phi\mapsto \int \phi \, dE$ mapping the $*$-algebra of bounded and measurable functions on $\sigma(N)$ to $B(\mc H)$ in the Hilbert space case. The functional calculus in \cite{Ka2015} can also be seen as a generalization of Heinz Langers spectral theorem on definitizable selfadjoint operators on Krein spaces; see \cite{langer1982}, \cite{KaPr2014}. Unfortunately, there are unsatisfactory phenomenons with this concept of definitizability in \cite{Ka2015}. For example, it is not clear, whether for a bijective, normal definitizable $N$ also $N^{-1}$ definitizable.
In the present paper we choose a more general concept of definitizability. We shall say that a normal $N$ on a Krein space $\mc K$ is definitizable if $[p(A,B)u,u] \geq 0$ for all $u \in \mc K$ for some, so-called definitizing, $p(x,y)\in\bb C[x,y]\setminus\{0\}$ with real coefficients. Then we study the ideal $I$ generated by all definitizing polynomials with real coefficients in $\bb C[x,y]$, and assume that $I$ is large in the sense that it is zero-dimensional, i.e.\ $\dim \bb C[x,y]/I<\infty$. By the way, if $N$ is definitizable in the sense of \cite{Ka2015}, then $I$ is always zero-dimensional.
Using results from algebraic geometry, under the assumption that $I$ is zero-dimensional, the variety $V(I)=\{a\in \bb C^2: f(a)=0 \ \text{ for all } \ f \in I\}$ is a finite set. We split this subset of $\bb C^2$ up as \[
V(I) = (V(I)\cap \bb R^2) \dot \cup (V(I)\setminus \bb R^2) \,, \] and interpret $V_{\bb R}(I):=V(I)\cap \bb R^2$ in the following as a subset of $\bb C$ by consider the first entry as the real and the second entry as the imaginary part.
By the ascending chain condition the ideal $I$ is generated by real definitizing polynomials $p_1,\dots,p_m$. With the help of the positive semidefinite scalar products $[p_j(A,B).,.]$, $j=1,\dots,m$ and $\sum_{k=1}^m [p_k(A,B).,.]$ we construct Hilbert spaces $\mc H_j$, $j=1,\dots,m$ and $\mc H$ together with bounded and injective $T_j: \mc H_j \to \mc K$ and $T: \mc H \to \mc K$. We consider $\Theta_j: (T_jT_j^+)' \to (T_j^+T_j)'$ and $\Theta: (TT^+)' \to (T^+T)'$ by $\Theta_j(C):= (T_j\times T_j)^{-1}(C)$ and $\Theta(C):= (T\times T)^{-1}(C)$, as studied in \cite{KaPr2014}. Here $(T_jT_j^+)', (TT^+)'\subseteq B(\mc K)$ and $(T_j^+T_j)'\subseteq B(\mc H_j), (T^+T)'\subseteq B(\mc H)$ denote the commutant of the respective operators.
The proper family $\mc F_N$ of functions suitable for the aimed functional calculus are functions defined on \[
\big(\sigma(\Theta(N)) \cup V_{\bb R}(I) \big) \dot\cup (V(I)\setminus \bb R^2) \,. \] Moreover, the functions $\phi\in \mc F_N$ assume values in $\bb C$ on $\sigma(\Theta(N)) \setminus V_{\bb R}(I)$ and values in a certain finite dimensional $*$-algebras $\mc A(z)$ at $z\in V_{\bb R}(I)$ and $\mc B((\xi,\eta))$ at $(\xi,\eta)\in V(I)\setminus \bb R^2$. On $\sigma(\Theta(N)) \setminus V_{\bb R}(I)$ we assume $\phi$ to be bounded and measurable. Finally, $\phi\in \mc F_N$ satisfies a growth regularity condition at all $w$ points from $V_{\bb R}(I)$ which are not isolated in $\sigma(\Theta(N)) \cup V_{\bb R}(I)$. Vaguely speaking, this growth regularity condition means that around $w$ the function $\phi$ admits an approximation by a Taylor polynomial, which is determined by $\phi(w)\in \mc A(w)$. Any polynomial $s(x,y)\in \bb C[x,y]$ can be seen as a function $s_N\in \mc F_N$ in a natural way.
For each $\phi\in\mc F_N$ we will see that there exists $p\in\bb C[x,y]$ and bounded, measurable $f_1,\dots,f_m: \sigma(\Theta(N)) \cup V_{\bb R}(I) \to \bb C$ with $f_j(z)=0$ for $z\in V_{\bb R}(I)$ such that \begin{equation}\label{decompact2}
\phi(z) = p_N(z) + \sum_j f_j(z) \, (p_j)_N(z) \end{equation} for all $z\in \sigma(\Theta(N)) \cup V_{\bb R}(I)$, and that $\phi((\xi,\eta)) = p_N((\xi,\eta))$ for all $(\xi,\eta) \in V(I)\setminus \bb R^2$. We then define \[
\phi(N):=p(A,B) + \sum_{k=1}^m T_k \int_{\sigma(\Theta_k(N))} f_k \, dE \, T_k^+ \,, \] and show that this operator does not depend on the actual decomposition \eqref{decompact2} and that $\phi \mapsto \phi(N)$ is indeed a $*$-homomorphism satisfying $\phi(N) = s(A,B)$ for $\phi=s_N$.
\section{Multiple embeddings}
In the present section $(\mc K,[.,.])$ will be a Krein space and $(\mc H,(.,.))$, $(\mc H_j,(.,.)), \ j=1,\dots,m$, will denote Hilbert spaces. Moreover, let $T: \mc H \to \mc K$, $T_j: \mc H_j \to \mc K$ and $R_j: \mc H_j \to \mc H$ bounded, linear and injective mappings such that $TR_j=T_j$. By $T^+: \mc K \to \mc H$ and $T_j^+: \mc K \to \mc H_j$ we denote the respective Krein space adjoints.
If $D$ is an operator on a Krein space, then we shall denote by $D'$ the commutant of $D$, i.e.\ the algebra of all operators commuting with $D$. For a selfadjoint $D$ this commutant is a $*$-algebra with respect to forming adjoint operators.
For $j=1,\dots,m$ we shall denote by $\Theta_j: (T_jT_j^+)' \ (\subseteq B(\mc K)) \to (T_j^+T_j)' \ (\subseteq B(\mc H_j))$, and by $\Theta: (TT^+)' \ (\subseteq B(\mc K)) \to (T^+T)' \ (\subseteq B(\mc H))$ the $*$-algebra homomorphisms mapping the identity operator to the identity operator as in \thref{thetadefeig} from \cite{KaPr2014} corresponding to the mappings $T_j$ and $T$: \[
\Theta_j(C_j) = (T_j\times T_j)^{-1}(C_j) = T_j^{-1}C_jT_j, \ \ C_j\in (T_jT_j^+)' \,, \] \begin{equation}\label{thetaVdef}
\Theta(C) = (T\times T)^{-1}(C) = T^{-1}CT, \ \ C\in (TT^+)' \,. \end{equation} We can apply \thref{thetadefeig} in \cite{KaPr2014} also to the bounded linear, injective $R_j: \mc H_j \to \mc H$, and denote the corresponding $*$-algebra homomorphisms by $\Gamma_j : (R_jR_j^*)' \ (\subseteq B(\mc H)) \to (R_j^*R_j)' \ (\subseteq B(\mc H_j))$: \[
\Gamma_j(D) = (R_j\times R_j)^{-1}(D) = R_j^{-1}DR_j, \ \ D \in (R_jR_j^*)' \,. \] For the following note that due to $(\ran T^+)^{[\bot]} = \ker T = \{0\}$ the range of $T^+$ is dense in $\mc H$.
\begin{lemma}\thlab{comreg}
For $j=1,\dots,m$ we have
$\Theta((T_jT_j^+)'\cap (TT^+)') \subseteq (R_jR_j^*)' \cap (T^+T)'$,
where in fact \begin{equation}\label{zuef}
\Theta(C) R_j R_j^* = R_j \Theta_j(C) R_j^*=R_j R_j^* \Theta(C), \ \ C\in (T_jT_j^+)'\cap (TT^+)' \,. \end{equation}
Moreover, \begin{equation}\label{zuefvor}
\Theta_j(C) = \Gamma_j \circ \Theta(C), \ \ C\in (T_jT_j^+)'\cap (TT^+)' \,. \end{equation} \end{lemma} \begin{proof}
According to \thref{thetadefeig} in \cite{KaPr2014} we have
$\Theta_j(C) T_j^+ = T_j^+ C$ and $T^+ C = \Theta(C) T^+$ for $C\in (T_jT_j^+)'\cap (TT^+)'$.
Therefore, \begin{align*}
T (\, R_j \Theta_j(C) R_j^* \, ) T^+ & = T_j \Theta_j(C) T_j^+ = T_j T_j^+ C \\ & =
T R_j R_j^* T^+ C = T ( \, R_j R_j^* \Theta(C) \, ) T^+ \,. \end{align*} $\ker T=\{0\}$ and the density of $\ran T^+$ yield $R_j \Theta_j(C) R_j^*=R_j R_j^* \Theta(C)$. Applying this equation to $C^+$ and taking adjoints yields $R_j \Theta_j(C) R_j^*=\Theta(C) R_j R_j^*$. In particular, $\Theta(C) \in (R_jR_j^*)'$. Therefore, we can apply $\Gamma_j$ to $\Theta(C)$ and get \[
\Gamma_j \circ \Theta(C) = R_j^{-1} T^{-1}C T R_j = T_j^{-1} C T_j = \Theta_j(C) \,. \] \end{proof}
For the following assertion note that by \eqref{zuefvor} and by the fact that $\Gamma_j$ is a $*$-algebra homomorphism mapping the identity operator to the identity operator, for $j=1,\dots,m$ we have \begin{equation}\label{4ucd75}
\sigma(\Theta(C)) \subseteq \sigma(\Theta_j(C)) \ \ \text{ for all } \ \ C\in (T_jT_j^+)'\cap (TT^+)' \,. \end{equation}
\begin{corollary}\thlab{normtransf}
For a $j\in\{1,\dots,m\}$ let $N\in B(\mc K)$ be normal such that $N\in(T_jT_j^+)'\cap (TT^+)'$.
Then $\Theta(N)$ is a normal operator in the Hilbert space $\mc H$, and
$\Theta_j(N)$ is a normal operator in the Hilbert space $\mc H_j$.
Denoting by $E$ ($E_j$) the spectral
measure of $\Theta(N)$ ($\Theta_j(N)$), we have $E(\Delta) \in (R_jR_j^*)' \cap (T^+T)'$ and
\[
\Gamma_j(E(\Delta)) = E_j(\Delta) \,,
\]
for all Borel subsets $\Delta$ of $\bb C$, where $E_j(\Delta) \in (R_j^*R_j)' \cap (T_j^+T_j)'$.
Moreover, $\int h \, dE \in (R_jR_j^*)' \cap (T^+T)'$
and
\[
\Gamma_j\left(\int h \, dE \right) = \int h \, dE_j
\]
for any bounded and measurable $h: \sigma(\Theta(N)) \to \bb C$, where $\int h \, dE_j
\in (R_j^*R_j)' \cap (T_j^+T_j)'$. \end{corollary} \begin{proof}
The normality of $\Theta(N)$ and $\Theta_j(N)$ is clear, since $\Theta$ and $\Theta_j$ are
$*$-homomorphisms. From \thref{comreg} we know that $\Theta(N) \in (R_jR_j^*)' \cap (T^+T)'$.
According to the well known properties of $\Theta(N)$'s spectral measure we obtain
$E(\Delta) \in (R_jR_j^*)' \cap (T^+T)'$ and, in turn,
$\int h \, dE \in (R_jR_j^*)' \cap (T^+T)'$. In particular, $\Gamma_j$ can be applied
to $E(\Delta)$ and $\int h \, dE$.
Similarly, $\Theta_j(N)\in (T_j^+T_j)'$ implies $E_j(\Delta), \int h \, dE_j \in (T_j^+T_j)'$
for a bounded and measurable $h$.
Recall from \thref{thetadefeig} in \cite{KaPr2014} that $\Gamma_j(D) R_j^*x = R_j^* D$
for $D \in (R_jR_j^*)'$. Hence, for $x\in \mc H$ and $y\in \mc H_j$ we have
\[
(\Gamma_j(E(\Delta)) R_j^*x, y) = (R_j^* E(\Delta) x,y) = (E(\Delta) x, R_j y)
\]
and, in turn,
\begin{multline*}
\int_{\bb C} s(z,\bar z) \, d(\Gamma_j(E) R_j^*x, y) =
\int_{\bb C} s(z,\bar z) \, d(E x, R_j y) = (s(\Theta(N),\Theta(N)^*)x, R_j y) \\
= (R_j^* s(\Theta(N),\Theta(N)^*)x, y) = (\Gamma_j\big(s(\Theta(N),\Theta(N)^*)\big) R_j^* x,y)
\end{multline*}
for any $s(z,w) \in \bb C[z,w]$.
By \eqref{zuefvor} and the fact, that $\Gamma_j$ is a $*$-homomorphism,
we have $\Gamma_j(s(\Theta(N),\Theta(N)^*)) = s(\Theta_j(N),\Theta_j(N)^*)$.
Consequently,
\[
\int_{\bb C} s(z,\bar z) \, d(\Gamma_j(E) R_j^*x, y) = \int_{\bb C} s(z,\bar z) \, d(E_j R_j^*x, y) \,.
\]
Since $E(\bb C\setminus K)=0$ and $E_j(\bb C\setminus K)=0$ for a certain compact $K\subseteq \bb C$ and since
the set of all $s(z,\bar z), \ s\in \bb C[z,w]$, is densely contained in $C(K)$, we obtain from the uniqueness assertion
in the Riesz Representation Theorem
\[
(\Gamma_j(E(\Delta)) R_j^*x, y) = (E_j(\Delta) R_j^*x, y) \ \ \text{ for all } \ \ x\in \mc H, \, y\in \mc H_j \,,
\]
for all Borel subsets $\Delta$ of $\bb C$.
Due to the density of $\ran R_j^*$ in $\mc H_j$ we even have
$(\Gamma_j(E(\Delta)) z, y) = (E_j(\Delta) z, y)$ for all $y,z \in \mc H_j$, and in turn
$\Gamma_j(E(\Delta))=E_j(\Delta)$. Since $\Gamma_j$ maps into $(R_j^*R_j)'$, we have
$E_j(\Delta)\in (R_j^*R_j)'$. This yields $\int h \, dE_j\in (R_j^*R_j)'$ for any bounded and measurable $h$.
If $h: \sigma(\Theta(N)) \to \bb C$ is bounded and measurable, then by \eqref{4ucd75} also
its restriction to $\sigma(\Theta_j(N)) = \sigma(\Gamma_j\circ\Theta(N))$ is bounded and measurable.
Due to $E_j(\Delta) R_j^*= \Gamma_j(E(\Delta)) R_j^* = R_j^*E(\Delta)$, for
$x\in\mc H$ and $y\in\mc H_j$ we have
\begin{multline*}
(\Gamma_j\left(\int h \, dE \right) R_j^*x,y) =
(R_j^* \left(\int h \, dE \right) x,y) =
(\left(\int h \, dE \right) x, R_j y) \\
= \int h \, d(E x,R_j y) = \int h \, d(E_j R_j^* x, y) =
(\left(\int h \, dE_j\right) R_j^*x,y) \,.
\end{multline*}
The density of $\ran R_j^*$ yields $\Gamma_j\left(\int h \, dE \right) = \int h \, dE_j$. \end{proof}
Recall from \thref{Xidefeig} in \cite{KaPr2014} the mappings ($j=1,\dots,m$) \[
\Xi_j : B(\mc H_j) \to B(\mc K), \ \
\Xi_j(D_j) = T_j D_j T_j^{+} \,, \] and $\Xi : B(\mc H) \to B(\mc K)$ with $\Xi(D) = T D T^{+}$. By ($j=1,\dots,m$) \[
\Lambda_j: B(\mc H_j) \to B(\mc H),
\ \ \Lambda_j(D_j) = R_j D_j R_j^{*} \,, \] we shall denote the corresponding mappings outgoing from the mappings $R_j: \mc H_j \to \mc H$. Due to $T_j=T R_j$ we have $\Xi_j = \Xi \circ \Lambda_j$.
According to \thref{Xidefeig} in \cite{KaPr2014}, $\Lambda_j\circ \Gamma_j(D) = D R_jR_j^*$ for $D\in (R_jR_j^*)'$. Hence, using the notation from \thref{normtransf} \begin{equation}\label{zuef2}
\Xi_j(\int h \, dE_j) = \Xi\Big(\Lambda_j\circ \Gamma_j\left(\int h \, dE \right)\Big) =
\Xi(R_jR_j^{*} \int h \, dE) \,. \end{equation}
\begin{lemma}\thlab{existtreanspost}
Assume that for $j\in\{1,\dots,m\}$ the operator $T_jT_j^+$ commutes with $TT^+$ on $\mc K$.
Then the operators $R_j R_j^*$, $T^+ T$ commute on $\mc H$ and $R_j^*R_j$, $T_j^+ T_j$ commute on $\mc H_j$. Moreover,
\begin{equation}\label{zuef3}
\Theta(T_jT_j^+) = R_j R_j^* T^+ T = T^+ T R_j R_j^* \,.
\end{equation} \end{lemma} \begin{proof} If $T_jT_j^+$ and $TT^+$ commute on $\mc K$, then \[
T ( \, T^+ T R_j R_j^* \, )T^+ = TT^+ T_jT_j^+ = T_jT_j^+ TT^+ = T ( \, R_j R_j^* T^+ T \, )T^+ \,. \] Employing $T$'s injectivity and the density of $\ran T^+$, we see that $R_j R_j^*$ and $T^+ T$ commute. From this we derive \[
T_j^+ T_j R_j^*R_j = R_j^* (T^+ T R_j R_j^*) R_j = R_j^* (R_j R_j^* T^+ T) R_j = R_j^*R_j T_j^+ T_j \,. \] \eqref{zuef3} follows from \[
T^{-1} T_jT_j^+ T = T^{-1} T R_j R_j^* T^+ T = R_j R_j^* T^+ T \,. \] \end{proof}
\section{Definitizability} \label{definitiza}
In \cite{Ka2015} we said that a normal $N\in B(\mc K)$ is definitizable, if its real part $A:=\frac{N+N^+}{2}$ and its imaginary part $B:=\frac{N-N^*}{2i}$ are definitizable in the sense that there exist real polynomials $p,q\in\bb R[z]\setminus \{0\}$ such that $[p(A)v,v] \geq 0$ and $[q(B)v,v] \geq 0$ for all $v\in \mc K$. In the present note we will relax this condition.
\begin{definition}\thlab{definidef} For a normal $N\in B(\mc K)$ we call $p(x,y) \in \bb C[x,y]\setminus \{0\}$ a definitizing polynomial for $N$, if \begin{equation}\label{defalgl}
[p(A,B) v,v] \geq 0 \ \ \text{ for all } \ \ v\in \mc K \,. \end{equation} where $A=\frac{N+N^+}{2}$ and $B=\frac{N-N^+}{2i}$. If such a definitizing $p \in \bb C[x,y]\setminus \{0\}$ exists, then we call $N$ definitizable normal. \end{definition}
Clearly, we could also write $p$ as a polynomial of the variables $N$ and $N^+$. But because of $A=A^+$ and $B=B^+$, writing $p$ as a polynomial of the variables $A$ and $B$ has some notational advantages.
\begin{remark}\thlab{definidefrem}
According to \eqref{defalgl} the operator $p(A,B)\in B(\mc K)$ must be selfadjoint;
i.e.\ $p(A,B)^+ = p^\#(A,B)$, where $p^\#(x,y) = \overline{p(\overline{x},\overline{y})}$.
Hence, $q:=\frac{p_j+ p_j^\#}{2}$ is real, i.e.\ $q(x,y) \in \bb R[x,y]\setminus \{0\}$, and
satisfies $q(A,B) = p(A,B)$.
Thus, we can assume that a definitizing polynomial is real. \end{remark}
In the present section we assume that $p_j(x,y) \in \bb R[x,y]\setminus \{0\}$, $j=1,\dots,m$, are real, definitizing polynomial for $N$.
\begin{proposition}\thlab{defNspaces}
With the above assumptions and notation
there exist Hilbert spaces $(\mc H,(.,.))$, $(\mc H_j,(.,.))$, $j=1,\dots,m$ and
bounded linear and injective operators $T: \mc H \to \mc K$, $T_j: \mc H_j \to \mc K$,
such that
\[
T_j T_j^+ = p_j(A,B), \ \ \text{ and } \ \
T T^+ = \sum_{k=1}^m T_k T_k^+ = \sum_{k=1}^m p_k(A,B) \,.
\] \end{proposition} \begin{proof} Let $(\mc H_j,(.,.))$ be the Hilbert space completion of $\mc K/\ker p_j(A,B)$ with respect to $[p_j(A,B).,.]$ and let $T_j: \mc H_j \to \mc K$ be the adjoint of the factor mapping $x\mapsto x+ \ker p_j(A,B)$ of $\mc K$ into $\mc H_j$. Since $T_j^+$ has dense range, $T_j$ must be injective. Similarly, let $(\mc H,(.,.))$ be the Hilbert space completion of $\mc K/(\ker \sum_{k=1}^m p_k(A,B))$ with respect to $[\big(\sum_{k=1}^m p_k(A,B)\big).,.]$ and let $T: \mc H \to \mc K$ be the injective adjoint of the factor mapping of $\mc K$ into $\mc H$.
From $[T T^+ x,y] = (T^+ x,T^+ y) = (x,y) = [\big(\sum_{k=1}^m p_k(A,B)\big) x,y]$ and $[T_j T_j^+ x,y] = (T_j^+ x,T_j^+ y) = (x,y) = [p_j(A,B)x,y]$ for all $x,y\in\mc K$ we conclude
\[
T_j T_j^+ = p_j(A,B) \ \ \text{ and } \ \ T T^+ = \sum_{k=1}^m p_k(A,B) \,,
\] where the operators $T_j T_j^+ = p_j(A,B)$, $j=1,\dots,m$, pairwise commute, because $A$ and $B$ do. \end{proof}
\thref{defNspaces} in particular yields \begin{equation}\label{ghqw73si}
TT^+ = \sum_{k=1}^m T_kT_k^+ \end{equation} Since for $x\in \mc K$ and $j\in \{1,\dots,m\}$ we have \[
(T^+x,T^+x) = [TT^+x,x] = \sum_{k=1}^m [T_kT_k^+x,x] =
\sum_{k=1}^m (T_k^+x,T_k^+x) \geq (T_j^+x,T_j^+x) \,, \] one easily concludes that $T^+ x \mapsto T_j^+x$ constitutes a well-defined, contractive linear mapping from $\ran T^+$ onto $\ran T_j^+$. By $(\ran T^+)^\bot = \ker T=\{0\}$ and $(\ran T_j^+)^\bot = \ker T_j=\{0\}$ these ranges are dense in the Hilbert spaces $\mc H$ and $\mc H_j$. Hence, there is a unique bounded linear continuation of $T^+ x \mapsto T_j^+x$ to $\mc H$, which has dense range in $\mc H_j$.
Denoting by $R_j$ the adjoint mapping of this continuation we clearly have $T_j = T R_j$ and $\ker R_j \subseteq \ker T_j = \{0\}$. From \eqref{ghqw73si} we conclude \[
T( \, I_{\mc H} \, )T^+ = TT^+ = \sum_{k=1}^m T R_k R_k^+T^+ = T( \, \sum_{k=1}^m R_k R_k^+ \, )T^+ \,. \] $\ker T=\{0\}$ and the density of $\ran T^+$ yield $\sum_{k=1}^m R_k R_k^* = I_{\mc H}$.
\begin{lemma}\thlab{existtreans}
With the above notations and assumptions for $j=1,\dots,m$ there exist injective contractions $R_j: \mc H_j \to \mc H$
such that $T_j = T R_j$ and $\sum_{k=1}^m R_k R_k^* = I_{\mc H}$. Moreover, we have
\begin{equation}\label{zfv6lkc}
\{N,N^+\}'=\{A,B\}' \subseteq
\bigcap_{k=1,\dots,m} (T_kT_k^+)' \subseteq (TT^+)'
\end{equation}
for all $j\in\{1,\dots,m\}$. Finally, \begin{equation}\label{heaybab} \begin{aligned}
p_j(\Theta(A),\Theta(B)) & = R_j R_j^* \big( \sum_{k=1}^m p_k(\Theta(A),\Theta(B)) \big) \\ & =
\big( \sum_{k=1}^m p_k(\Theta(A),\Theta(B)) \big) R_j R_j^*
\,, \end{aligned} \end{equation} and for any $u \in \bb C[x,y]$ \begin{equation}\label{tt2479}
p_j(A,B)u(A,B) = \Xi_j\big(u(\Theta_j(A),\Theta_j(B))\big) = \Xi\big(R_jR_j^{*} u(\Theta(A),\Theta(B))\big) \,, \end{equation}
where $\Theta: (TT^+)' \ (\subseteq B(\mc K)) \to (T^+T)' \ (\subseteq B(\mc H))$ is as in
\eqref{thetaVdef} and $\Xi : B(\mc H) \to B(\mc K)$ with $\Xi(D) = T D T^{+}$. \end{lemma} \begin{proof}
The first part was shown above, and \eqref{zfv6lkc} is clear from \thref{defNspaces}.
From \eqref{zuef3} and \thref{thetadefeig} in \cite{KaPr2014} we get \begin{align*}
p_j(\Theta(A),\Theta(B)) & = \Theta(p_j(A,B)) = \Theta(T_jT_j^+) = R_j R_j^* \, T^+ T = R_j R_j^* \, \Theta(TT^+) \\ & =
R_j R_j^* \, \Theta( \sum_{k=1}^m p_k(A,B) ) = R_j R_j^* \, \big( \sum_{k=1}^m p_k(\Theta(A),\Theta(B)) \big) \,, \end{align*}
where $R_j R_j^*$ commutes with $T^+ T= \sum_{k=1}^m p_k(\Theta(A),\Theta(B))$ by \thref{existtreanspost}.
Finally, \eqref{tt2479} follows from (see \thref{Xidefeig} in \cite{KaPr2014}) \begin{align*}
p_j(A,B)u(A,B) & = \Xi_j\big(\Theta_j(u(A,B))\big) = \Xi\circ\Lambda_j\circ\Gamma_j\big(\Theta(u(A,B))\big) \\
\nonumber & =
\Xi\big(R_jR_j^{*} u(\Theta(A),\Theta(B))\big) \,. \end{align*} \end{proof}
By \eqref{zfv6lkc} we can apply \thref{normtransf} in the present situation. In particular, $\Theta(N)$ is a normal operator on the Hilbert space $\mc H$. Condition \eqref{defalgl} for $p=p_j, \ j=1,\dots,m$, implies certain spectral properties of $\Theta(N)$.
\begin{lemma}\thlab{speknorm}
With the above assumptions and notation for $j\in\{1,\dots,m\}$ we have \[
\{z\in \bb C : |p_j(\RE z,\IM z)| > \| R_j R_j^* \| \cdot |\sum_{k=1}^m p_k(\RE z,\IM z)| \} \subseteq \rho(\Theta(N)) \,. \] In particular, the zeros of $\sum_{k=1}^m p_k(\RE z,\IM z)$ in $\bb C$ are contained in $\rho(\Theta(N)) \cup \{z\in \bb C: p_j(\RE z,\IM z)=0 \ \text{ for all } \ j=1,\dots,m\}$. \end{lemma} \begin{proof} Let $n\in \bb N$ and set \[
\Delta_n:= \{z\in \bb C : |p_j(\RE z,\IM z)|^2 > \frac{1}{n} + \| R_j R_j^* \|^2 \cdot |\sum_{k=1}^m p_k(\RE z,\IM z)|^2 \} \,. \] For $x\in E(\Delta_n)(\mc H)$, where $E$ denotes $\Theta(N)$'s special measure, we then have \begin{multline*}
\| p_j(\Theta(A),\Theta(B)) x \|^2 = \int_{\Delta_n} |p_j(\RE \zeta,\IM \zeta)|^2 \, d(E(\zeta)x,x) \geq \\
\int_{\Delta_n} \frac{1}{n} \, d(E(\zeta)x,x) +
\| R_j R_j^* \|^2 \int_{\Delta_n} |\sum_{k=1}^m p_k(\RE \zeta,\IM \zeta)|^2 \, d(E(\zeta)x,x) \\ \geq
\frac{1}{n} \|x\|^2 + \| R_j R_j^* \big(\sum_{k=1}^m p_k(\Theta(A),\Theta(B))\big)x \|^2 \,. \end{multline*} By \eqref{heaybab} this inequality can only hold for $x=0$. Since $\Delta_n$ is open, by the Spectral Theorem for normal operators on Hilbert spaces we have $\Delta_n \subseteq \rho(N)$. The asserted inclusion now follows from \[
\{z\in \bb C : |p_j(\RE z,\IM z)| > \| R_j R_j^* \| \cdot |\sum_{k=1}^m p_k(\RE z,\IM z)| \} =
\bigcup_{n\in\bb N} \Delta_n \,. \] \end{proof}
In the following let $I$ the ideal $\langle p_1,\dots,p_m\rangle$ generated by the real definitizing polynomials $p_1,\dots,p_m$ in the ring $\bb C[x,y]$. The variety $V(I)$ is the set of all common zeros $a=(a_1,a_2) \in \bb C^2$ of all $p\in I$. Clearly, $V(I)$ coincides with the set of all $a\in \bb C^2$ such that $p_1(a_1,a_2) = \dots = p_m(a_1,a_2) = 0$. $V_{\bb R}(I)$ is the set of all $a\in \bb R^2$, which belong to $V(I)$. It is convenient for our purposes, to consider $V_{\bb R}(I)$ as a subset of $\bb C$: \begin{align}\label{nullstmereal}
V_{\bb R}(I) :&=
\{z\in \bb C: f(\RE z,\IM z) = 0 \ \text{ for all } \ f\in I\}
\\ & \nonumber = \{z\in \bb C: p_k(\RE z,\IM z) = 0 \ \text{ for all } \ k \in \{1,\dots,m\}\} \,. \end{align}
\begin{corollary}\thlab{korvda}
Let $E$ denote the special measure of $\Theta(N)$. Then we have \[
R_jR_j^* \, E(\bb C \setminus V_{\bb R}(I)) = E(\bb C \setminus V_{\bb R}(I)) \, R_jR_j^* =
\int_{\bb C \setminus V_{\bb R}(I)} \frac{p_j(\RE z,\IM z)}{\sum_{k=1}^m p_k(\RE z,\IM z)} \, dE(z) \,. \] \end{corollary} \begin{proof}
First note that the integral on the right hand side exists as a bounded operator, because
by \thref{speknorm} we have $|p_j(\RE z,\IM z)| \leq \| R_j R_j^* \| \cdot |\sum_{k=1}^m p_k(\RE z,\IM z)|$
for $z\in\sigma(\Theta(N))$.
The first equality is known from \thref{normtransf}.
Concerning the second equality, note that both sides vanish on the range of
$E(V_{\bb R}(I))$.
Its orthogonal complement $\mc Q:= \ran E(\bb C \setminus V_{\bb R}(I))$ is invariant under
\[
\int \big(\sum_{k=1}^m p_k(\RE z,\IM z)\big) \, dE(z) = \sum_{k=1}^m p_k(\Theta(A),\Theta(B)) \,.
\]
By \thref{speknorm} the restriction of this operator to $\mc Q$
is injective, and hence, has dense range in $\mc Q$.
If $x$ belongs to this dense range, i.e.\
$x=\big(\sum_{k=1}^m p_k(\Theta(A),\Theta(B))\big) y$ with $y\in \mc Q$, then
\begin{align*}
\int_{\bb C \setminus V_{\bb R}(I)} & \frac{p_j(\RE z,\IM z)}{\sum_{k=1}^m p_k(\RE z,\IM z)} \, dE(z) x =
\int_{\bb C \setminus V_{\bb R}(I)} p_j(\RE z,\IM z) \, dE(z) y \\ & =
p_j(\Theta(A),\Theta(B))y = R_j R_j^* \big(\sum_{k=1}^m p_k(\Theta(A),\Theta(B))\big) y = R_j R_j^* x \,. \end{align*}
By a density argument the second asserted equality of the present corollary holds true on $\mc Q$
and in turn on $\mc H$. \end{proof}
\begin{remark}\thlab{nullrem}
In \thref{defNspaces} the case that $p_j(A,B) = 0$ for some $j$, or even for all $j$, is not excluded, and
yields $\mc H_j=\{0\}$, $T_j=0$ and $R_j=0$ (in \thref{existtreans}), or even $\mc H=\{0\}$ and $T=0$.
Also the remaining results hold true, if we interpret $\rho(R)$ as $\bb C$ and $\sigma(R)$ as $\emptyset$
for the only possible linear operator $R=(0\mapsto 0)$ on the vector space $\{0\}$. \end{remark}
\section{An Abstract Functional Calculus} \label{abstrfunccal}
In this section let $\mc K$ be again a Krein space, $N\in B(\mc K)$ be a definitizable normal operator. Let $I$ be the ideal in $\bb C[x,y]$, which is generated by all real definitizing polynomials. By the ascending chain condition for the ring $\bb C[x,y]$ (see for example \cite{CLO1}, Theorem 7, Chapter 2, \S5) $I$ is generated by finitely many real definitizing polynomials $p_1,\dots,p_m$, i.e.\ $I = \langle p_1,\dots,p_m \rangle$. We employ the same notion as in the previous sections for these polynomials $p_1,\dots,p_m$. In particular, $E_j$ ($E$) denotes the spectral measure of $\Theta_j(N)$ on $\mc H_j$ ($\Theta(N)$ on $\mc H$).
We also make the convention that for $p\in \bb C[x,y]$ and $z\in \bb C$ we write $p(z)$ short for $p(\RE z,\IM z)$.
\begin{lemma}\thlab{lre0}
For a bounded and measurable $f: \sigma(\Theta(N)) \to \bb C$ and
$j\in \{1,\dots,m\}$ we have
\begin{multline*}
\Xi_j\left( \int_{\sigma(\Theta_j(N))} f \, dE_j \right) = \\
\Xi\left( \int_{\sigma(\Theta(N))\setminus V_{\bb R}(I)} f\frac{p_j}{\sum_{l=1}^m p_l} \, dE
+ R_jR_j^* \int_{\sigma(\Theta(N))\cap V_{\bb R}(I)} f \, dE \right) \,.
\end{multline*} \end{lemma} \begin{proof}
By \eqref{zuef2} the left hand side coincides with
\[
\Xi\left( R_jR_j^* \int_{\sigma(\Theta(N)) \setminus V_{\bb R}(I)} f \, dE
+ R_jR_j^* \int_{\sigma(\Theta(N)) \cap V_{\bb R}(I)} f \, dE \right)
\,.
\]
$\int_{\sigma(\Theta(N)) \setminus V_{\bb R}(I)} f \, dE =
E(\bb C \setminus V_{\bb R}(I)) \int_{\sigma(\Theta(N)) \setminus V_{\bb R}(I)} f \, dE$
together with \thref{korvda} prove the equality. \end{proof}
\begin{lemma}\thlab{lre1}
Let $f,g: \sigma(\Theta(N)) \to \bb C$ be bounded and measurable,
and let $r\in \bb C[x,y]$. For $j,k\in \{1,\dots,m\}$
we then have
\begin{align}\label{rthz23}
r(A,B) \, \Xi_j( \int_{\sigma(\Theta_j(N))} f \, dE_j ) & =
\Xi_j( \int_{\sigma(\Theta_j(N))} f \, dE_j ) \, r(A,B) \\ \nonumber & =
\Xi_j( \int_{\sigma(\Theta_j(N))} r f \, dE_j ) \,,
\end{align}
and
\begin{equation}\label{rthz23b}
\Xi_j\left(\int_{\sigma(\Theta_j(N))} f \, dE_j \right)
\Xi_k\left(\int_{\sigma(\Theta_k(N))} g \, dE_k\right)
\end{equation}
\[
= \Xi\left(\int_{\sigma(\Theta(N))} fg \, \frac{p_jp_k}{\sum_{l=1}^m p_l} \, dE\right)
\]
\[
= \Xi_j\left(\int_{\sigma(\Theta_j(N))} fg \, p_k \, dE_j \right) =
\Xi_k\left(\int_{\sigma(\Theta_k(N))} fg \, p_j \, dE_k \right) \,.
\] \end{lemma} \begin{proof}
By \thref{Xidefeig} in \cite{KaPr2014} we have
\[
r(A,B)\,\Xi_j(D)=\Xi_j(\Theta(r(A,B)) D) = \Xi_j(r(\Theta_j(A),\Theta_j(B)) D)\,,
\]
\[
\Xi_j(D)r(A,B)=\Xi_j( D \, \Theta_j(r(A,B))) = \Xi_j(D \, r(\Theta_j(A),\Theta_j(B)))
\]
for $D\in (T^+ T)'$. For $D=\int_{\sigma(\Theta_j(N))} f \, dE_j$ this
implies \eqref{rthz23}.
According to \eqref{zuef2} the expression in \eqref{rthz23b} coincides with
\[
\Xi\left(R_jR_j^* \int_{\sigma(\Theta(N))} f \, dE \right)
\Xi\left(R_kR_k^* \int_{\sigma(\Theta(N))} g \, dE\right) \,.
\]
By \thref{Xidefeig} and \thref{thetadefeig} in \cite{KaPr2014}, we also know that
$\Xi(D_1)\Xi(D_2)=\Xi(T^+T D_1D_2)=\Xi(\Theta(TT^+) D_1D_2)$, where
(see \thref{defNspaces} and \eqref{nullstmereal})
\[
\Theta(TT^+) = \sum_{l=1}^m p_l(\Theta(A),\Theta(B)) = \int \sum_{l=1}^m p_l \, dE =
(\int \sum_{l=1}^m p_l \, dE) \, E(\bb C \setminus V_{\bb R}(I)) \,.
\]
Therefore, by \thref{korvda} and the fact, that
$E(\bb C \setminus V_{\bb R}(I))$ commutes with $\int_{\sigma(\Theta(N))} f \, dE$,
\eqref{rthz23b} can be written as
\[
\Xi\left( (\int \sum_{l=1}^m p_l \, dE)
(\int \frac{p_j}{\sum_{l=1}^m p_l} \, dE) (\int f \, dE)
(\int \frac{p_k}{\sum_{l=1}^m p_l} \, dE) (\int g \, dE)\right) =
\]
\[
\Xi\left(\int_{\sigma(\Theta(N))} fg \, \frac{p_jp_k}{\sum_{l=1}^m p_l} \, dE\right) \,.
\]
The remaining equalities follow from \thref{lre0} since the respective integrands
vanish on $V_{\bb R}(I)$. \end{proof}
\begin{lemma}\thlab{lre2}
For a bounded and measurable $f: \sigma(\Theta(N)) \to \bb C$ and $j\in \{1,\dots,m\}$ the
operator $\Xi_j\left( \int_{\sigma(\Theta_j(N))} f \, dE_j \right)$ belongs to $\{N,N^+\}''$. \end{lemma} \begin{proof}
Take $C\in \{N,N^+\}'=\{A,B\}' \subseteq \bigcap_{j=1,\dots,m} (T_jT_j^+)'$; see \eqref{zfv6lkc}.
From \thref{Xidefeig} in \cite{KaPr2014} we conclude
\[
C \, \Xi_j\left( \int_{\sigma(\Theta_j(N))} f \, dE_j \right)
= \Xi_j\left( \Theta_j(C) (\int_{\sigma(\Theta_j(N))} f \, dE_j) \right) \,.
\]
Since $\Theta_j$ is a homomorphism, $\Theta_j(C)$ commutes with $\Theta_j(N)$ and, in turn, with
$\int_{\sigma(\Theta_j(N))} f \, dE_j$. Hence, employing \thref{Xidefeig} in \cite{KaPr2014} once
more, the above expression coincides with
\[
\Xi_j\left( (\int_{\sigma(\Theta_j(N))} f \, dE_j) \, \Theta_j(C) \right)
= \Xi_j\left( \int_{\sigma(\Theta_j(N))} f \, dE_j \right) C \,.
\] \end{proof}
\begin{definition}\thlab{Psidef} Denoting by $\mf B\big(\sigma(\Theta(N))\big)$ the $*$-algebra of complex valued, bounded and measurable functions on $\sigma(\Theta(N))$, for $(r,f_1,\dots,f_m) \in \mc R:=\bb C[x,y]\times \mf B\big(\sigma(\Theta(N))\big) \times \dots \times \mf B\big(\sigma(\Theta(N))\big)$ we set \[
\Psi(r,f_1,\dots,f_m) := r(A,B) + \sum_{k=1}^m \Xi_k\left( \int_{\sigma(\Theta_k(N))} f_k \, dE_k \right) \,. \] By $\mc N$ we denote the set of all $(r,f_1,\dots,f_m) \in \mc R$ such that \[
r +\sum_{k=1}^m f_k p_k = 0 \ \ \text{ on } \ \ \sigma(\Theta(N))\setminus V_{\bb R}(I) \] and such that there exist $u_1,\dots,u_m \in \bb C[x,y]$ with \ $r=\sum_{k=1}^m u_k p_k$ \ and \[
(f_j + u_j)(z)=0 \ \text{ for } \ j=1,\dots,m, \ z\in V_{\bb R}(I)\cap \sigma(\Theta(N)) \,. \] \end{definition}
\begin{remark}\thlab{compmist}
Obviously, $\Psi$ is linear. From $\Xi_j(D^*)=\Xi_j(D)^+$ we easily deduce
$\Psi(r^\#,\overline{f_1},\dots,\overline{f_m})=\Psi(r,f_1,\dots,f_m)^*$.
Moreover, $\mc N$ constitutes a linear subspace of $\mc R$ invariant under
$.^\#: (r,f_1,\dots,f_m)\mapsto (r^\#,\overline{f_1},\dots,\overline{f_m})$. \end{remark}
\begin{lemma}\thlab{lre3}
If $(r,f_1,\dots,f_m) \in \mc N$, then $\Psi(r,f_1,\dots,f_m)=0$. \end{lemma} \begin{proof}
Due to \eqref{tt2479} $r=\sum_{k=1}^m u_k p_k$ implies
\[
r(A,B) = \sum_{k=1}^m p_k(A,B)u_k(A,B)
= \sum_{k=1}^m \Xi_k\big( u_k(\Theta_k(A),\Theta_k(B))\big) \,.
\]
From this and \thref{lre0} we obtain
\[
\Psi(r,f_1,\dots,f_m) = \sum_{k=1}^m \Xi_k\big( \int_{\sigma(\Theta_k(N))} (f_k + u_k) \, dE_k \big) =
\]
\[
\Xi\left(\int\limits_{\sigma(\Theta(N))\setminus V_{\bb R}(I)}
\sum_{k=1}^m \frac{f_kp_k + u_kp_k}{\sum_{l=1}^m p_l} \, dE
+ \sum_{k=1}^m R_kR_k^* \hspace{-2mm} \int\limits_{\sigma(\Theta(N))\cap V_{\bb R}(I)}
\hspace{-2mm} (f_k + u_k) \, dE \right) = 0 \,.
\] \end{proof}
\begin{lemma}\thlab{lre4}
For $(r,f_1,\dots,f_m), (s,g_1,\dots,g_m) \in \mc R$ have
\begin{align*}
\Psi(r,f_1,\dots, & f_m) \, \Psi(s,g_1,\dots,g_m) =
\\ & = \Psi(rs, rg_1+sf_1 + f_1 \sum_{k=1}^m g_k p_k,\dots, rg_m+sf_m + f_m \sum_{k=1}^m g_kp_k )
\\ & = \Psi(rs, rg_1+sf_1 + g_1 \sum_{k=1}^m f_k p_k,\dots, rg_m+sf_m + g_m \sum_{k=1}^m f_kp_k ) \,.
\end{align*} \end{lemma} \begin{proof}
By \thref{lre1} we have
\[
\Psi(r,f_1,\dots,f_m)\Psi(s,g_1,\dots,g_m) = r(A,B)s(A,B)
\]
\[
+ \sum_{k=1}^m r(A,B)\Xi_k( \int_{\sigma(\Theta_k(N))} g_k \, dE_k ) +
\sum_{j=1}^m \Xi_j( \int_{\sigma(\Theta_j(N))} f_j \, dE_j ) s(A,B)
\]
\[
+
\sum_{j,k=1}^m \Xi_j\left( \int_{\sigma(\Theta_j(N))} f_j \, dE_j \right)
\Xi_k\left(\int_{\sigma(\Theta_k(N))} g_k \, dE_k\right)
\]
\[
= (rs)(A,B) + \sum_{k=1}^m \Xi_k( \int_{\sigma(\Theta_k(N))} r g_k \, dE_k ) +
\sum_{j=1}^m \Xi_j( \int_{\sigma(\Theta_j(N))} s f_j \, dE_j )
\]
\[
+
\sum_{j=1}^m \Xi_j\left( \sum_{k=1}^m \int_{\sigma(\Theta_j(N))} f_jg_k \, p_k \, dE_j \right) \,,
\]
where this last term can also be written as
\[
\sum_{j=1}^m \Xi_j\left( \sum_{k=1}^m \int_{\sigma(\Theta_j(N))} f_k g_j \, p_k \, dE_j \right) \,.
\] \end{proof}
\begin{definition}\thlab{Psidefpost} We provide $\mc R$ with a multiplication in the following way: \begin{multline*}
(r,f_1,\dots,f_m) \cdot (s,g_1,\dots,g_m) :=
\\
(rs, rg_1+sf_1 + f_1 \sum_{j=1}^m g_j p_j,\dots, rg_m+sf_m + f_m \sum_{j=1}^m g_jp_j ) \,. \end{multline*} \end{definition}
\begin{remark}\thlab{compmistpost}
Obviously, $\cdot$ is bilinear and compatible with $.^\#$ as defined in \thref{compmist}.
It is elementary to check its associativity.
Moreover, for $(r,f_1,\dots,f_m) \in \mc N$ and $(s,g_1,\dots,g_m)\in \mc R$ we have
$rs +\sum_{j=1}^m p_j (rg_j + sf_j + f_j \sum_{k=1}^m g_k p_k) =
(r +\sum_{j=1}^m f_j p_j) (s +\sum_{k=1}^m g_k p_k) = 0$ on $\bb C \setminus V_{\bb R}(I)$.
For the corresponding $u_1,\dots,u_m \in \bb C[x,y]$ with
$r=\sum_{j=1}^m u_j p_j$ and $(f_j + u_j)(z)=0$ for all $z\in V_{\bb R}(I)$ we have
$rs = \sum_{j=1}^m (u_j s) p_j$ and
\[
rg_j+sf_j + f_j \sum_{k=1}^m g_k p_k + u_j s = r g_j + f_j \sum_{k=1}^m g_k p_k = 0
\]
on $V_{\bb R}(I)$ since $r$ and the $p_j$ vanish there.
Hence, $\mc N$ is a right ideal. Similarly, one shows that it is also a left ideal.
Finally, the commutator
\[
(r,f_1,\dots,f_m) \cdot (s,g_1,\dots,g_m) - (s,g_1,\dots,g_m) \cdot (r,f_1,\dots,f_m) =
\]
\[
(0,\sum_{j=1}^m (f_1g_j - g_1f_j) p_j, \dots, \sum_{j=1}^m (f_mg_j - g_mf_j) p_j)
\]
belongs to $\mc N$. Consequently, $\mc R/\mc N$ is a commutative $*$-algebra. \end{remark}
Gathering the previous results we obtain the final result of the present section.
\begin{theorem}\thlab{homo1}
$\Psi/\mc N: (r,f_1,\dots,f_m) + \mc N \mapsto \Psi(r,f_1,\dots,f_m)$
is a well-defined $*$-homomorphism from $\mc R/\mc N$ into $\{N,N^+\}'' \subseteq B(\mc K)$. \end{theorem}
\section{Algebra of Zero-dimensional Ideals}
By the Noether-Lasker Theorem (see for example \cite{CLO1}, Theorem 7, Chapter 4, \S7) any ideal $I$ in $\bb C[x,y]$ admits a minimal primary decomposition \begin{equation}\label{primdec}
I = Q_1\cap \dots \cap Q_l \,. \end{equation} $Q_j$ being a primary ideal means that $fg\in Q_j$ implies $f\in Q_j$ or $g^k \in Q$ for some $k\in \bb N$, and minimal means that $Q_j \not\supseteq \bigcap_{i\neq j} Q_i$ for all $j=1,\dots,l$ and $P_j \neq P_i$ for $i\neq j$, where $P_j$ denotes the radical \[
\sqrt{Q_j} := \{f\in \bb C[x,y]: f^k \in Q_j \ \text{ for some } \ k \in \bb N\} \,. \] For an ideal $I$ in $\bb C[x,y]$ such a decomposition is in general not unique. Nevertheless, the First Uniqueness Theorem on minimal primary decompositions states that the number $l\in \bb N$ and the radicals $P_1,\dots,P_l$ are uniquely determined by $I$; see for example \cite{BW}, Theorem 8.55 on page 362. Moreover, the Second Uniqueness Theorem on minimal primary decompositions says that if $Q_1'\cap \dots \cap Q_l'=I=Q_1\cap \dots \cap Q_l$ are minimal primary decompositions ordered such that $P_j=\sqrt{Q_j} = \sqrt{Q_j'}$ for $j=1,\dots,l$ and if $P_k$ is minimal in $\{P_1,\dots,P_l\}$ with respect to $\subseteq$, then $Q_k'=Q_k$; see for example \cite{BW}, Theorem 8.56 on page 364.
Assume now that $I$ is a zero-dimensional ideal in $\bb C[x,y]$, i.e.\ \[
\dim \bb C[x,y] / I < \infty \,. \] For necessary and sufficient conditions see for example \cite{BW}, Theorem 6.54 and Corollary 6.56 on pages 274 and 275 and \cite{CLO2}, page 39 and 40. Let \eqref{primdec} be a minimal primary decomposition. Then any $Q_j$, and in turn $P_j \supseteq Q_j$, is also zero-dimensional. In particular, $\bb C[x,y]/P_j$ is a finite integral domain, and hence, a field. In turn, the radicals $P_1,\dots,P_l$ of $Q_1,\dots,Q_l$ are maximal ideals. By \cite{CLO1}, Theorem 11, Chapter 4, \S5, this means that the $P_j$ are generated by $x-a_{x,j}, y-a_{y,j}$, i.e.\ $P_j=\langle x-a_{x,j}, y-a_{y,j} \rangle$, for pairwise distinct $a_j=(a_{x,j},a_{y,j}) \in \bb C^2$. Consequently, any $P_k$ is minimal in $\{P_1,\dots,P_l\}$, and by what was said above, \eqref{primdec} is the unique minimal primary decomposition of $I$.
By Hilbert's Nullstellensatz (see for example \cite{CLO1}, Theorem 2, Chapter 4, \S1) the set $V(Q_j)$ of common zeros in $\bb C^2$ of all $f\in Q_j$ coincides with $V(P_j)=\{a_j\}$. By \cite{CLO1}, Theorem 7, Chapter 4, \S3, we also have \[
V(I)= V(Q_1) \cup \dots \cup V(Q_l) = \{a_1,\dots,a_l\} \,. \] Since $V(Q_j + Q_i) = V(Q_j)\cap V(Q_i) = \{a_j\}\cap \{a_i\}=\emptyset$ (see \cite{CLO1}, Theorem 4, Chapter 4, \S3) for $i\neq j$, the weak Nullstellensatz (see for example \cite{CLO1}, Theorem 1, Chapter 4, \S1) yields $Q_j + Q_i=\bb C[x,y]$. Hence, by the Chinese Remainder Theorem the mapping \begin{equation}\label{chreth}
\phi: \ \left\{
\begin{array}{lcr}
\bb C[x,y]/ I & \to & ({\bb C}[x,y]/ Q_1)\times \dots \times ({\bb C}[x,y]/ Q_l) \,, \\
x+I & \mapsto & (x+Q_1,\dots,x+Q_l)
\end{array}\right. \end{equation} constitutes an isomorphism, and $I=\prod_{j=1}^l Q_j$.
\begin{remark}\thlab{qbem}\hspace*{0mm}
\begin{enumerate}
\item Since the ring ${\bb C}[x,y]/ Q_j$ is finite dimensional, its invertible elements $f+Q_j$ are exactly
those, for which $fg \in Q_j$ implies $g\in Q_j$. $Q_j$ being primary this is equivalent to
$f\not\in P_j$. Hence, $f+Q_j$ is invertible in ${\bb C}[x,y]/ Q_j$ if and only if $f(a_j)\neq 0$.
\item
As $\sqrt{Q_j}=P_j$ we have $(x-a_{x,j})^m,(y-a_{y,j})^n\in Q_j$ for sufficiently large
$m, n \in \bb N$. Therefore, the ideal $P_j\cdot Q_j$ contains $(x-a_{x,j})^{m+1},(y-a_{y,j})^{n+1}$.
Thus, $P_j\cdot Q_j$ is also zero-dimensional and $\sqrt{P_j\cdot Q_j}=P_j$.
\end{enumerate} \end{remark}
\begin{definition}
For $a\in V(I)$ we set by $Q(a):=Q_j$ and $P(a):=P_j$,
where $j$ is such that $a=a_j$. By $d_x(a)$ ($d_y(a)$)
we denote the smallest natural number $m$ ($n$) such that $(x-a_{x})^{m} \in Q(a)$
($(y-a_{y})^{n} \in Q(a)$).
Moreover, for $a\in V(I)$ we set
\[
\mc A(a) := {\bb C}[x,y]/ (P(a)\cdot Q(a)) \ \ \text{ and } \ \ \mc B(a) := {\bb C}[x,y]/ Q(a) \,.
\] \end{definition}
Since $P(a)\cdot Q(a)$ and $Q(a)$ are ideals with finite codimension satisfying $P(a)\cdot Q(a) \subseteq Q(a)$, $\mc A(a)$ and $\mc B(a)$ are finite dimensional algebras with $\dim \mc A(a) \geq \dim \mc B(a)$.
\begin{remark}\thlab{sternop1}
Assume that $I$ is invariant under $.^\#$, where $f^\#(x,y):=\overline{f(\bar x,\bar y)}$.
This is for sure the case if $I$ is generated by real polynomial $p_1,\dots,p_m$.
Then $V(I) \subseteq\bb C^2$ is invariant under $(z,w) \mapsto (z,w)^\#:=(\bar z,\bar w)$.
Moreover, it is elementary to check that with $Q$ also $Q^\#$ is a primary ideal. Hence, with
$I=Q_1\cap \dots \cap Q_l$ also $I=I^\#=Q_1^\#\cap \dots \cap Q_l^\#$ is a minimal primary decomposition.
By the uniqueness of the minimal primary decomposition for our zero dimensional ideal $I$ one has
$Q(a)^\# = Q(a^\#)$ for all $a\in V(I)$.
Consequently, $f\mapsto f^\#$ induces a conjugate linear bijection from $\mc A(a)$ ($\mc B(a)$)
onto $\mc A(a^\#)$ ($\mc B(a^\#)$). \end{remark}
For the following note that if we conversely start with primary and zero-dimensional ideals $Q_1,\dots,Q_l$ with $\sqrt{Q_i}\neq \sqrt{Q_j}$ for $i\neq j$, then $I:=Q_1\cap \dots \cap Q_l$ is also zero-dimensional, and by the above mentioned uniqueness statement, $Q_1\cap \dots \cap Q_l$ is indeed the unique minimal primary decomposition of $I$.
\begin{proposition}\thlab{algle1}
Let $I$ be a zero-dimensional ideal in ${\bb C}[x,y]$ which is generated by $p_1,\dots,p_m$, and
let $I=\bigcap_{a\in V(I)} Q(a)$ be its unique primary decomposition. Assume that $W$ is a subset of $V(I)$.
Then
\[
J:=\bigcap_{a\in V(I)\setminus W} Q(a) \cap \bigcap_{a\in W} (P(a)\cdot Q(a))
\]
is also a zero-dimensional ideal satisfying $J\subseteq I$. The mapping
\[
\psi: \ \left\{
\begin{array}{lcr}
\bb C[x,y]/ J & \to & \varprod\limits_{a\in V(I)\setminus W} \big({\bb C}[x,y]/ Q(a)\big) \times
\varprod\limits_{a\in W} \big({\bb C}[x,y]/ (P(a)\cdot Q(a))\big) \,, \\
x+I & \mapsto & ((x+Q(a))_{a\in V(I)\setminus W},(x+(P(a)\cdot Q(a)))_{a\in W})
\end{array}\right.
\]
is an isomorphism, and any $p\in J$ can be written in the form $p=\sum_{j} u_j p_j$, where
$u_j(a)=0$ for all $a\in W$. \end{proposition} \begin{proof}
We already mentioned that $P(a)\cdot Q(a)$ is zero-dimensional with
$\sqrt{P(a)\cdot Q(a)}=P(a)$ and that the intersection
$J = \bigcap_{a\in V(I)\setminus W} Q(a) \cap \bigcap_{a\in W} P(a)\cdot Q(a)$ is the unique primary
decomposition of the zero-dimensional $J$. The isomorphism property of $\psi$ is a special case
of the corresponding fact concerning $\phi$; see \eqref{chreth}. We also have
\begin{align*}
J & =\prod_{a\in V(I)\setminus W} Q(a) \cdot \prod_{a\in W} P(a)\cdot Q(a) =
\prod_{a\in V(I)} Q(a) \cdot \prod_{a\in W} P(a) \\ & = I \cdot \prod_{a\in W} P(a) =
\Big\langle p_1\cdot \prod_{a\in W} P(a), \dots, p_m \cdot \prod_{a\in W} P(a)\Big\rangle \,.
\end{align*}
This means that any $p\in J$ has a representation $p=\sum_{j} u_j p_j$ with
$u_j\in \prod_{a\in W} P(a)=\bigcap_{a\in W} P(a)$. Hence, $u_j(a)=0$ for all $a\in W$. \end{proof}
\begin{example}\thlab{vorteil1}
Assume that $I$ is generated by two polynomial $p_1, p_2 \in {\bb C}[x,y]$ such that
$p_1$ only depend on $x$ and $p_2$ only depends on $y$. The set $V(I)$ of common zeros of
$I$, or equivalently of $p_1$ and $p_2$, in $\bb C^2$ then consists of all points of the form
$(z,w)$, where $z\in \bb C$ is a zero of $p_1$ and $w\in \bb C$ is a zero of $p_2$,
i.e.\ $V(I)=p_1^{-1}\{0\} \times p_2^{-1}\{0\}$. For $z\in p_1^{-1}\{0\}$ denote by
$\mf d_1(z)$ $p_1$'s degrees of the zero at $z$, and for $w\in p_2^{-1}\{0\}$ denote by
$\mf d_2(w)$ $p_2$'s degrees of the zero at $w$.
Given $p(x,y) \in {\bb C}[x,y]$ we can apply polynomial division in one variable twice,
once with respect to $x$ and once $y$, on order to see that
\[
p(x,y) = p_1(x)\cdot u(x,y) + p_2(y)\cdot v(x,y) + q(x,y)
\]
with $u(x,y), v(x,y), q(x,y) \in {\bb C}[x,y]$ such that the
degree of $q(x,y)$, seen as a polynomial on $x$, is less then the degree of $p_1$,
and such that the
degree of $q(x,y)$, seen as a polynomial on $y$, is less then the degree of $p_2$;
see \thref{einbett2pre} in \cite{Ka2015}. Hence, $I$ is zero-dimensional.
Moreover, writing $p_1(x)$ and $p_2(y)$ as products of linear factors,
it follows that $p\in I$ if and only if
\begin{equation}\label{jjww306}
p \in \langle (x-z)^{\mf d_1(z)},(y-w)^{\mf d_2(w)} \rangle :=Q((z,w)) \,,
\end{equation}
for all $z\in {p_1}^{-1}\{0\}, w\in {p_2}^{-1}\{0\}$.
Since $Q((z,w))$ is a primary ideal in ${\bb C}[x,y]$,
\[
I = \bigcap_{(z,w)\in p_1^{-1}\{0\} \times p_2^{-1}\{0\}} Q((z,w))
\]
is the minimal primary decomposition of $I$. For the respective radicals we have
$P((z,w))=\langle x-z, y-w \rangle$. Moreover, $P((z,w))\cdot Q((z,w))$
coincides with
\[
\langle (x-z)^{\mf d_1(z)+1}, (x-z)^{\mf d_1(z)}(y-w),
(x-z)(y-w)^{\mf d_2(w)} ,(y-w)^{\mf d_2(w)+1} \rangle \,.
\]
Therefore, $\mc A((z,w))={\bb C}[x,y]/(P((z,w))\cdot Q((z,w)))$
($\mc B((z,w))={\bb C}[x,y]/Q((z,w))$) is isomorphic to
$\mc A_{\mf d_1(z),\mf d_2(w)}$ ($\mc B_{\mf d_1(z),\mf d_2(w)}$) as introduced in
\thref{muldefb1}, \cite{Ka2015}. \end{example}
\section{Function classes}
In the present section we make the same assumptions and use the same notation as in Section \ref{abstrfunccal}. In addition, we assume that the ideal $I$ generated by all real definitizing polynomials is zero-dimensional. We fix real, definitizing polynomials $p_1,\dots,p_m$ which generate $I$. For the zero-dimensional $I$ we apply the same notation as in the previous section.
The variety $V(I)=\{a_1,\dots,a_l\} \subseteq \bb C^2$ of common zeros of all $f\in I$ will be split up as \[
V(I) = \underbrace{(V(I)\cap \bb R^2)}_{=V_{\bb R}(I)} \, \dot\cup \, (V(I)\setminus\bb R^2) \,, \] where we consider $V_{\bb R}(I)$ as a subset of $\bb C$; see \eqref{nullstmereal}.
\begin{definition}
By $\mc M_{N}$ we denote the set of functions $\phi$ defined on
\[
\underbrace{\big(\sigma(\Theta(N)) \cup V_{\bb R}(I) \big)}_{\subseteq \bb C} \, \dot\cup \,
\underbrace{(V(I)\setminus \bb R^2)}_{\subseteq \bb C^2}
\]
with $\phi(z) \in \bb C$ for $z\in \sigma(\Theta(N))\setminus V_{\bb R}(I)$,
$\phi(z) \in \mc A(z)$ for $z \in V_{\bb R}(I)$,
$\phi(z) \in \mc B(z)$ for $z \in V(I)\setminus \bb R^2$.
We provide $\mc M_{N}$ pointwise with scalar multiplication, addition and multiplication.
We also define a conjugate linear involution $.^\#$ on $\mc M_{N}$ by \begin{align*}
& \phi^\#(z) := \overline{\phi(z)} \ \ \text{ for } \ \ z\in \sigma(\Theta(N))\setminus V_{\bb R}(I), \\
& \phi^\#(z) := \phi(z)^\# \ \ \text{ for } \ \ z \in V_{\bb R}(I) \\
& \phi^\#(\xi,\eta) := \phi(\bar \xi,\bar \eta)^\# \ \ \text{ for } \ \ (\xi,\eta) \in V(I)\setminus \bb R^2 \,. \end{align*}
\end{definition}
With the operations introduced above $\mc M_{N}$ is a commutative $*$-algebra as can be verified in a straight forward manner; see \thref{sternop1}.
\begin{definition}\thlab{feinbetefab}
Let $f: \dom f \to \bb C$ be a function with $\dom f \subseteq \bb C^2$ such that
$\tau\big(\sigma(\Theta(N)) \cup V_{\bb R}(I)\big) \subseteq \dom f$, where
$\tau : \bb C \to \bb C^2, \ (x+iy)\mapsto (x,y)$, such that
$f\circ \tau$ is sufficiently smooth -- more exactly, at least
$d_x(z) + d_y(z) - 1$ times continuously differentiable --
on a sufficiently small open neighbourhood $z$ for each $z\in V_{\bb R}(I)$, and
such that $f$ is holomorphic on an open neighbourhood of $V(I)\setminus \bb R^2 \ (\subseteq \bb C^2)$.
Then $f$ can be considered as an element $f_{N}$ of $\mc M_{N}$ by setting
$f_{N}(z) := f\circ \tau(z)$ for $z\in \sigma(\Theta(N)) \setminus V_{\bb R}(I)$, by \begin{multline*}
f_{N}(z) := \sum_{(k,l)\in J(z)}
\frac{1}{k!l!} \, \frac{\partial^{k+l}}{\partial a^k\partial b^l}
f\circ \tau(a+ib)\vert_{a+ib=z} \cdot \\ \cdot (x-\RE z)^k (y-\IM z)^l + (P(z)\cdot Q(z)) \in \mc A(z) \end{multline*}
for $z\in V_{\bb R}(I)$, where
\[
J(z)= (\{0,\dots,d_x(z)-1\}\times \{0,\dots,d_y(z)-1\}) \cup \{(d_x(z),0),(0,d_y(z))\} \,,
\]
and by \begin{multline*}
f_{N}(\xi,\eta) := \sum_{k=0}^{d_x(\xi,\eta)-1} \sum_{l=0}^{d_y(\xi,\eta)-1}
\frac{1}{k!l!} \, \frac{\partial^{k+l}}{\partial z^k\partial w^l}
f(z,w)\vert_{(z,w)=(\xi,\eta)} \cdot \\ \cdot (x-\xi)^k (y-\eta)^l + Q((\xi,\eta)) \in \mc B((\xi,\eta)) \,, \end{multline*}
for $(\xi,\eta)\in V(I)\setminus \bb R^2$. \end{definition}
\begin{remark}\thlab{bweuh30} By the Leibniz rule $f\mapsto f_{N}$ is compatible with multiplication. Obviously, it is also compatible with addition and scalar multiplication. If we define for a function $f$ as in \thref{feinbetefab} the function $f^\#$ by $f^\#(z,w) = \overline{f(\bar z,\bar w)}, \ (z,w) \in \dom f$, then we also have $(f^\#)_{N} = (f_{N})^\#$. \end{remark}
\begin{remark}\thlab{bweuh30po}
A special type of functions $f$ as in \thref{feinbetefab} are polynomials in
two variables, i.e.\ $f\in \bb C[x,y]$. Since for $z\in V_{\bb R}(I)$ and $(k,l)\not\in J(z)$
we have $(x-\RE z)^k (y-\IM z)^l \in P(z)\cdot Q(z)$,
\[
f_N(z)=f + (P(z)\cdot Q(z)) \in \mc A(z) \,.
\]
Similarly, $f_{N}(\xi,\eta) = f + Q((\xi,\eta))\in \mc B((\xi,\eta))$ for $(\xi,\eta)\in V(I)\setminus \bb R^2$.
In particular, for $f=\mathds{1}$ the element $f_N(z)$
is the multiplicative unite in $\mc A(z)$ or $\mc B(z)$
for all $z\in \big(\sigma(\Theta(N)) \cup V_{\bb R}(I) \big)\dot\cup (V(I)\setminus \bb R^2)$. \end{remark}
For the following recall for example from \cite{CLO1}, Theorem 4, Chapter 2, \S5, that any ideal in $\bb C[x,y]$ always has a finite number of generators.
\begin{definition}\thlab{abschaefu}
For any $w \in \sigma(\Theta(N)) \cap V_{\bb R}(I)$ such that
$w$ is not isolated in $\sigma(\Theta(N))$
let $h_1,\dots,h_n$ be generators of the ideal
$Q(w)$.
For a sufficiently small neighbourhood $U(w)$ of $w$
let $\chi_w: U(w)\setminus \{w\} \to [0,+\infty)$ be
\[
\chi_w(z):= \max_{j=1,\dots,n} |h_j(z)| \,,
\]
where $h_j(z)$, as usually, stands for $h_j(\RE z,\IM z)$. \end{definition}
Since $w$ is a common zero of all $h\in Q(w)$, we have $\chi_w(z) \to 0$ for $z\to w$. Moreover, for any $h\in Q(w)$ the fact, that $h_1,\dots,h_n$ are generators of $Q(w)$, yields $h=O(\chi_w)$ as $z \to w$.
Moreover, if $\chi_w'$ is defined in a similar manner starting with generators $h_1',\dots,h_{n'}'$, then $\chi_w' = O(\chi_w)$ and $\chi_w = O(\chi_w')$ as $z \to w$. Hence, as far as it concerns the order of growth towards $w$, the expression $\chi_w$ does not depend on the actually chosen generators.
\begin{definition}\thlab{FdefklM2}
We denote by $\mc F_{N}$ the set of all elements
$\phi \in \mc M_{N}$ such that
$z\mapsto \phi(z)$ is Borel measurable and bounded on
$\sigma(\Theta(N)) \setminus V_{\bb R}(I)$, and such that
for each $w \in \sigma(\Theta(N)) \cap V_{\bb R}(I)$, which
is not isolated in $\sigma(\Theta(N))$,
\begin{equation}\label{fn8qw3bab}
\phi(z) - \phi(w)\vert_{x=\RE z, y = \IM z} = O(\chi_w(z)) \ \ \text{ as } \ \
\sigma(\Theta(N)) \setminus V_{\bb R}(I) \ni z \to w \,.
\end{equation} \end{definition}
Note that in \eqref{fn8qw3bab} $\phi(w) \in \mc A(w)$ is a coset $p(x,y) + (P(w)\cdot Q(w))$ from ${\bb C}[x,y]/ (P(w)\cdot Q(w))$, and $\phi(w)\vert_{x=\RE z, y = \IM z}$ stands for any representative of this coset $\phi(w)$ considered as a function of $z$. In \eqref{fn8qw3bab} it does not matter what representative we take since $q = O(\chi_w)$ as $z \to w$ for any $q\in Q(w)$, and hence, for any $q\in (P(w)\cdot Q(w))$.
\begin{remark}\thlab{vorteil2}
Assume that our zero-dimensional ideal $I$ is generated by two
definitizing polynomials $p_1 \in \bb R[x], p_2\in \bb R[y]$ as in \thref{vorteil1}.
For $w\in V_{\bb R}(I)$, i.e.\ $(\RE w,\IM w)\in V(I)$, we conclude from
\eqref{jjww306} in \thref{vorteil1} that
\[
\chi_w(z):= \max(|(\RE z - \RE w)^{\mf d_1(\RE w)}|,|(\IM z - \IM w)^{\mf d_2(\IM w)}|) \,.
\]
Therefore, in this case the function class $\mc F_{N}$ here coincides exactly with
the function class $\mc F_{N}$ introduced in \thref{FdefklM}, \cite{Ka2015}. \end{remark}
\begin{example}\thlab{fedela33}
For $(\xi,\eta) \in V(I)\setminus \bb R^2$ and $a\in \mc B((\xi,\eta))$ the function
$a\delta_{(\xi,\eta)} \in \mc M_{N}$, which assumes the value $a$ at $(\xi,\eta)$ and the value zero on the rest of
$\big(\sigma(\Theta(N)) \cup V_{\bb R}(I) \big)\dot\cup (V(I)\setminus \bb R^2)$,
trivially belongs to $\mc F_{N}$.
Correspondingly, $a\delta_w \in \mc F_{N}$ for a $w \in V_{\bb R}(I)$, which is an isolated point of
$\sigma(\Theta(N)) \cup V_{\bb R}(I)$, and for $a\in \mc A(w)$. \end{example}
\begin{remark}\thlab{taylormehrdimremab}
Let $h$ be defined on an open subset $D$ of $\bb R^2$ with values in $\bb C$.
Moreover, assume that for given $m,n\in\bb N$ the function $h$ is
$m+n-1$ times continuously differentiable. Finally, fix $w\in D$.
The well-known Taylor Approximation Theorem from multidimensional calculus then yields
\[
h(z) = \sum_{j=0}^{m+n-2} \sum_{\stackrel{k,l\in \bb N_0}{k+l=j}}
\frac{1}{k!l!}
\frac{\partial^j h}{\partial x^{k} \partial y^{l}}(w)\RE(z-w)^k \IM(z-w)^{l}
+ O(|z-w|^{m+n-1})
\]
for $z\to w$. Since
\begin{align*}
|z-w|^{m+n-1} & \leq 2^{m+n-1}\max(|\RE(z-w)|^{m+n-1}, |\IM(z-w)|^{m+n-1})
\\ & = O(\max(|\RE(z-w)|^m, |\IM(z-w)|^{n})) \,,
\end{align*}
and since
$\RE(z-w)^k \IM(z-w)^{l} = O(\max(|\RE(z-w)|^m, |\IM(z-w)|^{n}))$ for
$k \geq m$ or $l\geq n$, we also have \begin{multline*}
h(z) = \sum_{k=0}^{m-1} \sum_{l=0}^{n-1} \frac{1}{k!l!}
\frac{\partial^{k+l} h}{\partial x^{k} \partial y^{l}}(w)\RE(z-w)^k \IM(z-w)^{l}
\\ + O(\max(|\RE(z-w)|^m, |\IM(z-w)|^{n})) \,. \end{multline*} \end{remark}
\begin{lemma}\thlab{gehzuFab}
Let $f: \dom f \ (\subseteq \bb C^2)\to \bb C$ be a function with the properties
mentioned in \thref{feinbetefab}. Then $f_N$ belongs to $\mc F_{N}$. \end{lemma} \begin{proof}
For a $w \in \sigma(\Theta(N)) \cap V_{\bb R}(I)$, which
is not isolated in $\sigma(\Theta(N))$, and $z\in \sigma(\Theta(N))\setminus V_{\bb R}(I)$
sufficiently near at $w$
by \thref{taylormehrdimremab} the expression
\[
f_N(z) - f_N(w)\vert_{x=\RE z, y = \IM z} =
\]
\[
f(\RE z, \IM z) - \hspace{-2mm} \sum_{(k,l) \in J(w)}
\frac{1}{k!l!} \, \frac{\partial^{k+l} f}{\partial x^k\partial y^l}(\RE w, \IM w)
\cdot (\RE z-\RE w)^k (\IM z-\IM w)^l
\]
is a $O(\max(|\RE(z-w)|^{d_x(w)}, |\IM(z-w)|^{d_y(w)}))$, and therefore a $O(\chi_w(z))$
as $z\to w$. Consequently $f_N\in \mc F_{N}$. \end{proof}
\begin{lemma}\thlab{einduF33}
If $\phi \in \mc F_{N}$ is such that $\phi(z)$ is invertible in $\bb C, \mc A(z)$ or $\mc B(z)$, respectively,
for all $z\in \big(\sigma(\Theta(N)) \cup V_{\bb R}(I) \big)\dot\cup (V(I)\setminus \bb R^2)$
and such that
$0\in\bb C$ does not belong to the closure of
$\phi\big(\sigma(\Theta(N)) \setminus V_{\bb R}(I)\big)$, then
$\phi^{-1}: z\mapsto \phi(z)^{-1}$ also belongs to $\mc F_{N}$. \end{lemma} \begin{proof}
By the first assumption $\phi^{-1}$ is a well-defined object belonging to $\mc M_{N}$.
Clearly, with $\phi$ also $z\mapsto \phi(z)^{-1}=\frac{1}{\phi(z)}$ is
measurable on $\sigma(\Theta(N)) \setminus V_{\bb R}(I)$.
By the second assumption of the present lemma
$z\mapsto \phi(z)^{-1}=\frac{1}{\phi(z)}$ is bounded on this set.
It remains to verify \eqref{fn8qw3bab} for $\phi^{-1}$ at each
$w \in \sigma(\Theta(N)) \cap V_{\bb R}(I)$, which is not isolated in $\sigma(\Theta(N)$.
To do so, first note that due to $\phi(w)$'s invertibility for $z\in \sigma(\Theta(N)) \setminus V_{\bb R}(I)$
sufficiently near at $w$ we have $\phi(w)\vert_{x=\RE z, y = \IM z} = p(z)\neq 0$,
where $p(x,y)$ is a representative of $\phi(w)$.
Now calculate \\
\begin{equation}\label{dfbz45pre}
\hspace*{-6cm} \phi^{-1}(z) - \phi(w)^{-1}\vert_{x=\RE z, y = \IM z} =
\end{equation}
\begin{equation}\label{dfbz45}
\hspace*{-2cm} = \frac{1}{\phi(z)} -
\frac{1}{\phi(w)\vert_{x=\RE z, y = \IM z}} +
\end{equation} \begin{equation}\label{fght33}
\hspace*{4cm} + \frac{1}{\phi(w)\vert_{x=\RE z, y = \IM z}} -
\phi(w)^{-1}\vert_{x=\RE z, y = \IM z} \,. \end{equation}
The expression in \eqref{dfbz45} can be written as
\[
\frac{1}{\phi(z) \cdot \phi(w)\vert_{x=\RE z, y = \IM z}} \cdot
\left( \phi(z) - \phi(w)\vert_{x=\RE z, y = \IM z} \right) \,.
\]
Here $\frac{1}{\phi(z)}$ is bounded by assumption.
The assumed invertibility of $\phi(w)$ implies the boundedness $\phi(w)\vert_{x=\RE z, y = \IM z}$
on a certain neighbourhood of $w$.
From $\phi\in \mc F_N$ we then conclude that \eqref{dfbz45} is a
$O(\chi_w(z))$ for $z\to w$.
\eqref{fght33} can be rewritten as
\[
- \frac{1}{\phi(w)\vert_{x=\RE z, y = \IM z}} \cdot
\Big(\phi(w)\vert_{x=\RE z, y = \IM z} \cdot \phi(w)^{-1}\vert_{x=\RE z, y = \IM z}- 1\Big) \,.
\]
The product in the brackets is a representative of
$\phi(w)\cdot \phi(w)^{-1}= 1 + (P(w)\cdot Q(w)) \in \mc A(w)$. Hence, \eqref{fght33} equals to
$\frac{1}{\phi(w)\vert_{x=\RE z, y = \IM z}} q(\RE z,\IM z)$ for a $q\in (P(w)\cdot Q(w))$, and is
therefore a $O(\chi_w(z))$ for $z\to w$. Altogether \eqref{dfbz45pre} is a $O(\chi_w(z))$ for $z\to w$.
Thus, $\phi^{-1}\in \mc F_N$. \end{proof}
\section{Functional Calculus for zero-dimensional $I$} \label{Funcalzero}
\begin{lemma}\thlab{uzgv24}
For each $\phi\in\mc F_N$ there exists $p\in\bb C[x,y]$ and complex valued
$f_1,\dots,f_m \in \mf B(\sigma(\Theta(N)) \cup V_{\bb R}(I))$ with
$f_j(z)=0$ for $z\in V_{\bb R}(I)$ such that
\[
\phi(z) = p_N(z) + \sum_j f_j(z) \, (p_j)_N(z)
\]
for all $z\in \sigma(\Theta(N)) \cup V_{\bb R}(I)$, and that
$\phi((\xi,\eta)) = p_N((\xi,\eta))$ for all $(\xi,\eta) \in V(I)\setminus \bb R^2$. \end{lemma} \begin{proof}
We apply \thref{algle1} to $W=V_{\bb R}(I)$. The fact, that $\psi$ is an isomorphism, then
yields the existence of a polynomial $p\in \bb C[x,y]$ such that $p + (P(w)\cdot Q(w)) = \phi(w)$
for all $w \in V_{\bb R}(I)$ and such that $p + Q((\xi,\eta)) =\phi((\xi,\eta))$
for all $(\xi,\eta) \in V(I)\setminus \bb R^2$.
By \thref{bweuh30po} we have $\phi(w)= p + (P(w)\cdot Q(w)) =p_N(w) \in \mc A(w)$
for $w\in V_{\bb R}(I)$.
For $(\xi,\eta) \in V(I)\setminus \bb R^2$ we have $\phi((\xi,\eta)) = p + Q((\xi,\eta))= p_N((\xi,\eta)) \in
\mc B((\xi,\eta))$.
For $j=1,\dots,m$ we set $f_j(z):=\frac{\phi(z) - p(z)}{\sum_k p_k(z)}$ if
$z\in \sigma(\Theta(N))\setminus V_{\bb R}(I)$
(see \thref{speknorm}), and $f_j(z) = 0$ if
$z\in V_{\bb R}(I)$. On $\sigma(\Theta(N))\cup V_{\bb R}(I)$ we then have
\[
\phi(z) = p_N(z) + \sum_j f_j(z) \, (p_j)_N(z) \,.
\]
It remains to verify that the functions $f_j$ are measurable and bounded on $\sigma(\Theta(N))\setminus V_{\bb R}(I)$.
The measurability easily follows
from the definition of $f_j$ and the measurability of $\phi$ on this set. Since there are only finitely many
points in $V_{\bb R}(I)$, the measurability of $f_j$ on $\sigma(\Theta(N))\cup V_{\bb R}(I)$ follows.
Concerning boundedness, note that by \thref{gehzuFab} $\phi-p_N$ belongs to $\mc F_N$. Since any representative
$(\phi-p_N)(w)\vert_{x=\RE z, y = \IM z}$ of $(\phi-p_N)(w)\in \mc A(w)$ belongs to $P(w)\cdot Q(w) \subseteq Q(w)$,
we have
$(\phi-p_N)(z)=O(\chi_w(z))$ as $z\to w$ for any $w\in \sigma(\Theta(N))\cap V_{\bb R}(I)$
which is not isolated on $\sigma(\Theta(N))$.
By \thref{speknorm} we have $\chi_w(z) = O(\sum_k p_k(z))$ as $z\to w$ for $z\in \sigma(\Theta(N))\setminus V_{\bb R}(I)$.
Therefore,
\[
f_j(z) = \frac{\phi(z) - p(z)}{\sum_k p_k(z)} = O(1) \ \ \text{ as } \ \ z\to w
\]
for $z\in \sigma(\Theta(N))\setminus V_{\bb R}(I)$. \end{proof}
\begin{definition}\thlab{deltadef} Let $\Delta$ be the set of all pairs $(\phi; (p,f_1\vert_{\sigma(\Theta(N))},\dots,f_m\vert_{\sigma(\Theta(N))}))$ such that all assertions from \thref{uzgv24} hold true for $\phi$ and $ (p,f_1,\dots,f_m)$. \end{definition}
\begin{remark}\thlab{deltadefpo} It is straight forward to check that $\Delta$ is a linear subspace of $\mc F_N \times \Big(\bb C[x,y] \times \mf B\big(\sigma(\Theta(N))\big) \times \dots \times \mf B\big(\sigma(\Theta(N))\big)\Big)$, i.e.\ a linear relations. Moreover, it is easy to check that with $(\phi; (p,f_1\vert_{\sigma(\Theta(N))},\dots,f_m\vert_{\sigma(\Theta(N))}))$ also $(\phi^\#; (p^\#, \overline{f_1\vert_{\sigma(\Theta(N))}},\dots,\overline{f_m\vert_{\sigma(\Theta(N))}}))$ belongs to $\Delta$; see \thref{compmist}. \end{remark}
$\Delta$ is also compatible with multiplication as will be shown next.
\begin{lemma}\thlab{ujr7984}
If both, $(\phi; (p,f_1\vert_{\sigma(\Theta(N))},\dots,f_m\vert_{\sigma(\Theta(N))}))$ and
$(\psi; (q,g_1\vert_{\sigma(\Theta(N))},\dots,g_m\vert_{\sigma(\Theta(N))}))$, belong to
$\Delta$, then also the pair $(\phi\cdot\psi; (r,h_1\vert_{\sigma(\Theta(N))},\dots,h_m\vert_{\sigma(\Theta(N))}))$
belongs to $\Delta$, where (see \thref{Psidefpost})
\begin{multline*}
(r,h_1\vert_{\sigma(\Theta(N))},\dots,h_m\vert_{\sigma(\Theta(N))}) = \\
(p,f_1\vert_{\sigma(\Theta(N))},\dots,f_m\vert_{\sigma(\Theta(N))}) \cdot
(q,g_1\vert_{\sigma(\Theta(N))},\dots,g_m\vert_{\sigma(\Theta(N))}) \,.
\end{multline*} \end{lemma} \begin{proof}
On $\sigma(\Theta(N)) \cup V_{\bb R}(I)$ we have
\[
\phi(z) = p_N(z) + \sum_j f_j(z) (p_j)_N(z) \ \text{ and } \
\psi(z) = q_N(z) + \sum_j g_j(z) (p_j)_N(z) \,.
\]
Moreover,
$f_j(z)=0=g_j$ for $z\in V_{\bb R}(I)$, and
$\phi((\xi,\eta)) = p_N((\xi,\eta))$, $\psi((\xi,\eta)) = q_N((\xi,\eta))$
for all $(\xi,\eta) \in V(I)\setminus \bb R^2$.
Since $p\mapsto p_N$ is compatible with multiplication, for $r=p\cdot q$ we have
$(\phi\cdot\psi)((\xi,\eta)) = r_N((\xi,\eta))$ for all $(\xi,\eta) \in V(I)\setminus \bb R^2$.
Clearly, $h_j = pg_j+qf_j + f_j \sum_{k=1}^m g_k p_k$ vanishes on $V_{\bb R}(I)$.
For $z\in \sigma(\Theta(N)) \cup V_{\bb R}(I)$ we have
\begin{multline*}
\phi(z) \, \psi(z) = p_N(z)\, q_N(z) + \\
\sum_j \Big(p_N(z) g_j(z) + q_N(z)f_j(z) + f_j(z) \sum_k g_k(z) (p_k)_N(z)\Big) \, (p_j)_N(z) \,,
\end{multline*}
which, for $z\in V_{\bb R}(I)$, coincides with $r_N(z) = r_N(z) + \sum_j h_j(z) (p_j)_N(z)$.
For $z\in \sigma(\Theta(N)) \setminus V_{\bb R}(I)$ the above equation can be written as
\begin{align*}
\phi(z)\, \psi(z) & = r(z) +
\sum_j \Big(p(z) g_j(z) + q(z)f_j(z) + f_j(z) \sum_k g_k(z) p_k(z)\Big) \, p_j(z) \\ & =
r_N(z) + \sum_j h_j(z) \, (p_j)_N(z) \,.
\end{align*} \end{proof}
We are going to determine the multivalued part $\mul \Delta$ of $\Delta$.
\begin{lemma}\thlab{ujr43094}
Assume that $p(x,y) \in\bb C[x,y]$ and
$f_1,\dots,f_m \in \mf B(\sigma(\Theta(N))\cup V_{\bb R}(I))$ with $f_j(z)=0$ for $z\in V_{\bb R}(I)$
such that
\[
0 = p_N(z) + \sum_j f_j(z) (p_j)_N(z)
\]
on $\sigma(\Theta(N))\cup V_{\bb R}(I)$ and that $\phi((\xi,\eta)) = 0$ for all
$(\xi,\eta) \in V(I)\setminus \bb R^2$. Then $(p,f_1\vert_{\sigma(\Theta(N))},\dots,f_m\vert_{\sigma(\Theta(N))})$ belongs to
the ideal $\mc N$ in $\mc R$ as defined in \thref{Psidef}. \end{lemma} \begin{proof}
Clearly, $p +\sum_{j=1}^m f_j p_j = 0$ on $\sigma(\Theta(N))\setminus V_{\bb R}(I)$.
According to \thref{bweuh30po} $p+(P(w)\cdot Q(w)) = 0 \in \mc A(w)$ for all $w\in V_{\bb R}(I)$ and
$p+Q((\xi,\eta)) = 0 \in \mc B((\xi,\eta))$ for all
$(\xi,\eta) \in V(I)\setminus \bb R^2$. Hence,
$p\in \bigcap_{(\xi,\eta)\in V(I)\setminus \bb R^2} Q((\xi,\eta)) \cap \bigcap_{w\in V_{\bb R}(I)} (P(w)\cdot Q(w))$.
By \thref{algle1} we therefore have $p = \sum_j u_jp_j$ with $u_j(w)=0$ for all $w\in V_{\bb R}(I)$.
We see that $(f_j + u_j)(z)=0$ for all $z\in V_{\bb R}(I)\cap \sigma(\Theta(N))$. Thus,
$(p,f_1\vert_{\sigma(\Theta(N))},\dots,f_m\vert_{\sigma(\Theta(N))}) \in \mc N$. \end{proof}
Since by \thref{lre3} $\mul \Delta \subseteq \mc N \subseteq \ker \Psi$ the composition $\Psi \Delta$ is a well-defined linear mapping from $\mc F_N$ into $B(\mc K)$.
\begin{definition}\thlab{calcueldef}
For $\phi\in\mc F_N$ we set $\phi(N):=(\Psi \Delta)(\phi)$. \end{definition}
By \thref{homo1}, \thref{ujr7984} and \thref{deltadefpo} the following result can be formulated.
\begin{theorem}\thlab{tu2989}
$\phi\mapsto \phi(N)$ constitutes a $*$-homomorphism from $\mc F_N$ into $\{N,N^*\}'' \subseteq B(\mc K)$.
It satisfies $p_N(N)=p(A,B)$ for all $p\in\bb C[x,y]$. \end{theorem} \begin{proof}
The final assertion is clear because of $(p_N; (p,0,\dots,0)) \in \Delta$. \end{proof}
\section{Spectral properties of the functional calculus}
For $w\in V_{\bb R}(I)$ we will need the following notation. By $\pi_w: \mc A(w) \to \mc B(w)$ we denote the mapping \[
\pi_w(f+(P(w)\cdot Q(w))) = f+ Q(w) \,. \]
\begin{lemma}\thlab{ehgtttpre}
If $\phi\in \mc F_N$ vanishes everywhere except at a fixed $w\in V_{\bb R}(I)$ and
if $\pi_w \phi(w) = 0$, then
\[
\phi(N) = \Psi(0;g_1,\dots,g_m)
\]
for $g_1,\dots,g_m \in \mf B\big(\sigma(\Theta(N))\big)$ which vanish on
$(\sigma(\Theta(N)) \cup V_{\bb R}(I)) \setminus \{w\}$. \end{lemma} \begin{proof}
Let $p(x,y)\in\bb C[x,y]$ and
$f_1,\dots,f_m \in \mf B(\sigma(\Theta(N)) \cup V_{\bb R}(I))$ with
$f_j(z)=0$ for $z\in V_{\bb R}(I)$ such that
\[
\phi(z) = p_N(z) + \sum_j f_j(z) \, (p_j)_N(z)
\]
for all $z\in \sigma(\Theta(N)) \cup V_{\bb R}(I)$, and that
$p_N((\xi,\eta)) = \phi((\xi,\eta)) = 0$ for all $(\xi,\eta) \in V(I)\setminus \bb R^2$.
The latter fact just means $p\in p((\xi,\eta)) \in Q((\xi,\eta))$.
From $0=\phi(z) = p_N(z) + \sum_j f_j(z) \, (p_j)_N(z)$ for $z\in V_{\bb R}(I)\setminus \{w\}$
we infer $p\in (P(z)\cdot Q(z))$. For $z=w$ this equation together with $\pi_w \phi(w) = 0$ yields
$p \in Q(w)$.
By \thref{algle1} $p=\sum_{j} u_j p_j$, where $u_j(z)=0$ for all $z\in V_{\bb R}(I) \setminus \{w\}$.
We define $g_j$ to be zero on $\big(\sigma(\Theta(N)) \cup V_{\bb R}(I)\big)\setminus \{w\}$ and set $g_j(w)=u_j(w)$.
The difference
\[
(p;f_1\vert_{\sigma(\Theta(N))},\dots,f_m\vert_{\sigma(\Theta(N))})
- (0;g_1,\dots,g_m) =
\]
\[
(p; f_1\vert_{\sigma(\Theta(N))}-\delta_w(.) u_1(w),\dots,f_m\vert_{\sigma(\Theta(N))}-\delta_w(.) u_m(w))
\]
satisfies $p + \sum_j (f_j(z)-\delta_w(z) u_j(w)) p_j(z) = \phi(z) =0$ for $z\in \sigma(\Theta(N)) \setminus V_{\bb R}(I)$
and $f_j(z)-\delta_w(z) u_j(w) + u_j(z) = 0$ for all $z\in V_{\bb R}(I)\cap \sigma(\Theta(N))$.
It therefore belongs to the ideal $\mc N$ of $\mc R$. Consequently,
\[
\phi(N) = \Psi(p;f_1\vert_{\sigma(\Theta(N))},\dots,f_m\vert_{\sigma(\Theta(N))})
= \Psi(0;g_1,\dots,g_m) \,.
\] \end{proof}
\begin{corollary}\thlab{ehgttt}
Assume that $E\{w\} = 0$ for a fixed $w\in V_{\bb R}(I)$, which surely happens if $w\not\in \sigma(\Theta(N))$.
Then $\phi(N)=\psi(N)$ for all $\phi, \psi$ that coincide on
$\big((\sigma(\Theta(N)) \cup V_{\bb R}(I))\setminus \{w\} \big)\dot\cup (V(I)\setminus \bb R^2)$
and that satisfy $\pi_w \phi(w) = \pi_w \psi(w)$. Here $\pi_w: \mc A(w) \to \mc B(w)$ is defined
by $\pi_w(f+(P(w)\cdot Q(w))) = f + Q(w)$. \end{corollary} \begin{proof}
By \thref{ehgtttpre} there exist $g_1,\dots,g_m \in \mf B\big(\sigma(\Theta(N))\big)$, which vanish on
$(\sigma(\Theta(N)) \cup V_{\bb R}(I)) \setminus \{w\}$, such that
\[
\phi(N) - \psi(N) = \Psi(0;g_1,\dots,g_m) = \sum_{k=1}^m \Xi_k\left( \int_{\sigma(\Theta_k(N))} g_k \, dE_k \right)
\]
According to \thref{lre0} together with our assumption $E\{w\} = 0$,
this operator vanishes. \end{proof}
\begin{remark}\thlab{rieszproj33} For $\zeta \in V(I)\setminus \bb R^2$ or a $\zeta \in V_{\bb R}(I)$, which is isolated in $\sigma(\Theta(N)) \cup V_{\bb R}(I)$, we saw in \thref{fedela33} that $a\delta_\zeta \in \mc F_{N}$. If $a$ is the unite $e$ in $\mc B(\zeta)$ or in $\mc A(\zeta)$, i.e.\ the coset $1+Q(\zeta)$ for $\zeta\in V(I)\setminus \bb R^2$ or the coset $1+(P(\zeta)\cdot Q(\zeta))$ for $\zeta\in V_{\bb R}(I)$, then $(e\delta_\zeta)\cdot (e\delta_\zeta) = (e\delta_\zeta)$ together with the multiplicativity of $\phi\mapsto \phi(N)$ shows that $(e\delta_\zeta)(N)$ is a projection. It is a kind of Riesz projection corresponding to $\zeta$.
We set $\xi:=\RE \zeta, \ \eta:=\IM \zeta$ if $\zeta \in V_{\bb R}(I)$ and $(\xi,\eta):=\zeta$ if $\zeta \in V(I)\setminus \bb R^2$. For $\lambda \in \bb C\setminus \{\xi+i\eta\}$ and for $s(z,w):=z+iw-\lambda$ we then have $s_N \cdot (e\delta_\zeta) = \big(s_N(\zeta)\big)\delta_\zeta$. As $s(\xi,\eta)\neq 0$, $s_N(\zeta)$ does not belong to $P(\zeta) \supseteq Q(\zeta)$. Therefore, it is invertible in $\mc B(\zeta)$ or in $\mc A(\zeta)$. For its inverse $b$ we obtain \[
s_N \cdot (e\delta_\zeta) \cdot (b\delta_\zeta) = e\delta_\zeta \,. \] From $s_N(N) = N - \lambda$ we derive that $(N\vert_{\ran (e\delta_\zeta)(N)} - \lambda)^{-1} =(b\delta_\zeta)(N)\vert_{\ran (e\delta_\zeta)(N)}$ on $\ran (e\delta_\zeta)(N)$. In particular, $\sigma(N\vert_{\ran (e\delta_\zeta)(N)}) \subseteq \{\xi+i\eta\}$. \end{remark}
\begin{lemma}\thlab{deth5633}
If $\phi\in \mc F_{N}$ vanishes on
\[
\big(\sigma(\Theta(N)) \cup (V_{\bb R}(I) \cap \sigma(N))\big)
\dot\cup \{(\alpha,\beta)\in V(I)\setminus \bb R^2: \alpha+i\beta, \bar \alpha+i\bar \beta \in \sigma(N)\} \,,
\]
then $\phi(N)=0$. \end{lemma} \begin{proof}
Since any $w \in V_{\bb R}(I) \setminus \sigma(N)$ is isolated
in $\sigma(\Theta(N)) \cup V_{\bb R}(I)$, we saw in \thref{rieszproj33} that
for
\[
\zeta \in \underbrace{\big(V_{\bb R}(I) \setminus \sigma(N)\big)}_{=:Z_1} \dot\cup
\underbrace{\{(\alpha,\beta)\in V(I)\setminus \bb R^2: \alpha+i\beta \in \rho(N)\}}_{=:Z_2}
\]
the expression
$(e\delta_\zeta)(N)$ is a bounded projection commuting with $N$. Hence,
$(e\delta_\zeta)(N)$ also commutes with $(N - (\xi+i\eta))^{-1}$, where
$\xi:=\RE \zeta, \ \eta:=\IM \zeta$ if $\zeta \in Z_1$ and
$(\xi,\eta):=\zeta$ if $\zeta \in Z_2$.
Consequently, $N\vert_{\ran (e\delta_\zeta)(N)}- (\xi+i\eta)$ is invertible on $\ran (e\delta_\zeta)(N)$,
i.e.\ $\xi+i\eta \not\in \sigma(N\vert_{\ran (e\delta_\zeta)(N)})$. In
\thref{rieszproj33} we saw $\sigma(N\vert_{\ran (e\delta_\zeta)(N)}) \subseteq \{\xi+i\eta\}$.
Hence, $\sigma(N\vert_{\ran (e\delta_\zeta)(N)}) = \emptyset$, which is impossible for
$\ran (e\delta_\zeta)(N) \neq \{0\}$. Thus, $(e\delta_\zeta)(N) = 0$.
For $(\xi,\eta)\in Z_3:=\{(\alpha,\beta)\in V(I)\setminus \bb R^2: \bar\alpha+i\bar\beta \in \rho(N)\}$
one has $(\bar \xi,\bar \eta)\in Z_2$. Hence,
\[
0=(e\delta_{(\bar\xi,\bar\eta)})(N)^*=(e^\#\delta_{(\xi,\eta)})(N)=(e\delta_{(\xi,\eta)})(N) \,.
\]
Since, by our assumption, $\phi$ is supported on $Z_1\cup Z_2 \cup Z_3$, we obtain
\[
\phi(N) = (\hspace{-2mm}\sum_{\zeta \in Z_1\cup Z_2 \cup Z_3} \hspace{-2mm}
\phi(\zeta)\delta_\zeta \hspace{+2mm})(N) =
\sum_{\zeta \in Z_1\cup Z_2 \cup Z_3} \phi(\zeta) (e\delta_\zeta)(N) = 0 \,.
\]
\end{proof}
As a consequence of \thref{deth5633} for $\phi \in \mc F_{N}$ the operator $\phi(N)$ only depends on $\phi$'s values on
\begin{multline}\label{eigmeng} \big(\sigma(\Theta(N)) \cup (V_{\bb R}(I) \cap \sigma(N))\big) \dot\cup \\ \{(\alpha,\beta)\in V(I)\setminus \bb R^2:
\alpha+i\beta, \bar \alpha+i\bar \beta \in \sigma(N)\} \,.
\end{multline} Thus, we can, and will from now on, re-define the function class $\mc F_{N}$ for our functional calculus so that the elements $\phi$ of $\mc F_{N}$ are functions on this set with values in $\bb C, \mc A(z)$ or $\mc B(z)$, such that $z\mapsto \phi(z)$ is measurable and bounded on $\sigma(\Theta(N)) \setminus V_{\bb R}(I)$ and such that \eqref{fn8qw3bab} holds true for every $w \in \sigma(\Theta(N)) \cap V_{\bb R}(I)$ which is not isolated in $\sigma(\Theta(N))$.
\begin{lemma}\thlab{wannboundinv33}
If $\phi \in \mc F_{N}$
is such that $\phi(z)$ is invertible in $\bb C, \mc A(z)$ or $\mc B(z)$, respectively, for all
$z$ in \eqref{eigmeng}, and such that
$0$ does not belong to the closure of
$\phi\big(\sigma(\Theta(N))\setminus V_{\bb R}(I) \big)$, then
$\phi(N)$ is a boundedly invertible operator on $\mc K$ with $\phi^{-1}(N)$
as its inverse. \end{lemma} \begin{proof}
We think of $\phi$ as a function on $\big(\sigma(\Theta(N)) \cup V_{\bb R}(I) \big)\dot\cup (V(I)\setminus \bb R^2)$
by setting $\phi(z)=e$ for all $z$ not belonging to \eqref{eigmeng}.
Then all assumptions of \thref{einduF33} are satisfied. Hence $\phi^{-1} \in \mc F_{N}$, and we conclude from
\thref{tu2989} and \thref{bweuh30po} that
\[
\phi^{-1}(N) \phi(N) = \phi(N) \phi^{-1}(N) = (\phi\cdot \phi^{-1})(N) = \mathds{1}_N(N) = I_{\mc K} \,.
\] \end{proof}
\begin{corollary}\thlab{sigmaN}
$\sigma(N)$ equals to
\begin{multline}\label{specform}
\sigma(\Theta(N)) \cup (V_{\bb R}(I) \cap \sigma(N))
\cup \\ \{\alpha + i\beta : (\alpha,\beta)\in V(I)\setminus \bb R^2,
\alpha+i\beta, \bar \alpha+i\bar \beta \in \sigma(N)\} \,.
\end{multline} In particular, $\sigma(N)\setminus \sigma(\Theta(N))$ is finite. \end{corollary} \begin{proof}
Since $\Theta$ is a homomorphism, we have $\sigma(\Theta(N))\subseteq \sigma(N)$.
Hence, \eqref{specform} is contained in $\sigma(N)$.
For the converse, consider the polynomial $s(z,w) = z+iw - \lambda$ for a $\lambda$
not belonging to \eqref{specform}. We conclude that for any
\[
\zeta\in (V_{\bb R}(I) \cap \sigma(N)) \cup \{(\alpha,\beta)\in V(I)\setminus \bb R^2:
\alpha+i\beta, \bar \alpha+i\bar \beta \in \sigma(N)\}
\]
the polynomial
$s$ does not belong to $P(\zeta) \supseteq Q(\zeta)$. Hence, $s_N(\zeta)$ is invertible
$\mc A(\zeta)$ or $\mc B(\zeta)$.
Clearly, $s_N(\zeta)\neq 0$ for $\zeta\in \sigma(\Theta(N)) \setminus V_{\bb R}(I)$.
Finally, $0$ does not belong to the closure of
\[
s_N\big(\sigma(\Theta(N))\setminus V_{\bb R}(I) \big) = s(\sigma(\Theta(N))\setminus V_{\bb R}(I))
\subseteq \sigma(\Theta(N)) - \lambda \,.
\]
Applying \thref{wannboundinv33}, we see that $s_N(N)=(N-\lambda)$ is invertible. \end{proof}
\begin{remark}\thlab{fhwr3435}
We set $K_r:=V_{\bb R}(I) \cap \sigma(N)$,
\[
Z:=\{(\alpha,\beta)\in V(I)\setminus \bb R^2: \alpha+i\beta, \bar \alpha+i\bar \beta \in \sigma(N)\} \,,
\]
and $K_i := \{\alpha + i\beta : (\alpha,\beta)\in Z\}$.
Using \thref{sigmaN} we could re-define once more the functions $\phi\in\mc F_N$
as functions $\phi$ on $\sigma(N)$ such that
\begin{enumerate}
\item $\phi$ is complex valued, bounded and measurable on $\sigma(N)\setminus (K_r \cup K_i)$,
\item $\phi(\zeta)\in \mc A(\zeta)$ for $\zeta \in K_r \setminus K_i$,
\item $\phi(\zeta)\in \varprod_{(\alpha,\beta) \in Z, \alpha + i\beta = \zeta} \mc A(\zeta)$ for $\zeta \in K_i \setminus K_r$,
\item $\phi(\zeta)\in \mc A(\zeta) \times \varprod_{(\alpha,\beta) \in Z, \alpha + i\beta = \zeta} \mc A(\zeta)$ for $\zeta \in K_r \cap K_i$;
\item for a $w\in K_r$, which is not isolated in $\sigma(N)$, we have
\[
\phi(z) - p(\RE z,\IM z) = O(\chi_w(z)) \ \ \text{ as } \ \
\sigma(N) \setminus (K_r \cup K_i) \ni z \to w \,,
\]
where $p$ is a representative of $\phi(w)$ for $w\in K_r \setminus K_i$ and
$p$ is a representative of the first entry of $\phi(w)$ for $w\in K_r \cap K_i$.
\end{enumerate} \end{remark}
\section{Special cases of definitizable operators}
Unitary and selfadjoint operators are special cases of normal operators on Hilbert spaces as well as on Krein spaces. We will show how some well-known facts on definitizable selfadjoint or unitary operators on a Krein space $\mc K$ can easily be obtain from the previously obtained results.
\subsection{Selfadjoint definitizable operators}
An operator $N\in B(\mc K)$ is by definition selfadjoint if $N=N^{+}$. Obviously, $N\in B(\mc K)$ is selfadjoint if and only if $N$ is normal and satisfies $p(A,B) = 0$, where $A=\frac{N+N^{+}}{2}, B=\frac{N-N^{+}}{2i}$ and \[
p(x,y) = y \in \bb R[x,y] \,. \] Therefore, according to \thref{definidef} any selfadjoint operator on a Krein space is definitizable normal, and the ideal $I$ generated by all real definitizing polynomials contains $p(x,y)=y$. Since the ideal generated by $p(x,y)=y$ is not zero-dimensional, the zero-dimensionality of $I$ implies the existence of at least one real definitizing polynomial of the form \begin{equation}\label{ndydivba}
y \cdot s(x,y) + t(x) \ \ \text{ with } \ \ s(x,y)\in \bb C[z,w], \ t(x)\in \bb C[x]\setminus \{0\} \,. \end{equation}
\begin{proposition}
The ideal $I$ is zero-dimensional if and only if
there exists a $t \in \bb R[x]\setminus \{0\}$ such that
$[t(A)u,u] \geq 0, \ u \in \mc K$, i.e.\ $N=A$ is definitizable in
the classical sense; see \cite{KaPr2014}. \end{proposition} \begin{proof}
Any $r(x,y) \in \bb C[z,w]$ can we written as
$r(x,y) = y\cdot s_r(x,y) + t_r(x)$ with unique
$s_r(x,y)\in \bb C[z,w], \, t_r(x)\in \bb C[x]$.
Hence, $r\in I$ if and only if $t_r(x) \in I$.
The set of $I_x:=\{t_r : r\in I\}$ forms an ideal in $\bb C[x]$.
If $I_x$ is the zero ideal, then $I=y\cdot\bb C[x,y]$ is not zero-dimensional.
If $I_x\neq \{0\}$, then, applying the polynomial division, we see that
$\dim \bb C[x]/I_x <\infty$. This implies the zero-dimensionality of $I$.
If $r(x,y)$ is a real definitizing polynomial as in \eqref{ndydivba}, then
\[
[t(A)u,u] = [r(A,B)u,u] \geq 0, \ u \in \mc K \,,
\]
i.e.\ $t(x)$ is a definitizing polynomial. \end{proof}
Assume that $N\in B(\mc K)$ is selfadjoint and that the ideal $I$ generated by all real definitizing polynomials is zero-dimensional. Consequently, we can apply the functional calculus developed in Section \ref{Funcalzero}. From $p(x,y)=y\in I$ we conclude \[
a=(a_x,a_y) \in V(I) \Rightarrow a_y=p(a) = 0 \,. \] Hence, the elements of $V_{\bb R}(I)$ are contained in $\bb R$, and $(\xi,\eta) \in V(I) \setminus \bb R^2$ yields $\eta=0$. Moreover, with $N$ also $\Theta(N)$ is selfadjoint in the Hilbert space $\mc H$; see \thref{defNspaces} and \eqref{thetaVdef}. In particular, $\sigma(\Theta(N)) \subseteq \bb R$. From \thref{sigmaN} we derive that $\sigma(N)$ is contained in $\bb R$ up to finitely many points which are located in $\bb C \setminus \bb R$ symmetric with respect to $\bb R$.
\subsection{Unitary definitizable operators}
An operator $N\in B(\mc K)$ is by definition unitary if $N^{+} N = N N^{+} = I_{\mc K}$. Obviously, $N\in B(\mc K)$ is unitary if and only if $N$ is normal and satisfies $p(A,B) = 0$, where $A=\frac{N+N^{+}}{2}, B=\frac{N-N^{+}}{2i}$ and \[
p(x,y) = (x+iy)(x-iy) - 1 = x^2 + y^2 - 1 \in \bb R[x,y] \,. \] Therefore, according to \thref{definidef} any unitary operator on a Krein space is definitizable normal, and the ideal $I$ generated by all real definitizing polynomials always contains $p(x,y)$. Since the ideal generated by $p$ is not zero-dimensional, the zero-dimensionality of $I$ implies the existence a definitizing polynomial different from $p$.
\begin{remark}
If, for example, there exists a polynomial $a\in \bb C[z]\setminus \{0\}$
such that $[a(N)u,u] \geq 0, \ u \in \mc K$, then the ideal $J$
generated by $a$ (considered as a polynomial in $\bb C[z,w]$) and
$b(z,w)=zw-1$ in $\bb C[z,w]$ is zero-dimensional. Indeed, it is easy to see that
the set $V(J)$ of common zeros of $a$ and $b$ is finite, which by \cite{CLO2}, page 39,
implies zero-dimensionality.
Since $c(z,w)\mapsto c(x+iy,x-iy)$ constitutes an isomorphism from
$\bb C[z,w]$ onto $\bb C[x,y]$, also the ideal generated by
$a(x+iy)$ and $p(x,y)$ in $\bb C[x,y]$ is zero-dimensional. Hence,
the same is true for $I$, and we can apply the functional
calculus developed Section \ref{Funcalzero}. \end{remark}
Assume that $N\in B(\mc K)$ is unitary and that the ideal $I$ generated by all real definitizing polynomials is zero-dimensional. Consequently, we can apply the functional calculus developed in Section \ref{Funcalzero}. From $p\in I$ we conclude \[
a \in V(I) \Rightarrow p(a) = 0 \,. \] Hence, the elements of $V_{\bb R}(I)$ are contained in $\bb T$, and $(\xi,\eta) \in V(I) \setminus \bb R^2$ yields \[
(\xi+i\eta)\overline{(\bar\xi+i\bar\eta)} = \xi^2+\eta^2=1. \] Moreover, with $N$ also $\Theta(N)$ is unitary in the Hilbert space $\mc H$; see \thref{defNspaces} and \eqref{thetaVdef}. In particular, $\sigma(\Theta(N)) \subseteq \bb T$. From \thref{sigmaN} we derive that $\sigma(N)$ is contained in $\bb T$ up to finitely many points which are located in $\bb C \setminus \bb T$ symmetric with respect to $\bb T$.
\section{Transformations of definitizable normal operators}
In this final section we examine, whether basic transformations, such as $\alpha N, N + \beta I_{\mc K}, N^{-1}$ with $\alpha, \beta \in\bb C, \ \alpha\neq 0$, of definitizable normal operators $N$ are again definitizable, and how the corresponding ideals $I$ behave.
For $\beta\in\bb C$ it is easy to see that $p(x,y)$ is a real definitizing polynomial for $N$ if and only if $p(x-\RE \beta,y-\IM \beta)$ is real definitizing for $N + \beta I_{\mc K}$. Since $r(x,y) \mapsto r(x-\RE \beta,y-\IM \beta)$ is a ring automorphism on $\bb C[x,y]$, the respective ideals $I$, corresponding to $N$ and $N + \beta I_{\mc K}$, are zero-dimensional, or not, at the same time.
Similarly, $p(x,y)$ is a real definitizing polynomial for $N$ if and only if $p(x \RE \frac{1}{\alpha} - y \IM \frac{1}{\alpha}, x\IM \frac{1}{\alpha} + y \RE \frac{1}{\alpha})$ is real definitizing for $\alpha N$. Also $r(x,y) \mapsto r(x \RE \frac{1}{\alpha} - y \IM \frac{1}{\alpha}, x\IM \frac{1}{\alpha} + y \RE \frac{1}{\alpha})$ is a ring automorphism on $\bb C[x,y]$. Hence, the ideal $I$ corresponding to $N$ is zero-dimensional if and only if the ideal $I$ corresponding to $\alpha N$ is zero-dimensional.
For the inverse $N^{-1}$ the situation is more complicated. We formulate two results that we will need. The first assertion is straight forward to verify. We omit its proof.
\begin{lemma}\thlab{trafos2}
The mapping $\Phi: p(x,y) \mapsto p(\frac{z+w}{2},\frac{z-w}{2i})$ from $\bb C[x,y]$ to
$\bb C[z,w]$ is an isomorphism, where $p$ is real, i.e.\ $p(\bar x, \bar y) = \overline{p(x,y)}$,
if and only if $\overline{\Phi(p)(z,w)} = \Phi(p)(\bar w,\bar z)$. \end{lemma}
Obviously, for a normal $N=A+iB$ and $p(x,y)\in \bb C[x,y]$ we have \begin{equation}\label{Phiwitz}
p(A,B) = \Phi(p)(N,N^+) \,. \end{equation}
For a polynomial $q\in\bb C[z,w]\setminus\{0\}$ let $d(q)$ be the maximum of
the $z$-degree of $q$ and the $w$-degree of $q$. Moreover, we set \[
\varpi(q)(z,w):= (zw)^{d(q)} q(\frac{1}{z},\frac{1}{w}) \in \bb C[z,w] \,. \]
\begin{lemma}\thlab{trafos3}
If $I=\langle q_1,\dots,q_m \rangle$ is zero-dimensional with
polynomials $q_1,\dots,q_m$ such that $\overline{q_j(z,w)} = q_j(\bar w,\bar z)$, then
the ideal $\langle \varpi(q_1),\dots,\varpi(r_m) \rangle$ is also zero-dimensional. \end{lemma} \begin{proof}
Let $(\zeta,\eta) \in V(\varpi(q_1),\dots,\varpi(r_m))$. For $\zeta\neq 0 \neq \eta$ we conclude
$q_j(\frac{1}{\zeta},\frac{1}{\eta}) = 0, \ j=1,\dots, m$, and in turn
$(\zeta,\eta) \in \{ (z,w) \in (\bb C\setminus\{0\})^2: (\frac{1}{z},\frac{1}{w}) \in V(I) \}$.
Assume that $\eta=0$ and $\zeta\neq 0$. If $q_j(z,w) = \sum_{k,l=0}^{d(q_j)} b_{k,l} z^k w^l$, then
$\overline{q_j(z,w)} = q_j(\bar w,\bar z)$ yields $b_{k,l} = \bar b_{l,k}$,
and we have $\varpi(q_j)(z,w) = \sum_{k,l=0}^{d(q_j)} b_{d(q_j)-k,d(q_j)-l} z^k w^l$. According to the choice
of $d(q_j)$ and by $b_{k,l} = \bar b_{l,k}$ the polynomial
\[
\rho_j(z):=\varpi(q_j)(z,0) = \sum_{k=0}^{d(q_j)} b_{d(q_j)-k,d(q_j)} z^k
\]
is non-zero and satisfies $\rho_j(\zeta)=0$, i.e.\ $(\zeta,\eta) \in \rho_j^{-1}(\{0\})\times\{0\}$.
From $\overline{q_j(z,w)} = q_j(\bar w,\bar z)$ we conclude $\rho_j(\bar w) = \overline{\varpi(q_j)(0,w)}$.
Hence, $\zeta=0$ and $\eta\neq 0$ yields
$(\zeta,\eta) \in \{0\}\times \overline{\rho_j^{-1}(\{0\})}$.
In any case $(\zeta,\eta)$ is contained in
\begin{multline*}
\{(0,0)\} \cup \{ (z,w) \in (\bb C\setminus\{0\})^2: (\frac{1}{z},\frac{1}{w}) \in V(I) \} \cup \\
\cup \bigcap_{j=1,\dots,m} \rho_j^{-1}(\{0\})\times\{0\}
\cup \bigcap_{j=1,\dots,m} \{0\}\times \overline{\rho_j^{-1}(\{0\})} \,.
\end{multline*}
Consequently, $V(\varpi(q_1),\dots,\varpi(r_m))$ is finite, and in turn $\langle\varpi(q_1),\dots,\varpi(r_m) \rangle$ is zero-dimensional; see
\cite{CLO2}, page 39. \end{proof}
\begin{proposition}\thlab{invdefbar}
Let $N$ be normal and bijective on the Krein space $\mc K$.
If $p(x,y)$ is real definitizing for $N$, then
$\Phi^{-1}\Big(\varpi\big(\Phi(p)\big)\Big)$ is definitizing for $N^{-1}$.
Moreover, if the ideal $I$ generated by all real definitizing $p(x,y)$ for $N$ is zero-dimensional, then
also the ideal generated by all real definitizing polynomials for $N^{-1}$ is zero-dimensional. \end{proposition} \begin{proof}
Let $p(x,y)$ be real definitizing for $N$.
By \thref{trafos2} we have $\overline{\Phi(p)(z,w)} = \Phi(p)(\bar w,\bar z)$, and in turn
$\overline{\varpi(\Phi(p))(z,w)} = \varpi(\Phi(p))(\bar w,\bar z)$.
We write $\Phi(p)(z,w) = \sum_{k,l=0}^{d(\Phi(p))} b_{k,l} z^k w^l$, and consequently
$\varpi(\Phi(p))(z,w) = \sum_{k,l=0}^{d(\Phi(p))} b_{d(\Phi(p))-k,d(\Phi(p))-l} z^k w^l$.
By \eqref{Phiwitz} for $u\in\mc K$ we have
\begin{align*}
[\Phi^{-1}\Big(\varpi\big(\Phi(p)\big) & (\RE N^{-1}, \IM N^{-1}) u,u] =
[\varpi(\Phi(p))(N^{-1},N^{-+}) u,u] \\ & =
[\sum_{k,l=0}^{d(\Phi(p))} b_{d(\Phi(p))-k,d(\Phi(p))-l} (N^{-1})^k (N^{-+})^l u,u] \\ & =
[\Phi(p)(N,N^+) \, (N^{-1})^{d(\Phi(p))} u,(N^{-1})^{d(\Phi(p))} \, u] \\ & = [p(A,B) \, (N^{-1})^{d(\Phi(p))} u,(N^{-1})^{d(\Phi(p))} \, u] \geq 0 \,.
\end{align*}
Hence, $\Phi^{-1}\Big(\varpi\big(\Phi(p)\big)\Big)$ is real definitizing for $N^{-1}$.
Finally, if $I$ is zero-dimensional and generated by real definitizing $p_1,\dots,p_m$, then
$\Phi(I)=\langle \Phi(p_1),\dots,\Phi(p_m) \rangle$ is zero-dimensional in $\bb C[z,w]$. According to \thref{trafos3}
$\langle \varpi\big(\Phi(p_1)\big),\dots,\varpi\big(\Phi(p_m)\big) \rangle$, and hence
also $\langle \Phi^{-1}\Big(\varpi\big(\Phi(p_1)\big)\Big),\dots,\Phi^{-1}\Big(\varpi\big(\Phi(p_m)\big)\Big) \rangle$ is zero-dimensional.
Since its generators are real definitizing for $N^{-1}$ also
the ideal generated by all real definitizing polynomials for $N^{-1}$ is zero-dimensional. \end{proof}
\end{document} | arXiv |
\begin{definition}[Definition:Irrational Number Space]
Let $\mathbb I := \R \setminus \Q$ be the set of irrational numbers.
Let $d: \mathbb I \times \mathbb I \to \R$ be the Euclidean metric on $\mathbb I$.
Let $\tau_d$ be the topology on $\mathbb I$ induced by $d$.
Then $\struct {\mathbb I, \tau_d}$ is the '''irrational number space'''.
\end{definition} | ProofWiki |
Stephen Siklos
Stephen Siklos (1950 – 17 August 2019) was a lecturer in the Faculty of Mathematics at the University of Cambridge. He is known for setting up the Sixth Term Examination Papers, used for undergraduate mathematics admissions at several British universities.[1]
Early life
Siklos was born in Epsom, Surrey, England in 1950.[2][1] His father, Theo Siklos, was an educator and his wife, Ruth Siklos, an almoner.[1] He was educated at Collyer's School before reading the Mathematical Tripos at Pembroke College, University of Cambridge, where he graduated with a masters in mathematics and was awarded the Tyson Medal.[2][1]
Academic career
In 1973, he began doing research in general relativity under Stephen Hawking, publishing his dissertation titled "Singularities, Invariants and Cosmology" in 1976.[3] From 1980 to 1999 he lectured at Cambridge and was the director of studies at Newnham College. In 1999, he joined Jesus College as a senior tutor, later becoming the college president.[1]
References
1. Clackson, James (23 September 2019). "Stephen Siklos obituary". the Guardian.
2. "Stephen Siklos, 1950-2019 | Features: Faculty Insights". www.maths.cam.ac.uk.
3. "Stephen Siklos - The Mathematics Genealogy Project". www.genealogy.math.ndsu.nodak.edu.
Authority control
International
• VIAF
Academics
• MathSciNet
• Mathematics Genealogy Project
• Scopus
• zbMATH
| Wikipedia |
Gödel numbering for sequences
In mathematics, a Gödel numbering for sequences provides an effective way to represent each finite sequence of natural numbers as a single natural number. While a set theoretical embedding is surely possible, the emphasis is on the effectiveness of the functions manipulating such representations of sequences: the operations on sequences (accessing individual members, concatenation) can be "implemented" using total recursive functions, and in fact by primitive recursive functions.
It is usually used to build sequential “data types” in arithmetic-based formalizations of some fundamental notions of mathematics. It is a specific case of the more general idea of Gödel numbering. For example, recursive function theory can be regarded as a formalization of the notion of an algorithm, and can be regarded as a programming language to mimic lists by encoding a sequence of natural numbers in a single natural number.[1][2]
Gödel numbering
Main article: Gödel numbering
Besides using Gödel numbering to encode unique sequences of symbols into unique natural numbers (i.e. place numbers into mutually exclusive or one-to-one correspondence with the sequences), we can use it to encode whole “architectures” of sophisticated “machines”. For example, we can encode Markov algorithms,[3] or Turing machines[4] into natural numbers and thereby prove that the expressive power of recursive function theory is no less than that of the former machine-like formalizations of algorithms.
Accessing members
Any such representation of sequences should contain all the information as in the original sequence—most importantly, each individual member must be retrievable. However, the length does not have to match directly; even if we want to handle sequences of different length, we can store length data as a surplus member,[5] or as the other member of an ordered pair by using a pairing function.
We expect that there is an effective way for this information retrieval process in form of an appropriate total recursive function. We want to find a totally recursive function f with the property that for all n and for any n-length sequence of natural numbers $\langle a_{0},\dots a_{n-1}\rangle $, there exists an appropriate natural number a, called the Gödel number of the sequence, such that for all i where $0\leq i\leq n-1$, $f(a,i)=a_{i}$.
There are effective functions which can retrieve each member of the original sequence from a Gödel number of the sequence. Moreover, we can define some of them in a constructive way, so we can go well beyond mere proofs of existence.
Gödel's β-function lemma
See also: Gödel's β function
By an ingenious use of the Chinese remainder theorem, we can constructively define such a recursive function $\beta $ (using simple number-theoretical functions, all of which can be defined in a total recursive way) fulfilling the specifications given above. Gödel defined the $\beta $ function using the Chinese remainder theorem in his article written in 1931. This is a primitive recursive function.[6]
Thus, for all n and for any n-length sequence of natural numbers $\langle a_{0},\dots a_{n-1}\rangle $, there exists an appropriate natural number a, called the Gödel number of the sequence such that $\beta (a,i)=a_{i}$.[7]
Using a pairing function
Main article: Pairing function
Our specific solution will depend on a pairing function—there are several ways to implement the pairing function, so one method must be selected. Now, we can abstract from the details of the implementation of the pairing function. We need only to know its “interface”: let $\pi $, K, and L denote the pairing function and its two projection functions, respectively, satisfying specification
$K\left(\pi \left(x,y\right)\right)=x$
$L\left(\pi \left(x,y\right)\right)=y$
We shall not discuss and formalize the axiom for excluding alien objects here, as it is not required to understand the method.
Remainder for natural numbers
We shall use another auxiliary function that will compute the remainder for natural numbers. Examples:
• $\mathrm {rem} (5,3)=2$
• $\mathrm {rem} (7,2)=1$
It can be proven that this function can be implemented as a recursive function.
Implementation of the β function
Using the Chinese remainder theorem, we can prove that implementing $\beta $ as
$\beta (s,i)=\mathrm {rem} \left(K\left(s\right),\left(i+1\right)\cdot L\left(s\right)+1\right)$
will work, according to the specification we expect $\beta $ to satisfy. We can use a more concise form by an abuse of notation (constituting a sort of pattern matching):
$\beta \left(\pi \left(x_{0},m\right),i\right)=\mathrm {rem} \left(x_{0},\left(i+1\right)\cdot m+1\right)$
Let us achieve even more readability by more modularity and reuse (as these notions are used in computer science[8]): by defining $\forall i<n$ the sequence $m_{i}=(i+1)\cdot m+1$, we can write
$\beta \left(\pi \left(x_{0},m\right),i\right)=\mathrm {rem} \left(x_{0},m_{i}\right)$.
We shall use this $m_{i}$ notation in the proof.
Hand-tuned assumptions
For proving the correctness of the above definition of the $\beta $ function, we shall use several lemmas. These have their own assumptions. Now we try to find out these assumptions, calibrating and tuning their strength carefully: they should not be said in an either superfluously sharp, or unsatisfactorily weak form.
Let $a_{0},\dots a_{n-1}$ be a sequence of natural numbers. Let m be chosen to satisfy
$\forall i\in {\overline {n}}\setminus \left\{0\right\}\left(i\mid m\right)$
$\forall i<n\left(a_{i}<m_{i}\right)$
The first assumption is meant as
$1\mid m\land \dots \land n-1\mid m$
It is needed to meet an assumption of the Chinese remainder theorem (that of being pairwise coprime). In the literature, sometimes this requirement is replaced with a stronger one, e.g. constructively built with the factorial function,[1] but the stronger premise is not required for this proof.[2]
The second assumption does not concern the Chinese remainder theorem in any way. It will have importance in proving that the specification for $\beta $ is met eventually. It ensures that an ${\tilde {x}}$ solution of the simultaneous congruence system
$x\equiv a_{i}{\pmod {m_{i}}}$ for each i where $0\leq i\leq n-1$
also satisfies
$a_{i}=\mathrm {rem} ({\tilde {x}},m_{i})$.[5][9]
A stronger assumption for m requiring $\forall i<n\;(a_{i}<m)$ automatically satisfies the second assumption (if we define the notation $m_{i}$ as above).
Proof that (coprimality) assumption for Chinese remainder theorem is met
In the section Hand-tuned assumptions, we required that
$\forall i\in {\overline {n}}\setminus \left\{0\right\}\left(i\mid m\right)$. What we want to prove is that we can produce a sequence of pairwise coprime numbers in a way that will turn out to correspond to the Implementation of the β function.
In detail:
$\forall i<n,j<n\;\left(i\neq j\rightarrow \mathrm {coprime} \left(m_{i},m_{j}\right)\right)$
remembering that $\forall i<n$ we defined $m_{i}=(i+1)\cdot m+1$.
The proof is by contradiction; assume the negation of the original statement:
$\exists i<n,j<n\;\left(i\neq j\land \lnot \mathrm {coprime} \left(m_{i},m_{j}\right)\right)$
First steps
We know what “coprime” relation means (in a lucky way, its negation can be formulated in a concise form); thus, let us substitute in the appropriate way:
$\exists i<n,j<n\;\left(i\neq j\land \exists p\in \mathrm {Prime} \;\left(p\mid m_{i}\land p\mid m_{j}\right)\right)$
Using a “more” prenex normal form (but note allowing a constraint-like notation in quantifiers):
$\exists i<n,j<n,p\in \mathrm {Prime} \;\left(i\neq j\land p\mid m_{i}\land p\mid m_{j}\right)$
Because of a theorem on divisibility, $p\mid m_{i}\land p\mid m_{j}$ allows us to also say
$p\mid m_{i}-m_{j}$.
Substituting the definitions of $m_{k}$-sequence notation, we get $m_{i}-m_{j}=(i-j)\cdot m$, thus (as equality axioms postulate identity to be a congruence relation[10]) we get
$p\mid (i-j)\cdot m$.
Since p is a prime element (note that the irreducible element property is used), we get
$p\mid i-j\lor p\mid m$.
Resorting to the first hand-tuned assumption
Now we must resort to our assumption
$\forall i\in {\overline {n}}\setminus \left\{0\right\}\left(i\mid m\right)$.
The assumption was chosen carefully to be as weak as possible, but strong enough to enable us to use it now.
The assumed negation of the original statement contains an appropriate existential statement using indices $i<n\land j<n\land i\neq j$; this entails $i-j\in {\overline {n}}\setminus \left\{0\right\}$, thus the mentioned assumption can be applied, so $i-j\mid m$ holds.
Using an (object) theorem of the propositional calculus as a lemma
We can prove by several means [11] known in propositional calculus that
$\left(A\land \left(A\rightarrow B\right)\right)\rightarrow B$
holds.
Since $i-j\mid m$, by the transitivity property of the divisibility relation, $p\mid i-j\rightarrow p\mid m$. Thus (as equality axioms postulate identity to be a congruence relation [10])
$p\mid m$
can be proven.
Reaching the contradiction
The negation of original statement contained
$p\mid m_{i}$
and we have just proved
$p\mid m$.
Thus,
$p\mid m_{i}-\left(i+1\right)\cdot m$
should also hold. But after substituting the definition of $m_{i}$,
$m_{i}-\left(i+1\right)\cdot m=1$
Thus, summarizing the above three statements, by transitivity of the equality,
$p\mid 1$
should also hold.
However, in the negation of the original statement p is existentially quantified and restricted to primes $\exists p\in \mathrm {Prime} $. This establishes the contradiction we wanted to reach.
End of reductio ad absurdum
By reaching contradiction with its negation, we have just proven the original statement:
$\forall i<n,j<n\;\left(i\neq j\rightarrow \mathrm {coprime} \left(m_{i},m_{j}\right)\right)$
The system of simultaneous congruences
We build a system of simultaneous congruences
$x\equiv a_{0}{\pmod {m_{0}}}$
$\vdots $
$x\equiv a_{n-1}{\pmod {m_{n-1}}}$
We can write it in a more concise way:
$\forall i<n\;\left(x\equiv a_{i}{\pmod {m_{i}}}\right)$
Many statements will be said below, all beginning with "$\forall i<n\;\left(\dots \right)$". To achieve a more ergonomic treatment, from now on all statements should be read as being in the scope of an $\forall i<n\;\left(\dots \right)$ quantification. Thus, $\forall i<n($ begins here.
Let us chose a solution $x_{0}$ for the system of simultaneous congruences. At least one solution must exist, because $m_{0},\dots m_{n-1}$ are pairwise comprime as proven in the previous sections, so we can refer to the solution ensured by the Chinese remainder theorem. Thus, from now on we can regard $x_{0}$ as satisfying
$x_{0}\equiv a_{i}{\pmod {m_{i}}}$,
which means (by definition of modular arithmetic) that
$\mathrm {rem} \left(x_{0},m_{i}\right)=\mathrm {rem} \left(a_{i},m_{i}\right)$
Resorting to the second hand-tuned assumption
Recall the second assumption, “$\forall i<n\;\left(a_{i}<m_{i}\right)$”, and remember that we are now in the scope of an implicit quantification for i, so we don't repeat its quantification for each statement.
The second assumption $a_{i}<m_{i}$ implies that
$\mathrm {rem} \left(a_{i},m_{i}\right)=a_{i}$.
Now by transitivity of equality we get
$\mathrm {rem} \left(x_{0},m_{i}\right)=a_{i}$.
QED
Our original goal was to prove that the definition
$\beta \left(\pi \left(x_{0},m\right),i\right)=\mathrm {rem} \left(x_{0},m_{i}\right)$
is good for achieving what we declared in the specification of $\beta $: we want $\beta \left(\pi \left(x_{0},m\right),i\right)=a_{i}$ to hold.
This can be seen now by transitivity of equality, looking at the above three equations.
(The large scope of i ends here.)
Existence and uniqueness
We have just proven the correctness of the definition of $\beta $: its specification requiring
$\forall a_{0},\dots ,a_{n-1}\;\exists s\;\forall i<n\;\beta (s,i)=a_{i}$
is met. Although proving this was most important for establishing an encoding scheme for sequences, we have to fill in some gaps yet. These are related notions similar to existence and uniqueness (although on uniqueness, “at most one” should be meant here, and the conjunction of both is delayed as a final result).
Uniqueness of encoding, achieved by minimalization
Our ultimate question is: what number should stand for the encoding of sequence $\left\langle a_{0},\dots ,a_{n-1}\right\rangle $? The specification declares only an existential quantification, not yet a functional connection. We want a constructive and algorithmic connection: a (total) recursive function that performs the encoding.
Totality, because minimalization is restricted to special functions
This gap can be filled in a straightforward way: we shall use minimalization, and the totality of the resulting function is ensured by everything we have proven till now (i.e. the correctness of the definition of $\beta $ by meeting its specification). In fact, the specification
$\forall a_{0},\dots ,a_{n-1}\;\exists s\;\forall i<n\;\beta (s,i)=a_{i}$
plays a role here of a more general notion (“special function”[12]). The importance of this notion is that it enables us to split off the (sub)class of (total) recursive functions from the (super)class of partial recursive functions. In brief, the specification says that a function f [13] satisfying the specification
$f\left(a_{0},\dots ,a_{n-1},s\right)=0\leftrightarrow \forall i<n\;\left(\beta (s,i)=a_{i}\right)$
is a special function; that is, for each fixed combination of all-but-last arguments, the function f has root in its last argument:
$\forall a_{0},\dots ,a_{n-1}\;\exists s\;\left(f\left(a_{0},\dots ,a_{n-1},s\right)=0\right)$
The Gödel numbering function g can be chosen to be total recursive
Thus, let us choose the minimal possible number that fits well in the specification of the $\beta $ function:[5]
$g:\mathbb {N} ^{n}\to \mathbb {N} $
$\left\langle a_{0},\dots ,a_{n-1}\right\rangle \longmapsto \mu a.\left[\forall i<n\;\left(\beta \left(a,i\right)=a_{i}\right)\right]$.
It can be proven (using the notions of the previous section ) that g is (total) recursive.
Access of length
If we use the above scheme for encoding sequences only in contexts where the length of the sequences is fixed, then no problem arises. In other words, we can use them in an analogous way as arrays are used in programming.
But sometimes we need dynamically stretching sequences, or we need to deal with sequences whose length cannot be typed in a static way. In other words, we may encode sequences in an analogous way to lists in programming.
To illustrate both cases: if we form the Gödel numbering of a Turing machine, then the each row in the matrix of the “program” can be represented with tuples, sequences of fixed length (thus, without storing the length), because the number of the columns is fixed.[14] But if we want to reason about configuration-like things (of Turing-machines), and specifically if we want to encode the significant part of the tape of a running Turing machine, then we have to represent sequences together with their length. We can mimic dynamically stretching sequences by representing sequence concatenation (or at least, augmenting a sequence with one more element) with a totally recursive function.[15]
Length can be stored simply as a surplus member:[5]
$g:\mathbb {N} ^{*}\to \mathbb {N} $
$\left\langle a_{0},\dots ,a_{n-1},a_{n}\right\rangle \longmapsto \mu a.\left[a_{0}=n\land \forall i<n\;\left(\beta \left(a,i+1\right)=a_{i}\right)\right]$.
The corresponding modification of the proof is straightforward, by adding a surplus
$x\equiv n{\pmod {m_{0}}}$
to the system of simultaneous congruences (provided that the surplus member index is chosen to be 0). Also, the assumptions have to be modified accordingly.
Notes
1. Monk 1976: 56–58
2. Csirmaz 1994: 99–100 (see online)
3. Monk 1976: 72–74
4. Monk 1976: 52–55
5. Csirmaz 1994: 100 (see online)
6. Smullyan 2003: 56 (= Chpt IV, § 5, note 1)
7. Monk 1976: 58 (= Thm 3.46)
8. Hughes 1989 (see online Archived 2006-12-08 at the Wayback Machine)
9. Burris 1998: Supplementary Text, Arithmetic I, Lemma 4
10. see also related notions, e.g. “equals for equals” (referential transparency), and another related notion Leibniz's law / identity of indiscernibles
11. either proof theoretic (algebraic steps); or semantic (truth table, method of analytic tableaux, Venn diagram, Veitch diagram / Karnaugh map)
12. Monk 1976: 45 (= Def 3.1.)
13. E.g. defined by
$f:\mathbb {N} ^{n+1}\to \mathbb {N} $
$f\left(a_{0},\dots ,a_{n-1},s\right)={\begin{cases}0&\mathrm {if} \;\forall i<n\;\left(\beta (s,i)=a_{i}\right)\\1&\mathrm {if} \;\exists i<n\;\left(\beta (s,i)\neq a_{i}\right)\end{cases}}$
14. Monk 1976: 53 (= Def 3.20, Lem 3.21)
15. Csirmaz 1994: 101 (=Thm 10.7, Conseq 10.8), see online
References
• Burris, Stanley N. (1998). "Supplementary Text, Arithmetic I". Logic for Mathematics and Computer Science. Prentice Hall. ISBN 978-0-13-285974-5.
• Csirmaz, László; Hajnal, András (1994). "Rekurzív függvények". Matematikai logika (postscript + gzip) (in Hungarian). Budapest: Eötvös Loránd University. {{cite book}}: |format= requires |url= (help) Each chapter is downloadable verbatim on author's page.
• Hughes, John (1989). "Why Functional Programming Matters". Computer Journal. 32 (2): 98–107. doi:10.1093/comjnl/32.2.98. Archived from the original on December 8, 2006.
• Monk, J. Donald (1976). Mathematical Logic. Graduate Texts in Mathematics. New York • Heidelberg • Berlin: Springer-Verlag. ISBN 9780387901701.
• Smullyan, Raymond Merrill (1992). Gödel's Incompleteness Theorems. Oxford University Press. ISBN 978-0-19-504672-4.
• Smullyan, Raymond Merrill (2003). Gödel nemteljességi tételei (in Hungarian). Budapest: Typotex. ISBN 978-963-9326-99-6. Translation of Smullyan 1992.
External links
• Burris, Stanley N. (1998). "Supplementary Text, Arithmetic I". Logic for Mathematics and Computer Science. Prentice Hall. ISBN 978-0-13-285974-5.
| Wikipedia |
Quantum double-double-slit experiment with momentum entangled photons
The Young-Feynman controlled double-slit electron interference experiment
Amir H. Tavabi, Chris B. Boothroyd, … Giulio Pozzi
Revisiting self-interference in Young's double-slit experiments
Sangbae Kim & Byoung S. Ham
Bell's inequality tests via correlated diffraction of high-dimensional position-entangled two-photon states
Wei Li & Shengmei Zhao
New quantum physics, solving puzzles of Wheeler's delayed choice and a particle's passing N slits simultaneously and quantum oscillator in experiments
Changyu Huang, Yong-Chang Huang & Yi-You Nie
Entanglement of orbital angular momentum in non-sequential double ionization
Andrew S. Maxwell, Lars Bojer Madsen & Maciej Lewenstein
Quantum wave–particle superposition in a delayed-choice experiment
Kai Wang, Qian Xu, … Xiao-song Ma
Experimental Limits of Ghost Diffraction: Popper's Thought Experiment
Paul-Antoine Moreau, Peter A. Morris, … Miles J. Padgett
Experimental Greenberger–Horne–Zeilinger entanglement beyond qubits
Manuel Erhard, Mehul Malik, … Anton Zeilinger
Indistinguishability of temporally separated pairwise two-photon state of thermal photons in Franson-type interferometry
Jiho Park, Heonoh Kim & Han Seb Moon
Manpreet Kaur1 &
Mandip Singh1
Double-double-slit thought experiment provides profound insight on interference of quantum entangled particles. This paper presents a detailed experimental realisation of quantum double-double-slit thought experiment with momentum entangled photons and theoretical analysis of the experiment. Experiment is configured in such a way that photons are path entangled and each photon can reveal the which-slit path information of the other photon. As a consequence, single photon interference is suppressed. However, two-photon interference pattern appears if locations of detection of photons are correlated without revealing the which-slit path information. It is also shown experimentally and theoretically that two-photon quantum interference disappears when the which-slit path of a photon in the double-double-slit is detected.
Wave nature of light was first experimentally demonstrated by the famous Young's double-slit experiment1,2. In quantum physics, light is quantised in the form of energy quanta known as photon. According to the statement of P.A.M. Dirac, "Each photon interferes only with itself"3. This self interference of a photon is a consequence of quantum superposition principle. If photons are incident on a double-slit one by one then the interference pattern of a photon gradually emerges. Where detection of each photon corresponds to a point on the screen. Young's double-slit experiment provides profound insight on the wave-particle duality if it is imagined for individual particles4. Interference pattern of a single particle is not formed if the path information of a particle i.e. a slit through which a particle has passed, is known. According to Copenhagen interpretation, an observation on the quantum superposition of paths of a particle corresponds to a measurement that collapses quantum superposition therefore, no interference pattern is formed. On the other hand, what happens if we modify the experiment in such a way that the which-path information of a particle is not available during its passage through a double-slit but can be obtained even after its detection. In this case, the which-path information can be carried out by the quantum state of another particle if total quantum state of particles is an entangled quantum state. By knowing its path by a measurement, the path information of the other particle is immediately determined. Because of path revealing quantum entanglement of particles the single particle interference is suppressed. However, quantum interference can be recovered even after completion of experiment by making correlated selection of measurement outcomes.
The first experiment to show the interference of light with very low intensity in the Young's double-slit experiment was performed in 1909 by G.I. Taylor5. Interesting experiments showing the Young's double-slit interference are performed with neutrons from the foundational perspective of quantum mechanics6, with electron beams7 and with a single electron passing through a double-slit8,9,10. Recently, a first experimental demonstration of interference of antiparticles with a double-slit is reported11. Interference of macromolecules is the subject of great interest in the quest to realise quantum superposition of mesoscopic and macroscopic objects12,13. In this context, number of interesting experiments have been performed to produce a path superposition of large molecules similar to the double-slit type interference experiments14,15,16.
The main concept of a quantum single double-slit experiment was extended to a quantum double-double-slit thought experiment by Greenberger, Horne and Zeilinger17 to provide foundational insight on the multiparticle quantum interference. In their paper. they have considered two double-slits and a source of particles placed in the middle of double-slits. Each particle is detected individually after it traverses a double-slit. Quantum entanglement of particles appears naturally in their considerations18,19 and it is shown, when single particle interference disappears and two-particle interference appears. An experimental realisation of quantum double-double-slit thought experiment showing a two-photon interference has been demonstrated with quantum correlated photons produced by spontaneous parametric down conversion (SPDC) process20. However, in this paper, we present a detailed experimental realisation of the quantum double-double-slit thought experiment with momentum entangled photon pairs, where a virtual double-double-slit configuration is realised with two Fresnel biprisms. This paper provides a detailed conceptual, theoretical and experimental analysis of the quantum double-double-slit experiment. In addition, an experiment of detection of a which-slit path of a photon is presented where it is shown that the two-photon interference disappears when a which-slit path of a photon is detected.
In this paper, experiments are presented in the context of a quantum double-double-slit thought experiment. However, experiments of foundational significance with polarization entangled photons21,22,23 and momentum entangled photons24 have been intensively studied. In addition, interesting experiments on delayed choice path erasure25,26,27,28,29,30 and two-photon interference31,32,33,34,35,36,37,38,39,40 are performed. Similar experiments have been proposed with Einstein–Podolsky–Rosen (EPR) entangled pair of atoms41.
Quantum double-double-slit experiment
Quantum double-double-slit experiment consists of two double-slits and a source of photon pairs. In this experimental situation, a single photon passes through each double-slit and detected individually on screens positioned behind the double-slits as shown in Fig. 1. However, interference of photons depends on the quantum state of two photons. To understand quantum interference of two photons in a double-double-slit experiment, consider a source is producing photons in pairs and both the photons have same linear polarisation. Double-slit 1 and double-slit 2 are aligned parallel to y-axis and positioned at distances \(l_{1}\) and \(l_{2}\), respectively along the x-axis from the source. Single slits \(a_{1}\) and \(b_{1}\) of double-slit 1 are separated by a distance \(d_{1}\) and single slits \(a_{2}\) and \(b_{2}\) of double-slit 2 are separated by a distance \(d_{2}\) as shown in Fig. 1 where each slit width is considered to be infinitesimally small. A single photon of a photon pair is detected on screen 1, which is positioned at a distance \(s_{1}\) from double-slit 1 and a second photon is detected on screen 2, which is positioned at a distance \(s_{2}\) from double-slit 2. There are four different possible paths by which photons can arrive at the respective screens i.e. a photon can arrive at a point \(o_{1}\) on screen 1 via double-slit 1 and the other photon can arrive at a point \(o_{2}\) on screen 2 via double-slit 2. Therefore, possible paths of photons are (i) a first photon can pass through slit \(a_{1}\) and the second photon can pass through slit \(a_{2}\), or (ii) a first photon can pass through slit \(b_{1}\) and the second photon can pass through slit \(b_{2}\), or (iii) a first photon can pass through slit \(a_{1}\) and the second photon can pass through slit \(b_{2}\), or (iv) a first photon can pass through slit \(b_{1}\) and the second photon can pass through slit \(a_{2}\). Since all the possible paths are indistinguishable and not revealing any which-path information therefore, total amplitude \(A_{12}\) to find a photon at \(o_{1}\) and a photon at \(o_{2}\) together is a quantum superposition of all the possible paths, which can be successively written as
$$\begin{aligned} A_{12}= & {} \langle o_{2}|a_{2}\rangle t_{a2} \langle a_{2}| \langle o_{1}|a_{1}\rangle t_{a1}\langle a_{1}|\psi \rangle + \langle o_{2}|b_{2}\rangle t_{b2} \langle b_{2}| \langle o_{1}|b_{1}\rangle t_{b1}\langle b_{1}|\psi \rangle \nonumber \\&+\langle o_{2}|b_{2}\rangle t_{b2} \langle b_{2}| \langle o_{1}|a_{1}\rangle t_{a1}\langle a_{1}|\psi \rangle +\langle o_{2}|a_{2}\rangle t_{a2} \langle a_{2}| \langle o_{1}|b_{1}\rangle t_{b1}\langle b_{1}|\psi \rangle \end{aligned}$$
where \(t_{a1}\), \(t_{b1}\), \(t_{a2}\), \(t_{b2}\) are amplitudes of transmission of slits \(a_{1}\), \(b_{1}\), \(a_{2}\), \(b_{2}\), respectively. Quantum states \(|a_{1}\rangle\), \(|b_{1}\rangle\) \(|a_{2}\rangle\), \(|b_{2}\rangle\) are position space basis states of locations on the slits on double-slit 1 and double-slit 2, respectively where a photon can be found. Similarly, \(|o_{1}\rangle\) and \(|o_{2}\rangle\) are the position space basis states of locations on the screens. However, position basis states corresponding to points on each double-slit and a screen form a different basis set such that \(\langle o_{1}|a_{1}\rangle\) represents the amplitude of transmitted photon to go from slit \(a_{1}\) to a location \(o_{1}\) on screen 1. Same terminology is applied for other amplitudes in Eq. 1.
A schematic diagram of a double-double-slit experiment. Photons are individually detected on screens after they pass through the double-slits separately. Which-slit path information of photons can be detected by blocking any single slit by closing the shutter.
Further, consider photon pairs produced by a source of finite size are emitted in opposite directions w.r.t. each other such that they are momentum entangled, their net momentum is zero and momentum of each photon is definitely unknown. Consider the spatial extension of source is much smaller than the slit separation but large to produce momentum entanglement. As a consequence of momentum entanglement, if a photon passes through slit \(a_{1}\) then the other photon passes through slit \(b_{2}\) and if a photon passes through slit \(b_{1}\) then the other photon passes through slit \(a_{2}\) after their transmission through the slits. For momentum entangled photons, both these possibilities are quantum superimposed, as result of it both the photons are path entangled via the slits and first two terms in the summation of Eq. 1 become zero. The last two terms in the summation are due to path entanglement via the slits, these two amplitudes interfere with each other and produce a two-photon interference of momentum entangled photons. When all four slits are opened, a two-photon path information is not revealed and a two-photon interference can be observed by recording detection locations of a photon corresponding to a particular location of detection of other photon on the other screen during each repetition of the experiment.
On the other hand, if one measures the direction of momentum of any single photon prior to its passage through double-slits then the momentum entangled state is collapsed. This measurement outcome reveals momentum direction of a photon on which a measurement is performed and the direction of momentum of the other photon is also revealed instantly after the collapse even without making any measurement on it. This measurement reveals which-slit path information of photons. On the other hand, which-slit path of photons in the double-double-slit can be detected by closing any single slit with a shutter. A shutter shown in Fig. 1 is considered as a photon measuring detector, if shutter is closed to block a slit \(a_{2}\) and a photon is detected on screen 2 then it reveals that a photon is passed through a slit \(b_{2}\) due to collapse of quantum entangled state caused by the shutter detector. As a consequence, one can find out that the other photon is passed through a slit \(a_{1}\) if it is detected at \(o_{1}\). Since a path of both photons is known therefore, two-photon interference is suppressed. Interesting situation appears when double-slit 2 and screen 2 are removed to allow a photon to propagate in space while other photon is passed through double-slit 1 and detected on screen 1. A single-photon interference not produced on screen 1 because of path entanglement the which-slit path information of photons can be obtained by measuring momentum of the propagating photon even after the detection of a photon on screen 1.
Furthermore, when all slits are opened and which-slit path is not detected, a photon can be detected at any location on a screen randomly during each repetition of the experiment and its detection location is not known prior to a measurement on screen. Once a photon is detected on a screen, its detection location instantly determines the amplitude to find other photon on other screen if it is not reached there. Individual photons show no interference on a screen because a well defined phase coherent amplitude to find a photon on a screen depends on a particular detection location of other photon. In this case, a single photon amplitude is completely incoherent. The information of detection location of a photon determines a particular two-photon interference pattern. In other words, in this type of joint and correlated registration of detection locations of photons, if a different detection location of a photon is selected the two-photon interference pattern exhibits a shift. If only single photons are registered on each screen without making any correlation between their detection locations then the interference pattern is not formed on each screen.
Two-photon interference
To find out a two-photon interference in the double-double-slit experiment for a finite width of each slit, consider a source of photons located at origin is producing a two-photon quantum state \(|\Psi \rangle\) as shown in Fig. 1. Double-slits can be defined by amplitude transmission functions \(t_{1}(y')\) and \(t_{2}(y'')\) of double-slit 1 and double-slit 2 respectively. Where \(y'\) and \(y''\) are the arbitrary points on double-slit 1 and double-slit 2, respectively such that the position basis states corresponding to these points located on the double-slits where a photon can be found are \(|l_{1}, y'\rangle\) and \(|l_{2}, y''\rangle\). Therefore, the amplitude \(A_{12}\) to find photons at points \(o_{1}\) and \(o_{2}\) together on screens can be written as
$$\begin{aligned} A_{12}= \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } \langle o_{2}| l_{2},y''\rangle t_{2}(y'') \langle l_{2},y''| \langle o_{1}|l_{1},y'\rangle t_{1}(y') \langle l_{1},y'|\Psi \rangle \mathrm {d}y' \mathrm {d}y'' \end{aligned}$$
Consider photon source has finite size and two-photon quantum state \(|\Psi \rangle\) is a momentum entangled quantum state, where both the photons have same linear polarisation and frequency. Such a two-photon quantum entangled state can be produced by degenerate noncollinear SPDC with type-I phase matching in a beta-barium-borate (BBO) crystal which is pumped by a laser beam propagating along the z-axis (longitudinal direction), where the z-axis (not shown in Fig. 1) is perpendicular to the xy-plane (transverse plane). Photons known as the signal and the idler photons are emitted from the source with opposite momenta with nearly equal in magnitude in the transverse plane such that their two-photon momentum entangled state in the transverse momentum space is42,43,44,45,46
$$\begin{aligned} |\Psi \rangle = N \int \int \mathrm {d}\mathbf {q_{s}}\mathrm {d}\mathbf {q_{i}} \Phi (\mathbf {q_{s},\mathbf {q_{i}}})|\mathbf {q_{s}}\rangle |\mathbf {q_{i}}\rangle \end{aligned}$$
where \(|\mathbf {q_{s}}\rangle\), \(|\mathbf {q_{i}}\rangle\) are the transverse momentum quantum states of the signal and the idler photons of momentum \(\mathbf {q_{s}}\) and \(\mathbf {q_{i}}\), respectively and N is a normalisation constant. Two-photon wavefunction \(\Phi (\mathbf {q_{s},\mathbf {q_{i}}})\) represents the amplitude to find a signal photon in momentum state \(|\mathbf {q_{s}}\rangle\) and an idler photon in momentum state \(|\mathbf {q_{i}}\rangle\). Quantum entanglement is manifested by non separability of \(\Phi (\mathbf {q_{s},\mathbf {q_{i}}})\). For the pump laser beam with gaussian intensity profile of finite width in the transverse plane, the two-photon wavefunction \(\Phi (\mathbf {q_{s},\mathbf {q_{i}}})\) is prominent only for momentum states of photons with opposite transverse momenta. Since source size is finite therefore, if momentum of a photon is measured precisely then the quantum state of the other photon corresponds to a momentum state of opposite momentum with finite uncertainty. Further detail on momentum entanglement of photons produced by degenerate noncollinear SPDC in the BBO crystal is given in methods.
There are two possibilities that can result in a joint detection of photons on screen 1 and screen 2. These indistinguishable possibilities are (i) the signal photon is passed through double-slit 1 and detected on screen 1 and the idler photon is passed through double-slit 2 and detected on screen 2 and (ii) the idler photon is passed through double-slit 1 and detected on screen 1 and the signal photon is passed through double-slit 2 and detected on screen 2. Any single photon (signal or idler) that can be detected after passing through the double-slit 1 is labeled as photon 1 and any single photon that can be detected after passing through the double-slit 2 is labeled as photon 2. Photon 1 and photon 2 are indistinguishable as they have same frequency and polarisation. Consider, transmission function of double-slit 1 is \(t_{1}(y')= a'_{t}\left( \frac{e^{-(y'-d_{1}/2)^2/2\sigma _{1}^2}}{(2\pi )^{1/2}\sigma _{1}}+\frac{e^{-(y'+d_{1}/2)^2/2\sigma _{1}^2}}{(2\pi )^{1/2}\sigma _{1}}\right)\), which represents two gaussian slits with separation between them \(d_{1}\) and slit width \(\sigma _{1}\) of each slit is such that \(d_{1}\) is considerably larger than \(\sigma _{1}\). Similarly, transmission function of double-slit 2 is \(t_{2}(y'')= a''_{t}\left( \frac{e^{-(y''-d_{2}/2)^2/2\sigma _{2}^2}}{(2\pi )^{1/2}\sigma _{2}}+\frac{e^{-(y''+d_{2}/2)^2/2\sigma _{2}^2}}{(2\pi )^{1/2}\sigma _{2}}\right)\), which represents two gaussian slits with separation between them \(d_{2}\) and slit width \(\sigma _{2}\) of each slit is such that \(d_{2}\) is considerably larger than \(\sigma _{2}\). Where, \(a'_{t}\) and \(a''_{t}\) are the complex multipliers of transmission functions, they include the phase shift introduced by the slits and limit the maximum transmission to one. For \(a'_{t}= a''_{t}=0\), the transmission of slits is zero. Each double-slit is positioned far away from the source as compared to its slit separation. Therefore, slits are located at close inclination with the x-axis such that photons coming from source are incident on slits almost close to the normal incidence. To have two-photon path entanglement via the slits the double-slits are positioned such that \(d_{1}/l_{1}=d_{2}/l_{2}\) and \(\sigma _{1}/l_{1}=\sigma _{2}/l_{2}\). In addition, uncertainty \(\Delta q_{\parallel }\) of momentum component of each photon parallel to the double-slits, provided momentum of other photon is precisely determined, is small such that \(\Delta q_{\parallel }/q\ll d_{1}/l_{1}= d_{2}/l_{2}\) to suppress single photon interference by each double-slit, where q is the magnitude of momentum of a photon41. However, \(\Delta q_{\parallel }/q\approx \sigma _{1}/l_{1}=\sigma _{2}/l_{2}\). These conditions implies, if a photon is passed through slit \(a_{1}\) then the other photon is most likely passed through slit \(b_{2}\) and if a photon is passed through slit \(b_{1}\) the other photon is most likely passed through slit \(a_{2}\). Therefore, photons contributing to the joint detection on screens are path entangled via the slits. However, if a photon is absorbed far away from slits at an arbitrary location \(y'\) on double-slit 1 then the other photon is most probably absorbed at \(y''=-y'l_{2}/l_{1}\) far away from slits of double-slit 2. Transmission of each slit is considered to be gaussian with very small width that allows a photon to pass through it. Under these considerations, the amplitude \(A_{12}\) of joint detection of photons on screens gets a major contribution from a small range of momentum states of quantum state \(|\Psi \rangle\). Remaining momentum states in \(|\Psi \rangle\) are absorbed at double-slits. Therefore, to evaluate \(A_{12}\) by using Eq. 2, a following approximation can be applied
$$\begin{aligned} t_{2} (y^{\prime\prime})t_{1} (y^{\prime})\langle l_{2} ,y^{\prime\prime}|\langle l_{1} ,y^{\prime}|\Psi \rangle \approx a^{\prime\prime}_{t} a^{\prime}_{t} c_{w} & \left( {e^{{iq(r_{{a1}} + r_{{b2}} )/\hbar }} \cdot \frac{{e^{{ - (y^{\prime} - d_{1} /2)^{2} /2\sigma _{1}^{2} }} }}{{(2\pi )^{{1/2}} \sigma _{1} }}\frac{{e^{{ - (y^{\prime\prime} + d_{2} /2)^{2} /2\sigma _{2}^{2} }} }}{{(2\pi )^{{1/2}} \sigma _{2} }}} \right. \\ & \quad \left. { +\, e^{{iq(r_{{b1}} + r_{{a2}} )/\hbar }} \cdot \frac{{e^{{ - (y^{\prime} + d_{1} /2)^{2} /2\sigma _{1}^{2} }} }}{{(2\pi )^{{1/2}} \sigma _{1} }}\frac{{e^{{ - (y^{\prime\prime} - d_{2} /2)^{2} /2\sigma _{2}^{2} }} }}{{(2\pi )^{{1/2}} \sigma _{2} }}} \right) \\ \end{aligned}$$
where \(c_{w}\) is a constant of proportionality that depends on the two-photon wavefunction. Since photons are incident on each slit close to the normal incidence therefore, \(e^{i q (r_{a1}+ r_{b2})/\hslash }\) is the two-photon amplitude of a photon to go from source to slit \(a_{1}\) located at a distance \(r_{a1}\) and other photon to go from source to slit \(b_{2}\) located at a distance \(r_{b2}\). Similarly, \(e^{i q (r_{b1}+ r_{a2})/\hslash }\) is the two-photon amplitude of a photon to go from source to slit \(b_{1}\) located at a distance \(r_{b1}\) and other photon to go from source to slit \(a_{2}\) located at a distance \(r_{a2}\). The transmitted amplitude of photons via the slits \(a_{1}\) and \(a_{2}\) or via the slits \(b_{1}\) and \(b_{2}\) is negligible because \(\Phi (\mathbf {q_{s},\mathbf {q_{i}}})\) is very small for these paths. Photons are path entangled via the slits and Eq. 4 represents the amplitude of transmitted photons on the double-slits that leads to the joint detection of photons.
Transmitted photon amplitude of a photon further emanates from a point on a double-slit such that it corresponds to an uniform probability distribution of the photon to be found on the screen. The amplitudes of transmitted photons to go from a point location on a double-slit to a point location on the nearest screen are \(\langle o_{1}| l_{1},y'\rangle \propto e^{iq|R'|/\hslash }/|R'|^{1/2}\) and \(\langle o_{2}| l_{2},y''\rangle \propto e^{iq|R''|/\hslash }/|R''|^{1/2}\) for photon 1 and photon 2, respectively. Where \(R'\) and \(R''\) are the distances of \(o_{1}\) and \(o_{2}\) from arbitrary points \(y'\) and \(y''\) located on double-slit 1 and double-slit 2, respectively. Since distances \(s_{1}\) and \(s_{2}\) of the screens from the nearest double-slits are much larger than the slit separations therefore, \(\langle o_{1}| l_{1},y'\rangle \propto e^{iq(r_{1}-y'\sin (\theta _{1}))/\hslash }/r^{1/2}_{1}\) and \(\langle o_{2}| l_{2},y''\rangle \propto e^{iq(r_{2}-y''\sin (\theta _{2}))/\hslash }/r^{1/2}_{2}\), where \(r_{1}\) and \(r_{2}\) are the distances of \(o_{1}\) and \(o_{2}\) from the middle points of double-slit 1 and double-slit 2, respectively as shown in Fig. 1. After solving Eq. 2 by using Eq. 4 the amplitude of joint detection of photons can be written as
$$\begin{aligned} A_{12}\,=\, & {} c_{n} \frac{e^{i q (r_{1}+r_{2})/\hslash } e^{i q (r_{a1}+r_{b2}+r_{a2}+r_{b1})/2 \hslash }}{(r_{1}r_{2})^{1/2}} e^{-q^{2}((\sigma _{1}\sin \theta _{1})^{2}+(\sigma _{2}\sin \theta _{2})^{2})/2\hslash ^{2}} \cos [q(d_{2}\sin \theta _{2}-d_{1}\sin \theta _{1})/2\hslash + \delta ] \end{aligned}$$
where \(\delta =q (r_{a1}+r_{b2}-r_{a2}-r_{b1})/2\hslash\) and \(c_{n}\) is a proportionality constant. Therefore, probability of coincidence detection \(p_{12}=|A_{12}|^{2}\) of photons is
$$\begin{aligned} p_{12}= \frac{ |c_{n}|^{2} }{r_{1}r_{2}} e^{-q^2((\sigma _{1}\sin \theta _{1})^{2}+(\sigma _{2}\sin \theta _{2})^{2})/\hslash ^2} \cos ^{2}[q(d_{2}\sin \theta _{2}-d_{1}\sin \theta _{1})/2\hslash + \delta ] \end{aligned}$$
Probability of coincidence detection of photons is a product of two functions, where the exponential functions corresponds to a single-photon diffraction of photons from single slits and a cosine function corresponds to two-photon interference from the double-double-slit. Since photons are path entangled via the slits therefore, Eq. 6 can not be written as a product of two separate functions of variables of photon 1 and photon 2, respectively. If only the single photon detection locations on each screen are recorded without making any correlation among them then no interference pattern is formed. A single photon interference is suppressed due to quantum entanglement of paths of photons in the double-double-slit. Both photons can be detected anywhere randomly on the respective screens however, a two-photon quantum interference pattern appears only in the position correlated measurements. Probability of detection of a single photon on the respective screens can be calculated by integrating all possible paths of a single photon. However, due to quantum entanglement of paths this integral results in an addition of probability of detection of a single photon via each slit of a double-slit. Therefore, probabilities \(p_{1}\) and \(p_{2}\) to find a single photon on screen 1 and screen 2 are
$$\begin{aligned} p_{1}\propto \frac{1}{r_{1}} e^{-q^2(\sigma _{1}\sin \theta _{1})^{2}/\hslash ^2} \end{aligned}$$
where each probability distribution of a single photon detection is gaussian and single photon interference pattern is not exhibited.
Actual experiment is performed in the three-dimensional position space, where momentum of photons and distances of detectors from double-slits are measured in the three-dimensional position space. Therefore, projection of momentum and distances onto the transverse plane should be considered in order to be consistent with Eqs. 6 and 7. In actual experiment the slits are located parallel to the transverse plane, detector displacement is parallel to the transverse plane and displacement range is such that \(y_{1}\ll s_{1}\), \(y_{2}\ll s_{2}\) therefore, \(\sin {\theta _{1}}\sim y_{1}/s_{1}\) and \(\sin \theta _{2}\sim y_{2}/s_{2}\). Under these considerations, terms in the form of a ratio, of transverse momentum and distance of a screen from a corresponding double slit, appears in Eqs. 6 and 7. Therefore, photon momentum and distances of detectors from double-slits measured in the three-dimensional position space can be placed in these equations to calculate the patterns.
Double-double-slit experiment presented in this paper is performed with momentum entangled photons produced by type-I degenerate noncollinear SPDC21,24,39,42,43,44,45,47. A BBO crystal is pumped by an extraordinary linearly polarised laser beam of wavelength 405 nm and down converted photon pairs of wavelength 810 nm with ordinary polarisation are produced in the forward direction in a conical emission pattern according to momentum and energy conservation as shown in Fig. 2. To produce a virtual double-double-slit configuration, two Fresnel biprisms are placed in the path of photons and photons are detected by single photon avalanche photodetectors \(D_{1}\) and \(D_{2}\). Optical narrow band pass filters are placed in front of each photon detector to stop the background light. Down converted photons have same frequency and linear polarisation, which is perpendicular to the polarisation of the pump laser beam. Pump laser intensity is such that probability of more than single photon pair production is extremely small. Number of photon counts of each single photon detector and their mutual coincidence photon counts are measured with a two channel single photon counting module. Transverse mode extension of the pump laser beam is reduced to keep the source size much smaller than the slit separation but it is large so that momentum entanglement of photons is preserved.
A schematic diagram of the experimental configuration of the double-double-slit experiment. Momentum entangled photon pairs are produced in a conical emission pattern by a nonlinear crystal. A double-double-slit configuration is realised with two Fresnel biprisms.
An unfolded diagram of the double-double-slit experiment realised with Fresnel biprisms. Virtual sources correspond to virtual slits. To detect which-slit path information of photons, a shutter can be placed in a path of photon 1 in such a way that a virtual slit \(b_{1}\) is blocked.
Two-photon interference pattern obtained by measuring the coincidence photon counts when measurement location \(y_{2}\) of photon 2 is stationary. Experimental measurements are represented by open circles and solid line interference pattern is the two-photon interference calculated from theory. There is no interference exhibited by the individual photons as shown by single photon counts of single photon detectors \(D_{1}\) and \(D_{2}\). Where (a) for \(y_{2}=~0~{\text {mm}}\) and (b) for \(y_{2}=~0.07~{\text {mm}}\). Two-photon interference pattern is shifted as the location \(y_{2}\) of photon detector \(D_{2}\) is displaced.
For type-I phase matching, the BBO emits two degenerate photons with opposite transverse momenta in the transverse plane as shown in Fig. 2 and quantum state of photons of a pair corresponds to a continuous variable momentum entangled quantum state. A two-dimensional unfolded diagram of the experimental schematic given in Fig. 2 is shown in Fig. 3, where a source S positioned at origin is a BBO crystal that emits momentum entangled photons pairs. Two virtual double-slits are realised with two Fresnel biprisms positioned in the path of both photons. Fresnel biprisms are aligned in such a way that after passing through each Fresnel biprism, paths of a photon can be extrapolated in the backward direction such that it appears as if the photon is originated from two virtual sources which are considered as slits. Each Fresnel biprism produces a virtual double-slit with gaussian slits of finite size. In this way, a double-double-slit configuration is realised with slit separation \(d_{1}\) and \(d_{2}\) of a virtual double-slit 1 and a virtual double-slit-2, respectively as shown in Fig. 3. The virtual double-double-slit is parallel to the transverse plane and both photon detectors are displaced parallel to the transverse plane. Photon 1 is detected at location \(o_{1}\) and photon 2 is detected at location \(o_{2}\) by single photon detectors. Shortest distance of \(D_{1}\), \(D_{2}\) are \(L_{1}\), \(L_{2}\) from double-slit 1 and double-slit 2, respectively as shown in Fig. 3. Single photon counts of single photon detectors positioned at different locations \(y_{1}\) and \(y_{2}\) and the corresponding coincidence photons counts are recorded. Experimental results on the double-double-slit interference of momentum entangled photons are shown in Fig. 4, where the coincidence and single photon counts of photons are measured at different \(y_{1}\) positions of single photon detector \(D_{1}\) when single photon detector \(D_{2}\) a kept stationary at a location \(y_{2}\). Single photon counts of each single photon detector and the coincidence photon counts are presented by open circles in Fig. 4a for \(y_{2}\)= 0 mm where each data point is the mean of photon counts acquired for 5 s and twenty five repetitions of the experiment. The coincidence photon counts represent a two-photon interference pattern and the corresponding theoretically calculated interference given by Eq. 6 with a consideration of finite size of photon detectors is shown by a solid line. Effect of finite size of detectors raises the minima of the interference pattern. Single photon counts show no interference pattern as presented by the theoretical analysis also. According to the experimental considerations, \(\sin \theta _{1}\sim y_{1}/s_{1}\) and \(\sin \theta _{2}\sim y_{2}/s_{2}\). The coincidence interference pattern exhibits a shift when measurement location \(y_{2}\) of photon 2 is shifted to another position by displacing single photon detector \(D_{2}\). A shift in the two-photon interference pattern is shown in Fig. 4b for \(y_{2}\)= 0.07 mm. In the opposite case, photon 1 is detected at a stationary location \(y_{1}\) and photon 2 is detected at different locations \(y_{2}\). Results of the coincidence measurements of photon counts and single photon counts are shown in Fig. 5a for \(y_{1}\)= 0 mm and Fig. 5b for \(y_{1}\)= 0.07 mm. Solid line in each plot of coincidence measurements is the two-photon interference calculated from Eq. 6 by including the effect of finite size of detectors. A two-photon interference shows a shift with the displacement of position of detection location \(y_{1}\) of photon 1, while single photon counts show no interference as theoretically shown in the previous section. In the experiment, each virtual double-slit has a same slit separation \(d_{1}=d_{2}=~0.67~\hbox {mm}\) and \(L_{1}=L_{2}=528\) mm.
It is evident from the probability of coincidence photon detection given in Eq. 6 that for \(d_{1}=d_{2}\), the fringe separation of the two-photon interference pattern will reduce to half if the coincidence photon counts are measured for \(y_{2}=-y_{1}\) i.e. when both single photon detectors are displaced in the opposite direction. For this case, a two-photon interference pattern and a single photon pattern are shown in Fig. 6, where each measured data point of photon counts is the mean of data acquired for 5 s and twenty five repetitions of the experiment. It is a different experimental set-up than the previous case and in this case \(d_{1}=d_{2}=~0.682~\hbox {mm}\) and \(L_{1}=L_{2}=520\) mm. Solid line represents a theoretically calculated two-photon interference by including the effect of finite size of photon detectors. It is evident that the fringe separation is reduced to half and therefore, the number of fringes are increased within the same gaussian envelop. There is no formation of coincidence interference pattern if both the single photon detectors are displaced in the same direction such that \(y_{2}=y_{1}\) as it is evident from Eq. 6.
In this experiment both the single photon detectors \(D_{1}\) and \(D_{2}\) are displaced in the opposite direction such that \(y_{2}=-y_{1}\). Fringe separation of two-photon interference is reduced and individual photons exhibit no interference.
Detection of which-slit path of photons
In the double-double-slit experiment, photons are momentum entangled and they can reveal the which-slit path information of each other if one of them is detected close to any double-slit. If one blocks a single slit of a double-slit then the which-slit path can be detected from the coincidence detection of photons. Consider a slit \(a_{2}\) is blocked by closing a shutter shown in Fig. 1.
Two-photon coincidence pattern when a virtual slit \(b_{1}\) is blocked. It is evident that two-photon interference is suppressed. Solid line is a theoretically calculated two-photon pattern.
If photons of a single pair are detected on screen 1 and screen 2 together then it is evident that photon 2 has passed through slit \(b_{2}\). One can consider a path blocking shutter as another single photon detector \(D_{3}\). If \(D_{3}\) detects a photon 2 then the path entangled state of photons collapses and the which-slit path of photon 1 in the double-slit 1 is also determined. In this case, the which-slit path of photon 1 is through the single slit \(b_{1}\). Since each photon is passed through a single slit therefore, neither a single photon nor a two-photon interference of joint detections of photons on screen 1 and a path blocking single photon detector \(D_{3}\) will occur. On the other hand, if photon 2 is detected on screen 2 then the path entangled state is collapsed by \(D_{3}\) such that photon 2 is passed though slit \(b_{2}\) and photon 1 is passed through slit \(a_{1}\). Single photon detection probability of photon 2 on screen 2 will reduce by half in comparison to the case when both slits were open. Probability of a single photon detection of photon 1 on screen 1 will remain unchanged because detection of photon 1 does not reveal any information whether a photon 2 is detected at screen 2 or by \(D_{3}\). In the experiment, shutter is placed after the Fresnel biprism 1 such that a virtual slit \(b_{1}\) shown in Fig. 3 is blocked. This configuration resembles to a double-double-slit schematic shown in Fig. 1 where the slit \(a_{2}\) can be blocked by a shutter. Photon counts are measured for different locations \(y_{2}\) of single photon detector \(D_{2}\) by keeping single photon detector \(D_{1}\) stationary at \(y_{1}\). Experimental results of a path detection experiment are shown in Fig. 7. Experimental parameters in this case are same as for the experiment described in the previous section. It is evident from the experimental results, if a which-slit path information of photons is extracted by blocking any single slit then both single and two-photon interferences are suppressed.
This paper has presented experimental and conceptual insights on the quantum double-double-slit thought experiment first introduced by Greenberger, Horne and Zeilinger17. Experiments presented in this paper are performed with momentum entangled photons produced by type-I degenerate noncollinear SPDC process in a BBO crystal. In the experiment, once both photons traverse the respective double-slits, they can be detected anywhere on screens randomly because when a photon strikes a screen its quantum state collapses to one location randomly. Patterns emerge in many repetitions of the same experiment. Since paths of photons in the double-double-slit configuration are quantum entangled, their individual quantum states are phase incoherent therefore, formation of a single photon interference is suppressed. However, if a photon is detected on a screen at a well defined location, the quantum state of other photon, which is not detected, corresponds to a phase coherent amplitude to find it on second screen. Therefore, knowledge of detection locations of a photon labels the different phase coherent amplitudes to find other photon on second screen. However, in subsequent repetitions of the experiment, detection locations of photons can vary randomly. For a given location of detection of a photon the other photon shows interference pattern which corresponds to the conditional interference pattern of two photons. As a detection location of a photon is varied the conditional interference pattern is shifted. On the other hand, if no correlations of detection locations of photons are made then there is no way to select a particular phase coherent amplitude in repeated measurements. Eventually, a single photon interference pattern does not appear. It is also shown experimentally and conceptually, if a which-slit path information of any one of the photons is detected then a single photon interference and a two-photon interference disappear because of random collapse of quantum superposition of paths.
Two-photon momentum entangled state
Two-photon momentum entangled state is produced by a negative uniaxial second order nonlinear BBO crystal by type-I SPDC process. A pump photon of frequency \(\omega _{p}\) is split into two photons known as the signal photon and the idler photon of frequency \(\omega _{s}\) and \(\omega _{i}\), respectively. A linearly polarised extraordinary pump laser beam propagating along the z-axis is incident on the crystal. A planar surface of the crystal is in the xy-plane with origin at the centre, where \(l_{x}\), \(l_{y}\), \(l_{z}\) are the spatial extensions of the crystal along each axis. Ordinary photons produced by SPDC are linearly polarised with propagation vectors in three dimensions \(\mathbf {k_{s}}\) and \(\mathbf {k_{i}}\). In type-I phase matching, due to dispersion and anisotropy of the crystal, the signal and the idler photons are produced with non zero angle of their propagation vectors with the propagation vector \(\mathbf {k_{p}}\) of the pump laser beam to conserve momentum of photons. For a thin crystal and a narrow pump laser beam, it produces a conical emission pattern of down converted photons. Pump laser beam is considered to be a continuous beam and due to low down conversion efficiency the pump laser beam amplitude is considered to be constant. The amplitude to produce more than one photon pair is extremely small, which is desirable in experiments with a single quantum entangled pair of photons during each cycle of the experiment. Pump laser beam is considered to be monochromatic, frequencies of the signal and the idler photons are same in the experiment and their propagation vectors are making a nonzero angle with the propagation direction of pump photons. Narrow band pass filters are placed after the crystal and prior to the detectors to increase coherence length. Due to sufficiently long interaction time, energy conservation condition is fulfilled such that \(\hslash \omega _{p}=\hslash \omega _{s}+\hslash \omega _{i}\). Since polarisation of down converted photons is same therefore, two-photon quantum state produced by degenerate type-I noncollinear SPDC process can be written as42,43,44,45,46,47
$$\begin{aligned} |\Psi \rangle _{spdc}\approx c_{0}|0\rangle +c_{1} \int \int \mathrm {d}\mathbf {p_{s}}\mathrm {d}\mathbf {p_{i}} \Phi (\mathbf {p_{s},\mathbf {p_{i}}})|\mathbf {p_{s}}\rangle |\mathbf {p_{i}}\rangle \end{aligned}$$
where \(c_{0}\), \(c_{1}\) are complex coefficients, \(c_{1}\) depends on the pump laser beam intensity and second order nonlinear coefficient of the crystal. The quantum states \(|\mathbf {p_{s}}\rangle\) and \(|\mathbf {p_{i}}\rangle\) represent single photon momentum states of the signal and the idler modes of momentum vectors \(\mathbf {p_{s}}=\hslash \mathbf {k_{s}}\) and \(\mathbf {p_{i}}=\hslash \mathbf {k_{i}}\), respectively. The quantum state \(|0\rangle\) is a vacuum state of the signal and the idler modes without any photon. A two-photon wavefunction \(\Phi (\mathbf {p_{s},\mathbf {p_{i}}})\) in the momentum space can be written as
$$\begin{aligned} \Phi (\mathbf {p_{s},\mathbf {p_{i}}})=c_{p}\int \mathrm {d}\mathbf {q_{p}} \mathbf {\nu }(\mathbf {q_{p}}) {{\,\mathrm{sinc}\,}}\left( \frac{\Delta p_{x}l_{x}}{2\hslash }\right) {{\,\mathrm{sinc}\,}}\left( \frac{\Delta p_{y}l_{y}}{2\hslash }\right) {{\,\mathrm{sinc}\,}}\left( \frac{\Delta p_{z}l_{z}}{2\hslash }\right) \end{aligned}$$
where \(c_{p}\) is a constant and the integration is carried out in transverse momentum space which is a projection of three-dimensional momenta onto the transverse two-dimensional xy-plane, \(\Delta p_{j}\) = \(p_{sj}+p_{ij}-p_{pj}\) for \(j\in \{x,y,z\}\) and \(p_{sj}\), \(p_{ij}\), \(p_{pj}\) represent components of momentum of the signal, the idler and pump photons along the j-axis, respectively. A function \(\mathbf {\nu }(\mathbf {q_{p}})\) is the normalised amplitude of pump laser beam corresponding to momentum projection \(\mathbf {q_{p}}\) in the transverse plane. For a plane wave, \(\mathbf {\nu }(\mathbf {q_{p}})\) is a Dirac delta function. If crystal extensions \(l_{x}\) and \(l_{y}\) are much larger than the wavelength of pump laser beam then \(\Delta p_{x}\) and \(\Delta p_{y}\) should be very small otherwise \(\Phi (\mathbf {p_{s},\mathbf {p_{i}}})\) diminishes. Therefore, \(\mathbf {q_{p}}\)=\(\mathbf {q_{s}}+\mathbf {q_{i}}\) for transverse momentum \(\mathbf {q_{s}}\) of the signal photon and transverse momentum \(\mathbf {q_{i}}\) of the idler photon. It corresponds to conservation of transverse momentum of photons. For a gaussian transverse momentum profile of the pump laser beam with radius \(\sigma _{p}\) in the position-space and for a very small angle between the pump photon momentum and the signal photon or the idler photon momentum, the two-photon wavefunction is given in Ref.44,
$$\begin{aligned} \Phi (\mathbf {q_{s},\mathbf {q_{i}}})= c_{\Phi }{{\,\mathrm{sinc}\,}}\left( \frac{l_{z}}{4 \hslash ^{2}|\mathbf {k_{p}}|} |\mathbf {q_{s}}-\mathbf {q_{i}}|^{2}\right) {{\,\mathrm{e}\,}}^{-\sigma ^{2}_{p}|\mathbf {q_{s}}+\mathbf {q_{i}}|^{2}/\hslash ^{2}} \end{aligned}$$
where, \(c_{\Phi }\) is a constant of proportionality. Two-photon wavefunction \(\Phi (\mathbf {q_{s},\mathbf {q_{i}}})\) is prominent if transverse momenta of photons are equal and opposite to each other.
Two-photon quantum entangled state in the transverse momentum space can be written as
where N is a normalisation constant and the vacuum state is not relevant in the context of present experiment. In general the momentum entanglement is manifested by non separability of two-photon wavefunction \(\Phi (\mathbf {q_{s},\mathbf {q_{i}}})\).
A linearly polarised pump laser light of wavelength 405 nm of gaussian beam profile is incident on a BBO crystal at room temperature. Two orthogonally polarised momentum entangled photons of wavelength 810 nm are emitted in a conical emission pattern and single photons are detected by avalanche single photon detectors. Each photon pair is passed through two Fresnel biprisms to realise a virtual double-double-slit configuration. After the crystal, pump light at 405 nm is blocked by an optical band pass filter with transmission window peak at 810 nm where the full-width-half-maximum of the transmission window is about 10 nm. Two multimode optical fibers carry photons from points \(o_{1}\) and \(o_{2}\) to each single photon detector. The other end of each optical fiber is mounted on separate three-dimensional precision displacement stages and photons are coupled to each optical fiber with an objective lens. Narrow apertures are positioned at \(o_{1}\) and \(o_{2}\) prior to the objective lens to allow photons to be detected at these two points only. Prior to each fiber coupler two optical band pass filters (filter 1 and filter 2) are placed in the path of each photon to block scattered photon of wavelength 405 nm and background photons reaching each single photon detector. Photon correlations are measured by counting electrical pulses produced by each single photon detector. Experimentally measured coincidence and single photon counts are shown by open circles data points in the figures. Each data point is acquired for 5 s with twenty five repetitions of the same experiment. Experimental results are compared with theoretical calculations considering the effect of finite size of photon detectors.
Young, T. The 1801 Bakerian Lecture: On the theory of light and colours. Philos. Trans. R. Soc. Lond. 92, 12–48 (1802).
ADS Google Scholar
Young, T. The 1803 Bakerian Lecture: Experiments and calculations relative to physical optics. Philos. Trans. R. Soc. Lond. 94, 1–16 (1804).
Dirac, P. The Principles of Quantum Mechanics 4th edn. (Oxford University Press, Oxford, 1930).
Feynman, M. A., Leighton, R. B. & Sands, M. L. The Feynman Lectures on Physics 2nd edn, Vol. 3 (Addison-Wesley, Boston, 1963).
Taylor, G. I. Interference fringes with feeble light. Proc. Camb. Philos. Soc. 15, 114–115 (1909).
Zeilinger, A., Gaehler, R., Shull, C. G., Treimer, W. & Mampe, W. Single and double-slit diffraction of neutrons. Rev. Mod. Phys. 60, 1067–1073 (1988).
ADS CAS Google Scholar
Jönsson, C. Elektroneninterferenzen an mehreren künstlich hergestellten feinspalten. Zeitschrift für Physik 161, 454–474 (1961).
Rodolfo, R. The merli-missiroli-pozzi two-slit electron-interference experiment. Phys. Perspect. 14, 178–195 (2012).
Frabboni, S. et al. The Young–Feynman two-slits experiment with single electrons: Build-up ofthe interference pattern and arrival-time distribution using a fast-readoutpixel detector. Ultramicroscopy 116, 73–76 (2012).
Roger, B., Damian, P., Sy-Hwang, L. & Herman, B. Controlled double-slit electron diffraction. New. J. Phys. 15, 033018 (2013).
Sala, S. et al. First demonstration of antimatter wave interferometry. Sci. Adv. 5, eaav7610 (2019).
ADS CAS PubMed PubMed Central Google Scholar
Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F. Cavity optomechanics. Rev. Mod. Phys. 86, 1391–1452 (2014).
Chan, J. et al. Laser cooling of a nanomechanical oscillator into its quantum ground state. Nature 478, 89–92 (2011).
ADS CAS PubMed Google Scholar
Cotter, J. P. et al. In search of multipath interference using large molecules. Sci. Adv. 3, e1602478 (2017).
ADS PubMed PubMed Central Google Scholar
Hornberger, K., Gerlich, S., Haslinger, P., Nimmrichter, S. & Arndt, M. Colloquium: Quantum interference of clusters and molecules. Rev. Mod. Phys. 84, 157–173 (2012).
Hornberger, K. & Arndt, M. Testing the limits of quantum mechanical superpositions. Nat. Phys. 10, 271–277 (2014).
Greenberger, D. M., Horne, M. A. & Zeilinger, A. Multiparticle interferometry and the superposition principle. Phys. Today 46, 8 (1993).
Horne, M. A. & Zeilinger, A. Microphysical Reality and Quantum Formalism (Kluwer Academic, Dordrecht, 1988).
Horne, M. A. Experimental Metaphysics (Kluwer Academic, Dordrecht, 1997).
Hong, C. K. & Noh, T. G. Two-photon double-slit interference experiment. J. Opt. Soc. Am. B. 15, 0740–3224 (1998).
Zeilinger, A. Experiment and the foundations of quantum physics. Rev. Mod. Phys. 71, S288–S297 (1999).
Kwiat, P. G. et al. New high-intensity source of polarization-entangled photon pairs. Phys. Rev. Lett. 75, 4337–4341 (1995).
Shadbolt, P., Mathews, J. C. F., Laing, A. & O'Brien, J. L. Testing foundations of quantum mechanics with photons. Nat. Phys. 10, 278–286 (2014).
Horne, M. A., Shimony, A. & Zeilinger, A. Two-particle interferometry. Phys. Rev. Lett. 62, 2209–2212 (1989).
Ma, X.-S., Kofler, J. & Zeilinger, A. Delayed-choice gedanken experiments and their realizations. Rev. Mod. Phys. 88, 015005 (2016).
Scully, M. O. & Drühl, K. Quantum eraser: A proposed photon correlation experiment concerning observation and "delayed choice" in quantum mechanics. Phys. Rev. A 25, 2208–2213 (1982).
Ma, X. .-S. et al. Quantum erasure with causally disconnected choice. Proc. Natl. Acad. Sci. 110, 1221–1226 (2013).
Kim, Y.-H., Yu, R., Kulik, S. P., Shih, Y. & Scully, M. O. Delayed, "choice" quantum eraser. Phys. Rev. Lett. 84, 1–5 (2000).
Kaiser, F., Coudreau, T., Milman, P., Ostrowsky, D. B. & Tanzilli, S. Entanglement-enabled delayed-choice experiment. Science 338, 637–640 (2012).
Peruzzo, A., Shadbolt, P., Brunner, N., Popescu, S. & O'Brien, J. L. A quantum delayed-choice experiment. Science 338, 634–637 (2012).
Strekalov, D. V., Sergienko, A. V., Klyshko, D. N. & Shih, Y. H. Observation of two-photon "ghost" interference and diffraction. Phys. Rev. Lett. 74, 3600–3603 (1995).
Howell, J. C., Bennink, R. S., Bentley, S. J. & Boyd, R. W. Realization of the Einstein–Podolsky–Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion. Phys. Rev. Lett. 92, 210403 (2004).
ADS PubMed Google Scholar
Fonseca, E. J. S., Machado da Silva, J. C., Monken, C. H. & Pádua, S. Controlling two-particle conditional interference. Phys. Rev. A 61, 023801 (2000).
Barbosa, G. A. Quantum images in double-slit experiments with spontaneous down-conversion light. Phys. Rev. A 54, 4473–4478 (1996).
Gatti, A., Brambilla, E. & Lugiato, L. A. Entangled imaging and wave-particle duality: From the microscopic to the macroscopic realm. Phys. Rev. Lett. 90, 133603 (2003).
Ribeiro, P. H. S., Pádua, S., da Silva, M. J. C. & Barbosa, G. A. Controlling the degree of visibility of Young's fringes with photon coincidence measurements. Phys. Rev. A 49, 4176–4179 (1994).
Souto Ribeiro, P. H. & Barbosa, G. A. Direct and ghost interference in double-slit experiments with coincidence measurements. Phys. Rev. A 54, 3489–3492 (1996).
Jaeger, G., Horne, M. A. & Shimony, A. Complementarity of one-particle and two-particle interference. Phys. Rev. A 48, 1023–1027 (1993).
Mandel, L. Quantum effects in one-photon and two-photon interference. Rev. Mod. Phys. 71, S274–S282 (1999).
Menzel, R., Puhlmann, D., Heuer, A. & Schleich, W. P. Wave-particle dualism and complementarity unraveled by a different mode. PNAS 109, 9314–9319 (2012).
Kofler, J. et al. Einstein–Podolsky–Rosen correlations from colliding Bose–Einstein condensates. Phys. Rev. A 86, 032115 (2012).
Monken, C. H., Ribeiro, P. H. S. & Pádua, S. Transfer of angular spectrum and image formation in spontaneous parametric down-conversion. Phys. Rev. A 57, 3123–3126 (1998).
Walborn, S. P., de Oliveira, A. N., Thebaldi, R. S. & Monken, C. H. Entanglement and conservation of orbital angular momentum in spontaneous parametric down-conversion. Phys. Rev. A 69, 023811 (2004).
Schneeloch, J. & Howell, J. C. Introduction to the transverse spatial correlations in spontaneous parametric down-conversion through the biphoton birth zone. J. Opt. 18, 053501 (2016).
Walborn, S. P., Monken, C., Pádua, S. & Souto Ribeiro, P. Spatial correlations in parametric down-conversion. Phys. Rep. 495, 87–139 (2010).
Hong, C. K. & Mandel, L. Theory of parametric frequency down conversion of light. Phys. Rev. A 31, 2409 (1985).
Joobeur, A., Saleh, B. E. A., Larchuk, T. S. & Teich, M. C. Coherence properties of entangled light beams generated by parametric down-conversion: Theory and experiment. Phys. Rev. A 53, 4360–4371 (1996).
Mandip Singh acknowledges research funding by the Department of Science and Technology, Quantum Enabled Science and Technology grant for project No. Q.101 of theme title "Quantum Information Technologies with Photonic Devices". DST/ICPS/QuST/Theme-1/2019 (General).
Department of Physical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector-81, Mohali, 140306, India
Manpreet Kaur & Mandip Singh
Mandip Singh
M.S. designed and setup the experiment, M.K. took data and also aligned the experiment. M.S. did theory, analysed data and wrote the manuscript, both authors discussed the experiment.
Correspondence to Mandip Singh.
Kaur, M., Singh, M. Quantum double-double-slit experiment with momentum entangled photons. Sci Rep 10, 11427 (2020). https://doi.org/10.1038/s41598-020-68181-1
Quantum imaging of a polarisation sensitive phase pattern with hyper-entangled photons
Top 100 in Physics | CommonCrawl |
Chair: Thomas Busch
08:45 - 09:00 Roderich Moessner & Scientific coordinators
09:00 - 09:40 Pietro Massignan (Universitat Politècnica de Catalunya)
Macroscopic coherent dressing of Bose polarons
09:40 - 10:20 Matteo Zaccanti (LENS - European Laboratory for Non-Linear Spectroscopy)
Ultracold Chromium-lithium Fermi mixtures with resonant few-body interactions
From quark and nuclear matter down to solid-state systems and ultracold gases, mass asymmetry is known to profoundly influence the behavior of strongly interacting fermionic particles, both at the few- and many-body level. In my talk I will discuss the prospects offered by a novel ultracold 53Cr - 6Li Fermi mixture, currently under development within the PoLiChroM experiment in Florence. I will first show how the unique few-body properties enabled by the peculiar chromium-lithium mass ratio render such a mixture a qualitatively new framework for the investigation of elusive phenomena of highly-correlated fermionic matter. I will then provide an overview of our new setup, discussing the current status and near future perspectives of the PoLiChroM machine.
11:00 - 11:40 Selim Jochim (Universität Heidelberg)
Juggling with Single Atoms to Observe Many-Body Physics
In our quest to obtain a deeper understanding of strongly interacting many body quantum systems, we turn to finite systems that can be prepared at very low entropies. Single-atom sensitive detection allow for the determination of high-order correlation functions. In our presentation we first show work with noninteracting spin polarized fermions and determine the correlations present solely due to indistinguishability, observing for the first time so-called Pauli crystals, geometric structures that form in the correlations between particles because of the antisymmetry of the wave function. We will also report on a significant breakthrough observing a precursor of many-body physics in a finite 2D system that due to degeneracies in the harmonic potential exhibits a shell structure: Increasing the attraction between particles, we observe a non-monotonous behavior of the excitation spectrum when the attraction becomes larger than the spacing between shells. In the many-body limit this can be interpreted as a quantum phase transition from normal to superfluid and associated with a long-lived Higgs mode. In the future we would like to use our newly established methods to measure correlations in (strongly) interacting systems. A first step could be to directly observe Cooper pairs by increasing the particle number.
Chair: Artur Widera
16:30 - 17:30 mcas20-Colloquium: Ana Maria Rey (chair: Matthew Eiles, mpipks) (JILA/NIST/University of Colorado)
Dynamical Phase Transitions in Cold Atomic Gases
Non-equilibrium quantum many-body systems can display fascinating phenomena relevant for various fields in science ranging from physics, to chemistry, and ultimately, for the broadest possible scope, life itself. The challenge with these systems, however, is that the powerful formalism of statistical physics, which have allowed a classification of quantum phases of matter at equilibrium does not apply. Therefore, using controllable cold atomic systems to shed light on the organizing principles and universal behaviors of dynamical quantum matter is highly appealing. One emerging paradigm is the dynamical phase transition (DPT) characterized by the existence of a long-time-average order parameter that distinguishes two non-equilibrium phases. I will report the observation of a DPT in two different but complementary systems: a trapped quantum degenerate Fermi gas and long lived arrays of atoms in an optical cavity. I will show how these systems can be used to simulate iconic models of quantum magnetism with tunable parameters and to probe the dependence of their associated dynamical phases on a broad parameter space. Besides advancing quantum simulation our studies pave the ground for the generation of metrologically useful entangled states which can enable real metrological gains via quantum enhancement.
17:50 - 18:30 David Weld (UC Santa Barbara)
Probing quantum dynamics in strongly driven lattices
Degenerate quantum gases in modulated optical lattices are a flexible testbed for the experimental study of quantum matter driven far from equilibrium. I will discuss results from a sequence of recent experiments in this area, starting with the tuning of band structure and transport properties in both rapidly-driven and slowly-driven lattices, moving on to experimental mapping of the properties of interacting prethermal Floquet matter, and concluding with probes of localization in driven quasicrystals and interacting quantum kicked rotors.
18:30 - 19:10 Christian Gross (Max-Planck-Institut für Quantenoptik)
The thinnest possible mirror - collective atom-light interaction in a structured atomic monolayer
Collective effects in atom-light interactions typically emerge when the scatterers are separated by much less than an optical wavelength. However, in structured materials they can dominate even for comparably low densities. Here we report on the observation of collective backscattering by ordered atoms in a 2d optical lattice. With a single atomic monolayer this realizes the thinnest possible mirror. We observe a significant narrowing of the spectral response compared to the lifetime-limited line width of individual atoms. This underlines the collective nature of the scattering process in our sample of about 200 atoms.
Chair: Doerte Blume
16:30 - 17:10 Miguel Angel Garcia-March (Polytechnic University of Valencia)
Open quantum systems approach to Bose polaron problems: bi-polaron, two-component systems, and applications to thermometry and phononics
I will discuss the dynamics of two impurities in a Bose-Einstein condensate (BEC) [1] and one impurity in a two-component BEC [2], when the problem is approached from the open quantum system perspective. For both cases and untrapped impurities, I will describe the long-time diffusive behavior. Particularly, I will show that this behavior can be controlled in the two component case. For two trapped impurities, I will discuss the realistic conditions for the final equilibrium state to be squeezed or to present entanglement. I will briefly discuss an application for quantum non-demolition thermometry in a BEC [3]; I will introduce in more detail an application in phononics, to construct a system that permits a heat current between two BECs, and evaluate this system as a thermal diode [4]. I will also discuss the effect of dimensionality in the dynamics described [5]. [1] C. Charalambous, et al., Sci. Post 6, 10 (2019). [2] C. Charalambous, et al., Quantum 4, 232 (2020) [3] M. Mehboudi, et al., Phys. Rev. Lett. 122, 030403 (2019). [4] C. Charalambous, et al., New Journal of Physics 21, 083037 (2019). [5] M. Miskeen Khan et al. arXiv:2007.04925.
17:10 - 17:50 Nikolaj Zinner (Aarhus University)
Strongly interacting particles in 1D traps at zero and finite temperature
17:50 - 18:30 Cindy Regal (University of Colorado)
Bosonic atoms in optical tweezers
18:30 - 19:10 Bruno Julia-Diaz (Universitat de Barcelona)
In memoriam Artur Polls, some recollections and recent results on droplets in optical lattices
We will briefly honour the memory of Prof. Artur Polls, who passed away in a completely unexpected way on Aug 12th 2020, see an obituary here [1]. We will share some recollections and pictures. Special attention will be taken to his last works related to mesoscopic quantum systems. We will finalize explaining one of his most recent work on the existence of quantum droplets of binary mixtures in 1D optical lattices [2]. [1] In memory of Artur Polls Martí, http://icc.ub.edu/news/616 [2] Quantum droplets of bosonic mixtures in a one-dimensional optical lattice Ivan Morera, Grigori E. Astrakharchik, Artur Polls, Bruno Juliá-Díaz Phys. Rev. Research 2, 022008 (R) (2020).
09:00 - 09:40 Meera Parish (Monash University)
The nature of the repulsive polaron
09:40 - 10:20 Hannah Williams (Institut d'Optique)
Using a reconfigurable array of Rydberg atoms to investigate the Ising Model
10:20 - 11:00 Li You (Tsinghua University)
Faster driving of adiabatic quantum phase transition by deep reinforcement learning
Driven quantum phase transition (QPT) occurs when a many body system is slowly or adiabatically swept across a quantum critical point (QCP). A recent example involves near deterministic production of an atomic Bose-Einstein condensate (BEC) in Twin-Fock as well as three-mode balanced Dicke states. The efficacy of the sweeping protocol, however, is limited by a compromise between finite coherence time (due to loss or decoherence) versus adiabaticity (from a finite-sized gap). This work presents a faster sweeping protocol with improved performance guided by deep reinforcement learning (DRL). Implemented in a BEC of upto 10000 87Rb atoms, the prepared balanced Dicke state ensemble exhibits an improved number squeezing of 13.08+/-0.29 dB within 750 ms, or in about half of the previously reported near adiabatic sweeping time. Our protocol highlights the potential of DRL to quantum dynamic control and quantum state preparation. With suitable adjustment, it can be applied to driving QPT along avoided crossing ground state over QCP in the broad class Lipkin-Meshkov-Glick (LMG) model of many coupled spins.
Breakout / Discussion Session
11:40 - 12:20 Giovanna Morigi (Universität des Saarlandes)
Topological effects in chains of interacting atoms
16:30 - 17:10 Ana Maria Rey (JILA/NIST/University of Colorado)
New Frontiers on Many-body Physics with Atomic Clocks
In this talk, I will report on recent developments on how to use ultracold fermionic alkaline-earth atoms (AEAs) –currently the basis of the most precise atomic clock in the world– for the investigation of complex many-body phenomena and magnetism. Specifically, I will discuss protocols that take advantage of the ultra long lifetime of AEAs dressed by an ultra-stable clock laser, which couples spin and motional degrees of freedom, to engineer superexchange spin models with high degree of tunability. I will show how these spin models can be a powerful resource for the generation of useful entangled states including cat states, cluster states and spin squeezed states. The proposed schemes are readily implementable in current state-of-the-art atomic clocks, and open a window to enhance their sensitivity beyond the standard quantum limit and a path to realize proof-of-principle demonstrations of one-way quantum computing with ultra-cold atomic systems.
17:10 - 17:50 Stephanie Reimann (Lund University)
Persistent Currents in Toroidal Dipolar Supersolids
We investigate the rotational properties of a dipolar Bose-Einstein condensate trapped in a toroidal geometry. Studying the ground states in the rotating frame and at fixed angular momenta, we observe that the condensate acts in distinctly different ways depending on whether it is in the superfluid or in the supersolid phase. We find that intriguingly, the toroidal dipolar condensate can support a supersolid persistent current which occurs at a local minimum in the ground state energy as a function of angular momentum, where the state has a vortex solution in the superfluid component of the condensate. The decay of this state is prevented by a barrier that in part consists of states where a fraction of the condensate mimics solid-body rotation in a direction opposite to that of the vortex. Furthermore, the rotating toroidal supersolid shows hysteretic behavior that is qualitatively different depending on the superfluid fraction of the condensate.
17:50 - 18:30 Przemysław Bienias (University of Maryland)
Scrambling and confinement with quantum simulators
In the first part of my talk, I will present our study of quantum information scrambling in spin models with both long-range all-to-all and short-range interactions. We argue that a simple global, spatially homogeneous interaction together with local chaotic dynamics is sufficient to give rise to fast scrambling, which describes the spread of quantum information over the entire system in a time that is logarithmic in the system size. In the second part, we propose a realization of mesonic and baryonic quasiparticle excitations in Rydberg atom chains with programmable interactions. By engineering a $\mathbb{Z}_3$-translational-symmetry breaking field on top of the Rydberg-blockaded Hamiltonian, we show that different types of defects experience confinement, and as a consequence form mesonic or baryonic quasiparticle excitations. We propose an experimental protocol involving out-of-equilibrium dynamics to directly probe the spectrum of the confined excitations.
18:30 - 19:10 Marcus Greiner (Harvard University)
Many-body localization under the microscope
09:00 - 09:40 Fabrice Gerbier (Laboratoire Kastler Brossel)
Anomalous momentum diffusion of strongly interacting bosons in optical lattices
I will describe experiments where we probe how a strongly-interacting gas of ultracold atoms reacts to dissipation. Specifically, I will discuss how the spatial coherence disappears when the gas is exposed to a near-resonant laser driving absorption-spontaneous emission cycles. Spontaneous emission introduces random momentum changes leading to diffusion in momentum space. This momentum diffusion is a well-known process is well-known in quantum optics which, for example, limits the temperature achievable in laser cooling. For strongly interacting bosons, we observed that the momentum space dynamics becomes anomalously slow, more precisely sub-diffusive. I will discuss the interpretation of this observation by the existence of slowly-relaxing states (somewhat analog of subradiant states in quantum optics) that dominate the long-times dynamics.
09:40 - 10:20 Jacqueline Bloch (Centre de Nanosciences et de Nanotechnologies)
Non-hermiticity and non-linearity in topological 1D polariton lattices
The discovery of topological materials has triggered a huge research effort to understand and control the link between symmetry properties and the emergence of novel physical properties. In particular, the existence of topological invariants, related to the band structure properties of the bulk material, is associated with the emergence of edge states, robust to local perturbation of the lattice. With the development of a variety of synthetic experimental platforms, it is now possible to engineer lattices and explore topology in a variety of contexts. Here we are interested in the physics of 1D topological lattices in the presence of both dissipation and non-linearity. I will describe recent experiments we have performed on photonic Su-Schrieffer-Heeger (SSH) lattices made of arrays of coupled polariton micropillars. The coupling between neighbor resonators is modulated so that a topological gap opens. Such polariton lattice is non-hermitian because of photon losses through the cavity mirrors, and highly non-linear because of polariton-polariton interactions. The physical properties of the system can be probed via optical spectroscopy using either incoherent pumping of the bands or resonant excitation of the lattice. I will show that it is possible to retrieve the value of the bulk topological invariants by monitoring the propagation of polaritons across the lattice in reciprocal space. This technics based on the experimental measure of the mean chiral displacement can be extended to 2D lattices. The second part of the presentation will be dedicated to the non-linear response of such a topological chain under resonant coherent drive. Gap solitons can be formed which properties are reminiscent of the underlying topological properties of the lattice. Their exponential tails present chirality symmetry properties, which are responsible for robustness of these solitons to defects when located on one sublattice. Moreover, engineering of the drive enables generating exotic topological solitons, which do not have any counterpart in conservative systems. Our results offer new insights to the rich phenomenology of topological systems subject to nonlinearities and non-hermicity. They open the door to the exploration of many-body topological effects in open systems.
10:20 - 11:00 Jean-Sebastien Caux (University of Amsterdam)
Quench dynamics in integrable atomic systems
This talk will present recent results on the post-quench time evolution of one-dimensional systems of relevance to atomic physics. A quick review will be given of various methods based on integrability, including the Quench Action, Generalized Hydrodynamics and Numerical Renormalization. Applications to interaction quenches and to spatially inhomogeneous quenches will be discussed.
11:00 - 11:40 David Luitz (Max-Planck-Institut für Physik komplexer Systeme)
Hierarchy of Relaxation Timescales in Local Random Liouvillians
To characterize the generic behavior of open quantum many-body systems, we consider random, purely dissipative Liouvillians with a notion of locality. We find that the positivity of the map implies a sharp separation of the relaxation timescales according to the locality of observables. Specifically, we analyze a spin-1/2 system of size ℓ with up to n-body Lindblad operators, which are n local in the complexity-theory sense. Without locality (n=ℓ), the complex Liouvillian spectrum densely covers a "lemon"-shaped support, in agreement with recent findings [S. Denisov et al., Phys. Rev. Lett. 123, 140403 (2019)]. However, for local Liouvillians (n
The times given in the schedule are according to the Central European Time (CET). | CommonCrawl |
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